**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

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Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters R. C. Ertekin and Z. -M. Qian University of Hawaii Honolulu, USA Abstract The shallow-water wave equations of Boussinesq type are employed to numerically solve the problem of a vertical strut moving in a channel. Since the most important parameter in soliton generation by moving disturbances is the blockage coefficient, a strut can be made equivalent to a finite draft ship. A boundary-fitted curvilinear coordinate system based on elliptic equations is generated to deal with the difficulties due to the body-boundary conditions in a channel containing an arbitrarily-shaped ship boundary which extends to sea floor. The strut problem is solved numerically in a transformed computational plane which contains uniform grid size. A finite-difference method is applied to the equations to march in time. The surface elevation and wave resistance are computed and compared with the available experimental data. The agreement between the calculations and experimental data is, in general, very good. 1. Introduction The phenomenon of ship-generated solitons was rediscovered experimentally (see Huang et al. [12]) during the experiments done by Sibul et al. t28] on an unrelated subject. Huang et al. tl2] observed that when a ship model is set into motion, starting from rest and quickly reaching a constant velocity, two-dimensional waves that precede the model are generated one after the other in addition to the usual three-dimensional waves behind the model. The waves that move ahead of the model were completely above the still water line and their speeds were critical or supercritical. These waves have been termed solitons or solitary waves which, unconventionally, refer to individual waves in a train of waves. The subcritical, critical and supercritical 421 wave speed refer to the depth Froude number, Fh =U/ ~ , being less than, equal to and greater than 1.0, respectively, where U is the ship-model speed, g is the gravitational acceleration and h is the undisturbed water depth which is constant. No published mention of the phenomenon of ship-generated solitons could be found until the reports by Thews and Landweber [30], Sturtzel and Graff t29], and Graff [9] that describe the continuous solitary wave generation experimentally were brought to attention in 1984. Wu and Wu t35] reported first on some numerical calculations of a two-dimensional pressure distribution moving steadily over the water surface in which the same phenomenon of soliton generation was predicted. These calculations were based on generalized Boussinesq equations derived earlier by Wu t34]. Some of the calculations were also reported in Huang et al. [13]. The most striking feature of these nonlinear waves is that they are almost perfectly two-dimensional, spanning the tank walls, even though the generating source is a three-dimensional ship model. Only a few qualitative features of these solitons could be observed during the experiments of Huang et al. [12] since the experiments were not systematic. Ertekin [3] carried out a series of experiments in which certain parameters such as water depth, model draft and tank width were changed systematically. During the experiments, a ship model (Series 60, Block 80) was towed along the centerline of the tank with a constant velocity. The total resistance experienced by the model and the run-away-soliton amplitudes were measured simultaneously in these experiments.

The experimental results of Ertekin [3] and Ertekin et al. t4] showed some very important qualities of the ship-generated solitons. Among several are: the dependence of soliton amplitudes on the blockage coefficient (the ratio of the cross-sectional area of the model at midships to the cross- sectional area of the water mass, see Eq. (33~; the phenomenon is not associated with equipment malfunctioning; at critical and supercritical speeds a steady-state flow cannot, in general, be obtained; below critical speed soliton amplitudes die out leaving a shelf in front of the ship model; and no solitons can be generated as blockage coefficient goes to zero. The last feature of the phenomenon implies that as the tank width or the water depth goes to infinity, no upstream waves can be generated. However, we note that a recent work by Pedersen [24] challenges this conjecture by en argument related to the existence of Mach stems as discussed in Ertekin t3]. Ertekin t3] also investigated the theoretical cases of a two-dimensional pressure distribution, and a two-dimensional bump on the sea floor moving with a constant velocity. Both the Green and Naghdi [10] equations for a thin fluid sheet (see also, Ertekin t6]), and the shallow-water equations derived by Wu [34] were used. Some of the results were included in Ertekin et al. t4]. Akylas [1] and Cole t2] have considered a two-dimensional bottom bump by using the Korteweg de Vries equation (KdV). Lee t16] and Lee et al. [17] considered a two-dimensional bottom bump both experimentally and numerically. Some of their preliminary results were included in the Discussion Section of Ertekin et al. [4]. First attempts to consider a three-dimensional disturbance in computations are due to Mei and Choi [18], Mei [19], Ertekin et al. tS] and Mei and Choi [20]. Mei [19] considered a vertical strut which is slender so that the rigid-boundary condition on the body can be applied at the center-line of the strut within the order of perturbation expansion. This approximation has resulted intwo-dimensionalwavesin both the upstream end downstream regions since the modified KdV equation derived is two-dimensional only. A remark may be necessary with regard to the terminology used here for the number of dimensions. In three-dimensional Cartesian coordinates where x and y are in the horizontal still-water plane and z is vertical pointing up, we use the terminology two-dimensional for flows confined to the x-y plane and three-dimensional for flows confined to 422 both the x-z plane and the x-y plane even though some flow quantities, such as the velocity potential, do not depend on the vertical coordinate z because of the mean-layer approximation. Going back to the discussion on Mei's results, we note that the two-dimensionality of downstream waves as calculated by Mei [19] was not observed, in general, in the experiments of Ertekin t3]. Ertekin et al. [5] considered a three-dimensional pressure distribution and solved the Green-Naghdi (G-N) equations in the time domain. The solution of this nonlinear initial-boundary-value problem also showed three-dimensional downstream waves, qualitatively agreeing with the experimental results. This confirmed that the application ofMei's formulation has a limited range, at least, as far as the downstream waves are concerned. The wave resistance is also found to be in qualitative agreement with the experimental data. More recently, Katsis and Akylas t14] and Wu and Wu [36] obtained results for a three-dimensional moving surface-pressure distribution by using forced nonlinear Kadomtsev-Petviashvili (K-P) and generalized Boussinesq (gB) equations, respectively. The stability of the forced KdV equation as it relates to run-away solitons is investigated by Wu [37]. In the present study, we investigate the nonlinear waves generated by a vertical strut by using the generalized Boussinesq equations as derived by Wu [34]. The no-flux boundary condition is satisfied by means of a n,~merical grid-generation technique (see for instance, Thompson et al. t31]. The nonlinear and unsteady results are directly compared with the experimental data. In Section 2, we formulate the fluid-dynamics problem to be solved with all the boundary and initial conditions to be satisfied. We also transform the equations from earth-bound coordinates to moving coordinates in this Section. In Section 3, the numerical grid-generation technique used is discussed and the equations of Section 2 are transformed to a regular rectangular computational domain. In Section 4, the numerical-solution method is given and wave resistance experienced by the strut is discussed. The finite-difference method employed is presented and sample results are shown. Preliminary results are also given in Ertekin and Qian t8], and the detailed derivations of the equations presented here and some other results can be found in Qian t25], and Ertekin and Qian t7].

2. Formulation of the problem In order to clarify the physical problem let us consider Fig. 1. This shows a ship model moving along the centerline of a shallow-water channel. The dimensionless speed of the ship-model is given by the depth Froude nether, Fh, which is not necessarily critical. The boundaries consist of the tank walls, the center-plane symmetry axis if only half of the physical region is to be considered due to symmetry (mirror-image problem), the two inlet and outlet boundaries (or "open" boundaries) ahead of and behind the model, and the no-flux condition on the model. The channel floor and the free-surface boundary conditions are not discussed since these are either exactly (in the case of G-N equations) or approximately (in the case of gB, KdV or K-P type equations) satisfied by the particle velocity vector. Then the problem can be solved by using a nonlinear and unsteady shallow-water wave equation to obtain the unknown particle velocities created by the movement of the model. One can use neither linear nor steady-state equations because of the nature of the phenomenon. In fact, it can be shown (Ertekin [3]), perhaps unsurprisingly, that the steady form of the gB equations used in this study predicts no disturbance in the upstream region. The same is true for other shallow-water wave equations. 2.1 Boussines~ equations The two different sets of shallow-water equations that have been applied frequently tosoliton-generation problems in the past, namely the Green-Naghdi equations and the generalized Boussinesq equations, have both advantages and disadvantages compared with the other. Even though the derivation of both of these equations can start with the assumption that the fluid is incompressible and inviscid, only the gB equations require that the flow be irrotational. This feature of the gB equations allows one to consider the layer mean value of the velocity potential and the surface elevation as the unknowns to be determined. On the other hand, such a potential does not exist in the case of the G-N equations since the flow is, in general, rotational. As a consequence, the G-N equations are expressed in terms of the unknown velocity components and the surface elevation. The apparent advantages of the G-N equations over the gB equations were discussed in Ertekin et al. t4, 5]. In a three-dimensional problem with a large domain, the gB equations are more efficient to solve computationally. Therefore, we choose to solve the following set ofgB equations for a constant water depth and zero atmospheric pressure (Wu [34]~: t°+V ((h+~°)Vl°~=O, (1) ¢,o+21V¢°1 2 + gtO = h v260 (2) where (x°,y°) are the coordinates of the fixed coordinate system in which x° specifies the direction opposite to the movement of the ship, <° is the surface elevation of the wave, {° is the layer mean value of the velocity potential defined by u =V4° in which u =(u°,v°) , h is the undisturbed water depth (constant) of the channel and V is the two-dimensional gradient vector in the horizontal plane. Eqs. (1) and (2) are the statements of conservation of mass and momentum, respectively. The general form of these equations in which the sea floor topography may depend on the x°,y° coordinates and time t° were obtained under the assumption that the Ursell number is of order unity (Ursell [32]~. However, they seem to be valid in a wide range of Ursell numbers as shown by Lee t16]. The velocity potential and the surface elevation in these equations depend on x°,y° and to. These equations satisfy approximately the nonlinear free-surface condition and the sea-floor condition. The configuration of the physical region is shown in Fig. 2. Before elaborating on the boundary conditions and initial conditions to be satisfied, we need to justify the use of a vertical strut to model the conditions of the experiments done by Ertekin [3]. In those experiments, the tank width, the model draft and the water depth were systematically changed to obtain 27 different blockage coefficients, Sb; S - To - (3) where Ao is the cross-sectional area of the underwater portion of the full model at midships at a given draft and W is the half-width of the tank. Also, Fh is varied to cover the range of 0.5-1.3. The most striking finding of these experiments was the dependence of the soliton amplitude, speed and the period of generation (the time that it takes for the second soliton to generate) on the blockage coefficient, Sb. Typical experimental results have been given in Ertekin et al. t4]. Therefore, it is clear that one can use a vertical strut 423

to model the same conditions which existed during the experiments, i.e., hull form is of secondary importance. Now, we can go back to the discussion of the boundary conditions and initial conditions. Because we assume that (see Fig. 2) AO and BC are part of the symmetry axis, only half of the physical domain needs to be considered. The computational advantage of this scheme is obvious. We then have the following boundary conditions. On the symmetry axis AO and BC, and the channel wall DE, the no-flux condition is ye = 0. On the ship boundary, which is moving in the negative x° direction, we have (Oonxo+tyonyo=-uonxo, (5) where U° is the speed of the moving boundary and n=(nX0,ny0) is the unit normal vector of the ship boundary pointing into the fluid. On the upstream and downstream open boundaries, we use Sommerfeld's radiation condition with constant shallow-water wave celerity, ides, (tO~<XO= 0 9 bto + ~ tOo = 0 on EO (-) and CD (+). The initial conditions are chosen such that there is no motion at time t=0-: t°=0, t°=0, (7) for all x0 and ye. The governing equations (1) and (2) will be solved subject to the boundary and initial conditions (4~-~7~. Since we will use a numerical grid-generation technique to map the physicalplane onto a regular rectangular computational plane to avoid interpolations as much as possible in satisfying the body-boundary condition, we must first transform the gB equations and the boundary and initial conditions to a moving coordinate system as shown in Fig. 3. 2.2 Equations in moving coordinates The equations of last section will be transformed from the fixed coordinate 424 system (x°,y°) to the left moving coordinate system (x',y') whose velocity is -U°. Since t°=t , x°=x _u°to y°=y (8) the continuity equation (1) becomes _ _ (t'+ V' [(<'+ h)V~ ]=o and the momentum equation (2) becomes +'+2(t ++ 2 _ Uo2) + A<' = h (V2¢ +U0V2~x). (10) Referring to Figs. 2 and 3, the boundary conditions and the initial conditions become =0 on AO, BC and DE, (11) .++y~ny'=o on AB, (12) Hi, + (U° ~ ~<x, = 0, hi, + (U° ~ m)(lx. - U°) = 0, on EO (-) and CD (+), and (13) =0, ~ =U°x , at t=0-. (14) Next, we nondimensionalize these equations in the moving coordinates by the new dimensionless variables given by x' y' A. x= h' Y=h' t=h, t=t~, U=U , l= h I/ h (15) The dimensionless forms of the continuity and momentum equations then become _ _ At + V · [( 1 + <)V4] = 0 9 (16) At + 2 (62X + by - U2) + ~ = 3(V2¢t+UV2lx). (17) And the dimensionless boundary and initial conditions (11)-(14) become ty=0 on AO9 BC and DE9 (18) (xnX+tyny=o on AB9 (19)

(t+(U~ 1)<X=o, At + (U ~ 1 ) (¢X - U) = 0 , on EO (-) and CD (+), and t=0, t=Ux, at i= 0~. (21) Eqs. (16)-(21) will be solved by a finite-difference method. Because we are dealing with a simply connected region which has a non-rectangular boundary in the plan view of the ship, it is preferable to use boundary-conforming coordinates in the fluid domain. In the next Section, we will describe the numerical generation of these boundary-conforming coordinates and transform the governing equations from the physical moving-plane onto a rectangular plane where a finite-difference method can be easily used within a uniform rectangular grid. 3. Numerical gr~-generation When there are irregular boundaries in a fluid domain, one can use several discretization schemes in a finite-difference method. One of them is the "irregular-star" technique which has been used for some time in ship-wave problems (see for instance, Ohring and Telste [22]), and another is the "staggered-mesh" system used by Miyata et al. [21] in a marker-and-cell method. Yet, another method suitable for the present problem is the numerical generation of boundary-conforming coordinates (see for instance, Thompson et al. [31]~. In the first two schemes mentioned above, the difficulties in dealing with an irregular boundary are: the use of unequal grid size in different regions of the computational domain which brings additional complexities and inaccuracies, the heavy use of various interpolation techniques on irregular boundaries for the evaluation of the higher-order derivatives of a function being solved for, and as a consequence, the need to use very small grid size which results in an increased computational time, and finally, serious questions concerning the assumption of the existence of a function being extrapolated outside the fluid domain. The last difficulty is again related to irregular boundaries. These difficulties do not exist if one uses a numerically generated grid along with a finite-difference method. However, due to the addition of extra terms in the governing equations, numerical calculations become more complicated and intensive when one transforms the 425 equations from a physical plane to a rectangular computational plane with a (20) uniform grid. Depending on the boundary conditions, one can use several different grid-generation systems, such as the elliptic, parabolic, hyperbolic or algebraic generation systems. An elliptic generation system may be chosen in our case since Dirichlet boundary conditions are specified on all the boundaries enclosing simply-connected physical region. Dirichlet conditions are simply boundary coordinates given in physical problem. 3.1 Elliptic Generation System the The the the Referring to Fig. 4, a uniform grid in the transformed plane can be generated by determining the coordinates x(5,~)and y(5,~), for A5=~=1.0. Because of the fact that the extreme of solutions of an elliptic system cannot occur inside the boundaries, one-to-one correspondence between points of the respective planes is guaranteed. This, of course, also ensures the single-valuedness of the coordinate functions. Then the equations of the elliptic generation system given by V25=o, V2~=o, can be replaced by a set of nonlinear equations in terms of x and y to be solved in the transformed plane: axis - 2p~x',, + YX,,T, = 0, ayes - 2,By'T, + yyT,r, = 0, where (22] a = X2 + Y2, ~ = XLX,] + YT\Y5, ~ = XL + YE . (23) The above equations can be solved by a finite-difference method based on second-order central-difference formulas, i.e., a(x~-2x',+x'`~)--(x~',`~-x',`+,- x'-,,,+X',~,)+y(x'',`-2x~+x',`)=o, (24 and a(y'`+,-2y'`+y'~ I)- P~(YJ',,`+,- yj I,`~- yj+,~,+y',~,)+y(y'.~i-2y',+y'~)=0. (25, This results in a system of nonlinear

equations which we solve by a Successive H =-(~+1)V26-V; Vt Over Relaxation (SOR) method. The solution is accomplished by a point iteration in which the initial guess is chosen as a weighted average of the boundary points. The boundary values of x and y in the physical plane are specified. A typical result for a parabolic strut whose dimensionless equation is given by y=2Aw(l- ~ At, where AW/2 is the half-width of the strut and L is the length of the strut, is shown in Fig. S. In this Figure AW/2= 0.3, L= 8.0, Sb= 7.5 %, and Ax=Ay=0.1. However, only every fifth grid line is plotted for clarity. The accuracy of the elliptic generation system used here has been checked against a simple ideal flow problem (doublet + uniform flow) which has an exact solution, and found to agree within a tolerance of 1 x 10-8. We note that x = constant lines and ~ = constant lines coincide in their respective planes, and as a result, no coordinate contraction has been used. The purpose of this scheme is to preserve the symmetry boundary AO and BC. If we had allowed x to depend upon ~ also, then we would have had to solve the problem in the entire physical region, not just in half of it. In that case, the boundary points would not have been uniform. 3.2 Governing equations in the transformed plane and the Jacobian, J. of the (26) transformation is J=x~y~-x~y'. Note that even though the general formulas are given here' x depends on ~ alone since y = constant and ~ = constant are parallel everywhere. = - J2 (<+13(a¢~-20~+yl~- 12 (Yet' Yt, n) (Yntt - Yttn) + J2 (Xq+t - x`~) (-x~' + x`~), Similarly, the momentum equation (17) becomes (! + At,`` + Bl'6T'+ Char,,, = Q. (28) where A=-3 2, B=3 ~2, C=-3y2, Q = _t, _ ~ (42 + 42 _ U2) + 3 UV Ax = ~< ~ ~ <ry ~ _y64052 (-xql6+Xt~q) up+ (A,,56+ Bay + C,tr,r, + DO + Edit,, + F f A,, + G¢',,,r,), 3J4( o JOB)' B~=-3 4(JBo- 2J p) Cal 3J4(JCo-JoY), D=3J3y~a, E = - 3 J3 (2YT, pin + yeah, F = 3 J3(y~+ Maya), The purpose of transforming the equations from an earth-bound system to a moving system was to avoid the generation of a numerical grid at each time step, thereby not allowing the time derivatives to produce extra terms. If we now apply the differential relations between the derivatives of a function in 1 (x,y) and (5,~) planes (see, Thompson et to 2(y~a~ Yeats al. [31]), the continuity equation (16) becomes At= H. where 426 Jo=J`Y~~J~Y' =y2x`~-2y~y~x'~+y~x~-x~y~y`~+ (gyp + x~y`)y`~ - x~y~y~, = X~ynXtn - X~y~xq~ + y2ytn _ y~y~ynn, (27) Bo=y~-y`~=x~y~x`~+(x~y~-x~y`)x`~- X5y~xq~ + y2yt'_ yt yen,

CO=2(Y~` yarn) = x~yr,x~` - x~y~x~r, + YLYT\Y\E - Y' YET] ~ And the dimensionless boundary and initial conditions (18)-(21) become as follows. t~=0 on AO, BC and DE (29) -~+~=0 on AB. (30) (t = ZI,2 ~ t! = Ply, where ZI.2 = - ~ (Yq<~- Yawn), PI,2=-(U~ 1~(~(Y~-Ytl~-U), on ED (subscript 1) and CD (subscript 2), t=0, t=Ux(~), at t=O-. As before, the subscripts ~ and ~ denote the derivatives with respect to these variables. We are now ready to solve equations (27) and (28) subject to the boundary conditions (29~-~32~. 4. Numerical solution of gB equations and wave resistance In this Section, we present the method employed to solve the nonlinear and unsteady shallow-water equations as given in the last Section, and also, give the formulation for wave resistance. Some of the results obtained will be presented and compared directly with the available experimental data. The accuracy of the equations and the numerical method will be revealed when we discuss the speed, amplitude and the period of generation of solitons, and wave resistance. 4.1 Finite-difference method Because of the success of the Modified Euler Method in previous applications, we use this two-step method to march in time. The spatial derivatives are approximated by second-order central-difference formulas. The truncation error of the resulting equations is O ~ A X 2 , ~ y 2 , Atop. In this method, the values of ~ and ~ at the present time step, indicated by the superscript (2), may be obtained by modifying the values obtained in the middle step, indicated by the superscript (m). The middle-step values are evaluated by modifying the previous-time-step values, indicated by the superscript (1). With these preliminaries, the continuity equation (27) becomes him) = t(~) + AtHtI) (33) t t2' = t t ~ ' + ^ t t H ` ~ ' + H tm, j , ( 34 ) where At is the time interval and H was 31 given by (27). Similarly the momentum equation (28) becomes + A lt~ ~ + B. `n ~ + Ct ~ ~ ~ = ho'+ Al `~, + Bait'+ Ctt''+ AtQt',, (353 lt2) + A+tt, + B+~2, + cash = t(~) + Attt'+ Bl t2'+ C¢t2, + ^t Its + Qtm,' ~ (36) where A, B. C and Q were defined by (28). The upstream and downstream open-boundary conditions become <;tm) = itch) + ~tzt~2, (37) tt2~=<t~+~tzt~' +ztm)' (38) = ~ ~ + ~ t p ti ,2 ~ (39) ¢,t2~=~1,t~+~\~`pt~, + pimps `40; The right-hand-side terms of these equations are calculated from the previous time step, and then considered as known variables in the present time step. During each time step, ~ can be directly solved from (33) for the middle step, and from (34) for the present step. and ~ can also be found directly on the open boundaries from (37) and (39) for the middle step, and from (38) and (40) for the present step as the solution of the initial-value problem. ~ will then be determined iteratively from (35) for the middle step, and from (36) for the present step as the solution of the boundary-value problem. As in the case of the numerical grid generation, we use the SOR method to solve the set of simultaneous equations which resulted by the application of the Modified Euler Method. A five-point filtering formula (see Shapiro [27], 427

Haussling [11], and Ertekin et al. t5]) is applied to reduce the effect of the numerical errors which are caused by high-frequency cross-channel waves. and ~ are filtered in the ~ and ~ directions in the transformed plane after each time step. The SOR parameter varied between I.5 and 1.9 in our calculations. 4.2 Wave resistance The wave resistance, R. experienced by the vertical strut can be obtained by evaluating the pressure, Ps, on the strut, i.e. -R(t)=pgh3= 2 J ( J Ps dZ)YxdX (41) The pressure on the strut can be obtained from Euler's integral once the velocity potential and the surface elevation are determined. Following Wu t34] and Wang et al. t33], we obtain PsdZ=~(1 +~)~1 I+ -h 2 (42+ 24-2)(~+2(62+~2-U2)+~), (42) in the moving coordinates. Equation (42) is obtained by expanding the pressure in a perturbation series where the error term is O(es) , and ~ is the dispersion parameter. One can now substitute (42) into (41) and obtain the wave resistance as a function of time after transforming the variables from the (x,y) system to (5,~) system. 4.3 Discussion on the numerical Accuracy and results When the water depth, h, is constant, the conservation of mass statement given by (~) is exact. Therefore, this equation can serve as a check on the numerical accuracy of the results. We note that the same argument cannot be made by using the conservation of linear momentum statement given by (2) since this equation is approximate. The dimensionless water mass, m/ph3, given by J ( 1 + t) Jd A5n + J :; Sd ~ - ~ Sd Id t, (43) where S(5, A, t ) = ~ ( 1 + t) (yntE - yips), and lo, tR represent the left and the right open boundaries, respectively, and 428 All denotes the surface of the computational region, remains constant within 1%, showing that numerical accuracy is very high. The gB equations are solved by a computer program which uses the grid points generated by another computer program. Two sets of numerical tests are conducted and the parameters of the problem are selected to approximately correspond to the experiments done by Ertekin [3]. In the first set of the numerical tests, two different blockage coefficients, 7.5% and 12.5%, are selected. The grid size on boundaries is chosen as Ax=Ay= 0.1, and the time interval is chosen as At=0.0Seven though Von Neumann's stability method applied shows that At can be the same as Ax without any stability problems. Two different speeds, Fh=U=0.8 and 1.0 , are used so that precursor solitons can be generated quickly. The perspective plots (see Fig. 6 a,b) as well as the contour plots (see Fig. 7 a,b) are shown here for Case 3. For other results see Qian [25]. In perspective plots the strut below the lowest wave surface could not be plotted because of the limitation of the graphics software that we used. The period between the first and second soliton or the period of generation, the amplitude and the speed of the first and second solitons are calculated (see Tables 1-3). The period of generation is obtained from the numerical moving-gauge results as shown in Fig. 8 (continuous lines). In Fig. 8 the location of the numerical gauges are at distances of0.625L (Gauge2) and L (Gauge 3) ahead of the strut on the symmetry axis. Note that in the first set of numerical tests (Cases 1,2 and 3), W=4, L=8 and [R-l' =64. Comparisons with the experiments (Ertekin t3] and Ertekin et al. t4] are also made in Tables 1-3. To understand the effect of strip length on the computational results, we conducted another numerical test by increasing the ship length to 15.2 but leaving all other parameters of the problem the same. This test is referred to as Case 4. Case 4 results are shown in Figs. 6 c,d and 7 c,d, and in Fig. 8 (dotted lines). As can be seen from these Figures, shorter strut has considerably more wave build-up around the bow and stern than the longer strut, and also, the downstream waves are smaller in the longer strut case. From Fig. 8 we see that it takes longer to generate the first soliton in the longer strut case than the shorter strut case. Also the amplitude of the first soliton is smaller, and the amplitude of the

second soliton is larger in Case 4 compared with Case 3 (see also, Tables 1-3~. In the second set of numerical tests, we used exactly the same Sb=14.2,W=6.1 and L=15.2 used in one set of the experiments of Ertekin [3]. In the experiments, h=10 cm, d=7.5 cm and Fh=1.0 were used (Test No. 1936~. The location of the moving gauges were given inErtekin etal. [4]. This numerical test is called Case 5. In Case 5, Ax=Ay= =0.15 and At=0.08 are used since the computational region is larger than before. Also! R - I ~ =137.25 in this case. The perspective plots of Case 5 results are shown in Fig. 9, and the contour plots in Fig. 10. The numerical wave-gauge results end the wave resistance results are shown in Fig. 11. The corresponding experimental results are shown in Fig. 12. The numerical results in Fig. 11 are shifted in time so that the moment the towing carriage started moving in Fig. 12 (see the sudden increase in resistance at t=2.5 s) corresponds to t=0 s in calculations. This shift in time in experiments is due to starting data acquisition before the carriage is put into motion. The quantitative differences between the experiments and calculations in Case 5 are given in Tables 1-3. The agreement can be considered quite good. The average total resistance measured during the experiments was 9.3 newtons for Case 5. In the calculations the average wave resistance is obtained as 9.6 newtons. We estimate the frictional resistance as 1.4 newtons from Schoenherr's flat plate skin friction formula. Of course, the eddy or form resistance must be included also. Thus, one can conclude that the total resistance predicted is slightly higher than the experimental datum. Some other cases which are not presented here also support this conclusion. In conjunction with the numerical schemes used in this study, we mention that the surface elevation and the velocity potential are filtered along both the ~ and ~ directions at each time step. The average number of iterations is 15 for the first-step solution (for a tolerance of 8 x 10-7) and 6 for the modified or second-step solution (for a tolerance of 5 x 10-7). The computations, which were performed on the X-MP/48 Cray Computer of the San Diego Supercomputer center, took about 90 minutes of CPU time for 2200 time steps in Case 3. Single precision is used on this 64 bit machine. Filtering was absolutely necessary in our calculations. Removal of the filtering scheme caused very high waves at the 429 stern, and eventually, the computations could not be continued. The unphysical waves in the absence of filtering occurred for various grid sizes and time intervals. From the results presented here, we see that general appearance of the waves agree with the results of earlier research (e.g., Huang et al. [13], Ertekin [3], and Ertekin et al. t5]~- For each soliton, including the first and the second one in all cases, the amplitude increases rapidly while the speed increases. The relation between the amplitude and speed of the solitons (computed) satisfies approximately the dispersion relation for a general two dimensional soliton propagation, such as the second-order formula proposed by Laitone [15], and a formula derived by Schember t26]. However, solitons generated by ships do not have an exact permanent form. The amplitude of each soliton gradually increases after generation. This, of course, may be partially due to the governing equations which satisfy the free-surface conditions only approximately. Also, the rather small differences between the computations and experiments may be also due to the neglect of viscosity (although this effect must be very small), and, of course, computational as well as experimental errors, especially in the case of period of soliton-generation. The soliton generation clearly depends on the blockage coefficient and speed of the moving strut, but it also slightly depends on the length of the strut. An increase in the Froude number, or the speed of the ship, can delay the generation of the first soliton, and thus increase the period between the first and second solitons. Soliton amplitudes and speeds increase correspondingly. Fast movement of the disturbance increases the period of soliton generation and larger blockage coefficient decreases the period of soliton generation. The second soliton almost always comes out with a smaller amplitude and speed than the first soliton. As can be seen in Table 2, the agreement between the experiments and computations for the amplitude and speed of the second soliton is very good. 5. Conclusions The waves generated by a ship in a shallow-water channel can be modeled by numerical calculations of Boussinesq equations, and two-dimensional solitons may be generated ahead of the ship. The shallow-water equations of gB type can successfully simulate this phenomenon. The generation of a boundary-fitted curvilinear coordinate system is very

effective for the configuration containing the ship boundary. The physical problem is solved in the transformed plane with a uniform grid size. The boundary conditions, especially on the ship boundary, can be satisfied without the need for a special treatment of the boundary values. 1. However, all equations become more complicated after the transformation, and thus computations become much more intensive. Since the grid size is chosen 2. to be uniform on the physical boundaries, equal ~ lines become straight and are perpendicular to the symmetry boundary 3. and wall boundary. On the symmetry boundary and wall boundary, the no-flux boundary condition can be simplified, and also the symmetry condition can be used to deal with the third-order derivatives. If we use coordinate system 4. control (attraction) for the grid generation, the grid lines will not be generally straight, and the no-flux boundary condition will become more complicated on all solid boundaries, and also, no symmetry condition can be used. The constant wave celerity used on the open boundaries work out perfectly. Therefore, it seems that one cannot easily justify the use of a numerical scheme in which the phase speed is calculated (see for instance, Orlanski 6. [23]~. More efficient methods of numerical grid generation can be investigated and used with three-dimensional ship surfaces to reduce the errors introduced 7. by numerical grid generation. Coordinate-system control is recommended in order that the coordinate lines can be attracted by the boundaries, especially around the ship boundary, so that the boundary effects can be considered more accurately. This 8. obviously requires that we can deal successfully with the higher-order derivatives on the boundaries. Nevertheless, the symmetry boundaries will be destroyed in this case, causing 9. the necessity to solve the problem in the entire physical domain. Acknowledgements We are grateful to Prof. John V. Wehausen for his invaluable comments throughout this research. Our appreciation also goes to Prof. Theodore Y.-T. Wu who made a very timely comment on the resistance formulation. We would like to thank the International Business Machines Corporation for providing the computer equipment used in this study. This material is based upon work supported by the National Science Foundation under Grant Numbers 430 MSM-8706910 and BCS-8958346. The computer time was provided under Account Numbers 941 HAW and 234 HAW awarded by the San Diego Supercomputer Center. Fences Akylas, T.R., "On the Excitation of Long Nonlinear Water Waves by a Moving Pressure Distribution," J. of Fluid Mech., Vol. 141, pp. 455-466, 1984. Cole,S.L., "Transient Waves Produced by Flow Past a Bump," Wave Motion, Vol. 7, pp. 579-587, 1985. Ertekin, R.C., "Soliton Generation by Moving Disturbances in Shallow Water: Theory, Computation and Experiment," Ph.D. Thesis, University of California, Berkeley, 1984. Ertekin, R.C.; Webster, W.C. and Wehausen, J.V., "Ship-Generated Solitons," Proc. 15Th. Symposium on Naval Hydrodynamics, Hamburg, pp. 347-361; Disc. pp. 361-364, 1984, (ONR, National Academy Press, Washington, D.C., 1985~. i. Ertekin, R.C.; Webster, W.C. and Wehausen, J.V., "Waves Caused by a Moving Disturbance in a Shallow Channel of Finite Width," Journal of Fluid Mechanics, Vol. 169, pp. 275-292, 1986. Ertekin, R.C., "Nonlinear Shallow Water Waves: The Green-Naghdi Equations," Proc. Pacific Congress on Marine Science and Technology, PACON '88, Honolulu, Hawaii, May, pp. OST6/42 - OST6/52, 1988. Ertekin, R.C. and Qian, Z.-M., "Soliton Generation and Ship Wave Resistance in Shallow Waters. Part I: Vertical Strut," Univ. of Hawaii, Dept. of Ocean Engng., Report No. UHMOE-88101, December, ii+50 pp., 1988. Ertekin, R.C. and Qian, Z.-M., "Solitary Waves Generated by Ships in Restricted Waters," The 4Th. Asian Congress of Fluid Mechanics, August, Hong Kong, 4 pp., 1989. Graff, W., "Untersuchungen uber die Ausbildung des Wellenwiderstandes im Bereich der StauwellengeschwindigReit in flachem, seitlich beschranktem Fahrwasser," Shiffstechnik, Vol. 9, pp. 110-122, 1962. 10. Green, A.E. and Naghdi, P.M., "A Derivation of Equations for Wave Propagation in Water of Variable Depth," Journal of Fluid Mechanics, Vol. 78, part 2, pp. 237-246, 1976. 11. Haussling, H.J., "Solution of nonlinear wave problems using boundary-fitted coordinate systems," In Numerical Grid Generation (Ed. J.F. Thompson), Elsevier Science Publishing Company, New York, pp. 385-407, 1982.

12. Huang, De-Bo; Sibul, O.J. and Wehausen, J.V., "Ships in Very Shallow Water," FestRolloquim - Dedicated to Professor Karl Wieghardt, March, Institut fur Schiffbau der Universitat Hamburg, 1982. 13. Huang, De-Bo; Sibul, O.J.; Webster, W.C.; Wehausen, J.V.; Wu, De-Ming and Wu, T.Y., "Ships Moving in the Transcritical Range," Proc. Conference on Behavior of Ships in Restricted Waters, November, Varna, Bulgaria, Vol. II, pp. 26-1 to 26-10, 1982. 14. Katsis, C. and Akylas, T.R., "On the Excitation of Long Nonlinear Water Waves by a Moving Pressure Distribution. Part 2: Three Dimensional Effects.," J. Fluid Mech., Vol. 177, pp. 49-65, 1987. 15. Laitone, E.V., "The Second Approximation to Cnoidal and Solitary Waves," J. Fluid Mech., Vol.9, pp. 430-444, 1960. 16. Lee, S.-J., "Generation of Long Water Waves by Moving Disturbances," Ph.D. Thesis, California Institute of Technology, Pasadena, California, 1985. 17. Lee, S.-J.; Yates, G.T. and Wu, T.Y., "A theoretical and Experimental Study of precursor Solitary Waves Generated by Moving Disturbances," IUTAM Symp. on Nonlinear Water Waves, Tokyo, pp. 62-63, 1987. 18. Mei, C.C. and Choi, H.S., "Forces on a Slender Ship Advancing Near Critical Speed in a Shallow Channel," 4Th. Int. Conf. on Numerical Ship Hydrodynamics, September, National Academy of Sciences, Washington, D.C., pp. 359-367, 1985. . Mei, C.C., "Radiation of Solitons by Slender Bodies Advancing in a Shallow Channel," J. of Fluid Mech., Vol. 162, pp. 53-67, 1986. 20. Mei, C.C. and Choi, H.S., "Forces on a Slender Ship Advancing Near the Critical Speed in a Wide Canal," J. Fluid Mech., Vol. 179, pp.59-76, 1987. 21. Miyata, H.; Nishimura, S. and Masuko, A., "Finite Difference Simulation of Nonlinear Waves Generated by Ships of Arbitrary Three-Dimensional Configuration," J. of Comput. Phys., Vol. 60, pp. 391-436, 1985. 22. Ohring, S. and Telste, J., "Numerical Solutions of Transient Three-Dimensional Ship-Wave Problems," Proc. 2nd Int. Conf. on Numerical Ship Hydrodyn., Berkeley, pp. 88-103, 1977. 23. Orlanski, I., " A Simple Boundary Condition for Unbounded Hyperbolic Flows," J. Comput. Phys., Vol. 21, pp. 251-269, 1976. 24. Pedersen, G., "Three-dimensional Wave Patterns Generated by Moving Disturbances at Critical Speeds," J. Fluid Mech., Vol. 196, pp. 39-63, 1988. 25. Qian, Z.-M., "Numerical Grid Generation and Nonlinear Waves Generated by a Strip in a Shallow-Water Channel," M.S. Thesis, University of Hawaii at Manoa, Dept. of Ocean Engineering, x+121 pp., 1988. 2 6 . Schember, H. R., "A New Model for Three-Dimensional Nonlinear Dispersive Long Waves," Ph.D. Dissertation, California Institute of Technology, Pasadena, California, 1982 . 27. Shapiro, R., "Linear filtering," Maths. Comput. Vol.29, pp. 1094-1097, 1975. 2 8 . Sibul, O.J.; Webster, W.C. and Wehausen, J.V., "A Phenomenon Observed in Transient Testing," Schiffstechnik, Vol . 2 6, pp . 179-2 00, 1979 . 29. Sturtzel, W. and Graff, W., "Untersuchungen der in stehendem und stromendem Wasser festgestellten Anderungen des Shiffswiderstandes durch Druckmessungen," Forschungsberichte des Wirtschafts- und Verkehrsministeriums Nordrhein-Westfalen, Nr. 618, 34 pp., 1958. 30. Thews, J.G. and Landweber, L., "The Influence of Shallow Water on the Resistance of a Cruiser Model," U.S. Experimental Model Basin, Navy Yard, Washington, D.C., Report No. 408, 1935. 31. Thompson, J.F.; Warsi, Z.U.A. and Mastin, C.W., "Numerical Grid Generation," Elsevier Sci. Pub. Co., New York, xv + 483 pp., 1985 32. Ursell, F., "The Long Wave Paradox in the Theory of Gravity Waves," Proc. Camb. Phil. Soc., Vol. 49, pp. 685-694, 1953. 33. Wang, K.H.; Wu, T.Y. and Yates, G.T., "Scattering and Diffraction of Solitary Waves by a Vertical Cylinder," 17Th. Symp. on Naval Hydrodyn., The Hague, The Netherlands, pp. 37-46, 1988 . 34. Wu, T.Y., "Long Waves in Ocean and Coastal Waters," Journal of Engng. Mech. Div., ASCE, Vol. 107, No. EM3, Proc. Paper No. 16346, June pp. 501-522, 1981. Wu, D.-M. and Wu, T.Y., "Three-Dimensional Nonlinear Long Waves Due to Moving Surface Pressure," Proc. 14Th. Symposium on Naval Hydrodynamics, Ann Arbor, pp. 103-125, 1982, (ONR, National Academy Press, Washington, D.C., 198 3 ~ . 431

36. Wu, D.-M. and Wu, T.Y., "Precursor Solitons Generated by Three-Dimensional Disturbances Moving in a Channel," IUTAM Symp. on Nonlinear Water Waves, Tokyo, pp. 18-19, 1987. 37. Wu, T.Y., "Generation of Upstream Advancing Solitons by Moving Disturbances," J. Fluid Mech., Vol. 184, pp. 75-99, 1987. Case Sb As us % L I ~ l | E | D | c T | ~ %, 1 0.8 7.5 0.18 0.15 20.0 1.05 1.07 -1.9 2 ~ 1.0 ~ 7.5 0.54 0.50 ~ 8.0 ~ 1.20 ~22 -1.6 3 ~ 0.8 12.5 0.29 0.27 7.4 1.12 - 1.11 - 4 0.8 12.5 0.25 0.27 -7.4 1.12 1.11 0.9 5 ~ 1.0 14.2 ~ 0.70 0.62 ~ 12.9 1.27 1.23 - - 3.2 Table 1. Dimensionless amplitude and speed of the first soliton. Case Sb As us No. ~ ~ (%) ~ ~ Ah I I ~1 T ~ %, T I | ~ %, 1 ~ -0.8 ~ _ 7.5 0.11 T 0.12 1 -8.8 ~ 1.02 1 1-03 1 -1~0 2 ~ 1.0 7.5 0.50 ~ 0.49 ~ 2.0 ~ 1.20 1 1.18 1 1.7 0.8 12.5 0.21 0.23 -8.7 - 1.09 1 1-09 T °~° 4 0.8 12.5 0.24 0.23 4.3 1.09 1.09 0.0 1.0 14.2 0.69 0.56 23.2 1.27 1.23 3.2 ,. ~ Table 2. Dimensionless amplitude and speed of the Notes on the Tables: C : Present computational results, E : Experimental data (Ertekin [3]), D : Difference between the calcula- tions and experimental data, Sb: Blockage coefficient, h : Depth of undisturbed water, g : Gravitational acceleration, U : Speed of the ship, As: Amplitude of the first or second soliton, Us: Speed of the first or second soli- ton in earth-bound coordinates, Period of generation of solitons. second soliton. Case No. 1 2 3 4 5 U (sib) UTV C ~ E | D 0.8 7.5 31 34 -8.8 1.0 7.5 44- 47 -6.4 0.8 12.5 21 ~ 23 -8.7 0.8 12.5 20 ~ ~23 -13.1 1.0 14.2 27 33 -18.2 Table 3. Dimensionless period between the first and the second soliton as observed in the moving coordinates. 432

- ~ ~ ~ c _ ,~ 9 L; ~t I , ~ ~ ~TAN~WALL ~ (a, ~ ~ , i,,,, ,, At, ~ =E s .... . l SOLITON CREST l l ~ " _. _ I j-N _ I ~ ~/~/~ _ _ ~ ( b ) , F ~ ~ _ _ _ ~ _ -. , . , , _. ~ Figure 1. (a) Sketches of crest patterns ~t during the emergence of a soliton in a moving reference of frame, (b) superposed sketches of (a) and a train of solitons (from Ertekin [3]~. yO ~11 ba~ndazy ~ /- ,,~ / ,,, / ~ ,~ / up 2~ O \ . s~~ bout_ ~ , 1 ~ / c x° Figure 2. Configuration of the physical region. Y' ~ Lit X' xo Figure 3. Fixed (x°,y°) and moving (x',y') coordinate systems. j,1 - 1 M1 Figure 4. Physical (x,y) and computational (hi) planes. D _ Dad 4.0- 3.0 A: \2.0 >I 1.0 0.0 4.0 x/h 8.0 Figure 5. Numerically generated grid for the parabolic strut, Cases 1 and 2. 433

(a) U * T / h = 20 A A ~ A ~ e ~ ~ ~ ~ -24.0 ~1 6.0 - 8.u u.u O.~ 1~.w ^~.v 32.0 40.0 . . . r ~ 0.0 I ~ l -2 t.0 -16.0 -8.0 0.0 8.0 16.0 24.0 t2.0 40.0 ~ ~) \ ~` 1 _ ~ ~ __ N~' _ ~ \ ~ ~ _ (~) U ~ T / h = 20 <~` ~- _ _ _ _ ~' _- `` \ ` , ~ ( b) U ~ T / h = 40 (b) U * T / h = 40 -24.0 ~ ~ ~ _Q ~ n n Rn 16.0 24.0 32.0 40.0 16.0 - 8.0 0.0 8.0 16.0 2J'.0 32.0 40.0 (c) U ~ T / h = 20 16.8 -8.0 O.B 9.6 18.4 27.2 36.0 44.8 1 ~ ( d) U ~ T / h = 40 Figure 6. Perspective plots of computed surface elevation (a) and (b): Case 3 (shorter strut), (c) and (d): Case 4 (longer strut). 434 (d) U * T / h = 40 ~ Pt q ~ 18.4 27.2 0.0- - ~ 6.S -~.0 0.S 9.6 ~ B.4 27.2 36.0 44. Figure 7. Contour plots of computed surface elevation (a) and (b): Case 3 (shorter strut), (c) and (d): Case 4 (longer strut).

0.65 \ 0.4 - o a) - 0.2 - O; 1~- m] ~O ~ ~ ·) ~ ~D d] 0.0 o.o 1 o.o 20.0 30.0 40.0 50.0 60.0 70.0 ~o.o (gauge2) U *T / h 0.6 5: \0.4 N 0.2 o.o /" \\N ,/ \ / ' ' 1 o.o 1 o.o 20.0 30.0 40.0 50.0 60.0 70.0 80.0 (gauge3) U * T / h Figure 8. Computed surface elevation as observed by numerical moving-wave-gauges. ~ ): Case 3, (-----~: Case 4. U ~T / h = 60 -45.75 -30.s5 -15.35 -0.15 15.05 30.2s 4s.4s 60.65 7s.85 -45.75 -30.5s - 1 s.35 -o. 1 5 ~ 5.~ 30.25 ~s.45 60.65 75.85 l) * T / h = 100 -4s.7s -30.5s -15.35 -0.15 15.05 30.2s ~s.4s 60.65 7s.8s ooo L v.w -45.75 -30.5s -15.35 -0.15 15.05 30.2s 45.4s 60.6s 7s.8s U * T / h = 140 -45.75 -30.5s -15.35 0. 1 5 1 5 ~ .~n 's -45.75 -30.55 -15.35 -0.15 15.05 30.25 4s.45 60.65 7s.8s Figure 1O. Contour plots of computed surface elevation, Case 5. ~ c U ~ T / h = 1 ~P . _ ,~,. U ~ T / h = 140 _ Figure 9. Perspective plots of computed surface elevation J Case 5 e 435

.o - ~ 8.0 _ 6.0 L~ _ 4.0 2.0 0.0 / i I l I l l .o ~ ~ ~ o.o 20.0 30.0 8.0 TIME (SEC) ~, _ <~, 60- ~\~M; 1 , o.o 0.0 Q ~ 10.0 20.0 Jo.o 8.0 Tl M E (SEC) ~ 2.0- J:~\ o.o ~ 10.0 Q ~ 10.0 20.0 3c.o :~ 8.0 Tl M E (S EC) =60- J~J o.o , , ~ 1 1 ~ 1 o.o 1 o.o 20.0 ~o.o r' l o.o 1 o.o 20.0 30.0 TIME (SEC) Figure 11. Computed surface elevation as observed at 4 numerical moving-wave-gauges and wave resistance, Case 5. 10 u 6 2 < o 10 6 2 N 10 20 30 {` ~__- 1 - I 1 ~ L' - :, 2 0 6 o 6 w 2 o o 20 :. w a O 10 20 30 ~1 340 t_ ~ - ~1 1 1 0 3 0 2 0 10T ~__~ 0 10 20 30 Time (see) Figure 12. The experimental data of surface elevation and total resistance (Test No. 1936), Fh=l.O, h=10 cm, draft=7.5 cm, W=61 cm, moving gauges. 436

DISCUSSION by J.W. Kim The authors should be congratulated on their successful extension of Boussinesque type equation to 3-dimensional free-surface flow problems. We would like to make comments on the following two points. Author's Reply The first is the time integration method adopted in the present paper. As pointed out in your earlier paper(1984) [Al] numerical solutions of the gB equation for a disturbance moving at near the critical speed show the gradual increase of the amplitude of upstream solitons whereas the ON equation shows nearly constant amplitude. We have also reproduced this numerical calculation based on the 4th order Runge Kutta method for the time integration and then we have obtained the constant amplitude for both equations. The only difference between ours and your previous calculation is in the time integration method, where you used the Modified Euler Method (MEM). In the present paper you also used MEM for the time integration. From our experience in both time integration methods, we believe that the difficulties you experienced in your present work before introducing the filtering process in mainly due to the inadequacy of MEM for this problem. The second point I would like to discuss is on the treatment of the convection term. Although the gB equation given in your paper has no convection term explicitly, the perturbed velocity potential and wave elevation will have convection effect im- plicitly since the basic flow is uniform stream. For the convection operator it is well-known that the central difference scheme in spatial discretization has a considerable phase error on short waves with wave length comparable to the mesh size. We have also experienced similar difficulties in our calculation by the finite element method. As a remedy for this difficulty we used the par- tial upwinding scheme. From the above two points, we believe that the filtering process in your computation scheme has presumably played a role of eliminating the overall combined effect of the two dif- ficulties due to MEM and convection. [Al] Ertekin, R.C., Webster, W.C. & Wehausen, J.V., Ship-generated Solitons, Proc. 15th Symp. Naval Hydrodynamics, Hamburg, 1984 pp.347-364. We would like to thank Mr. Kim for a very useful and timely discussion. Your comments on the use of MEM and its effects on the numerical predictions are agreeable. It is clear to us that a fourth- order method such as the RK4 method used will produce more accurate results. However, it is not very clear that reducing the truncation errors by using a higher order scheme will totally eliminate the continuous amplitude increases observed when one uses the gB equations. One must keep in mind that the momentum and, therefore, the energy is not exactly conserved in gB equations even if the sea floor is flat. When the disturbance is small, the amplitude increase may be reduced, perhaps to a minimum. But we do not think that it can be eliminated when large disturbances are used. The same is not true when the Green-Naghdi equations are used since these equations satisfy the conservation of mass and energy exactly, even if the sea floor is not flat. On the other hand, one may justifiably argue that if the disturbance is large, then the assumptions behind the derivation of the gB equations are violated, and therefore one should not expect that the amplitude does not grow. We agree with this agreement. It is true that the MEM causes high frequency waves because of the presence of central differencing. That is why we used filtering. Filtering eliminates high frequency waves which occur around and behind the hull. But, obviously, it does not effect the very low-frequency waves ahead of the hull in the upstream region. Since these waves are the primary concern to us, we did not worry about using MEM which proved to be a very valuable scheme because of its efficiency compared to a higher-order scheme. 437