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Three Dimensional, Unsteady Computations of Nonlinear Waves Caused by Underwater Disturbances Y. Cao, W. Schultz, R. Beck (The University of Michigan, USA) Abstract Thre~dimensional unsteady nonlinear waves gen- erated by underwater disturbances (such as a mooring sourc~sink pair or a moving body) are modeled us- ~ng a mixed Eulerian-Lagrang~an tune marching pros cedure combined with a desingularized boundary in- tegral method. The waves computed by the present method are compared with those for linear theory to find the nonlinear effects. The wave resistance, lift, moment and the pressure distributions on the body are also calculated. We find that the wave patterns of a spheroid and a relevant simple sourc~sink pair dm- turbance are very similar as long as the disturbances are not too close to the free surface. 1 Nomenclature CD CL CM Cp D Dm Fr F M drag coefficient lift coefficient moment coefficient pressure coefficient diameter of spheroid local mesh size Froude number hydrodynamic force acting on body hydrodynamic moment acting on body submerged depth of disturbance h _ i;' ,k unit vectors L Ld p - r S Sib Sf Sb Sf length ot mayor axis of spheroid desingularization distance 1~' factor of desingularization - M moment acting on body Nb node number on spheroid Nit number of elements on spheroid in ~ direction Nb number of elements on spheroid in x direction Of node number on free surface Nf N,,f n node number on free surface in x direction node number on free surface in y direction outward normal of body surface into fluid pressure position vector from center of body area of spheroid surface body boundary free-surface boundary singular surface inside body singular surface above free surface 417 t V V(t) 27 Xb x a(t) ab Q time velocity of body surface velocity of disturbance field point position vector of free surface= (Of, Of ~ Of ) point on body singular point of fundamental solution desingularization exponent startup inverse time constant wave length of 2-D linear wave wave elevation strength of source-sink disturbance steady value of a(t) strength of source distribution above free surface strength of source distribution inside body velocity potential fluid domain 2 Introduction Since Longuet-Higgins and Cokelet~ first developed the mixed Eulerian-Lagrangian method for two-dimensional sur- face waves on water, variations of this method have been used for a variety of nonlinear free surface problems in two dimensions (Baker,2 Vinje and Brevig,3 etc). The meth- ods require at each time step: 1) solve a boundary value problem in an Eulerian frame and 2) update the free sur- face points (which construct the free surface) by integrating the nonlinear kinematic and dynamic free surface boundary conditions with respect to time. More recently, the method has been used for three-dimensional nonlinear wave prob- lems. Dommermuth and Yue4 used this method to solve several axisymmetric problems. Extensions to fully three- dimensional nonlinear unsteady waves are given in Dommer- muth and Yue5 using a spectral expansion procedure that is limited to periodic problems without bodies. Jensen, Mi and Soding6 solved the steady nonlinear ship wave problem by using a simple source distribution above the free surface. Cao, Schultz and Beck7 used the time marching procedure for a preliminary study of the three-dimensional nonlinear wave pattern caused by a simple disturbance (a source-sink pair) moving under the free surface. To make the time marching procedure practical, it is im- portant to have an effective solution method for the bound

ary value problem since it requires most of the computa- tion time. A boundary integral method is powerful because it reduces the computational domain by one dimension. In conventional boundary integral formulations, singularities of the fundamental solution are placed on the domain bound- ary. When singularities of the fundamental solution are placed away from the boundary and outside the domain of the problem, a desingularized boundary integral equation is obtained. The first use of a desingularized method is the classi- cal work by Von Karmas for the flow about axisymmet- ric bodies using an axial source distribution. The strength of the source distribution is determined by the kinematic boundary condition on the body surface. Kupradze9 pro- poses locating the boundary nodes on an auxiliary boundary outside the problem domain. Heisted studies some numer- ical properties of integral equations in which the singular points are on an auxiliary boundary outside the solution domain for plane elastostatic problems. Han and Olsonii and Johnston and Fairweather~2 use an adaptive method in which the singularities are located outside the domain and allowed to move as part of the solution process. This adap- tive method requires considerably fewer singularities than the number of boundary nodes, but it results in a system of nonlinear algebraic equations for both the strength and the location of the singularities. For unsteady nonlinear waves, Schultz and Hong~3 use the desingularization technique in two dimensions. McIver and Peregrinei4 ~5 show that a two- dimensional overturning wave can be well modeled by only a few singularities outside the flow domain with desingu- larization. Webster uses a triangular mesh of a simple source distribution inside the surface of arbitrary, three- dimensional smooth bodies and improves the accuracy of solution. Cao, Schultz and Beck7~7 compute the unsteady waves caused by a simple source-sink pair using a source dis- tribution above the free surface as in the steady nonlinear computations of Jensen, Mi and Soding6. In the following sections, we describe the problem for- mulation in section 3 and a more detailed discussion on the desingularized boundary integral method in section 4. Finally we present the results of the computations for the waves caused by a simple disturbance moving below a free surface with forward speed and the results for a fully sub- merged spheroid moving below a free surface. 3 Problem Formulation For an irrotational, incompressible flow in an ideal fluid, the Laplace equation is the governing equation for the ve- locity potential ¢: A¢=0 (in Q) The boundary conditions are: Dt Zf + 2 Vat Vat (on Sf)' (2) ~ = V'> (on Sf), (3) ~ = V ~ n (on Sb), (4) Van O (es x ~ Do), (5) where Q is the fluid domain. Equations (2) and (3) are the dynamic and kinematic conditions on the free surface Sf in Lagrangian form. Here, Xf = (Xf, Of, Of) is the position vec- tor of a fluid particle on the free surface, Do iS the substantial derivative following the fluid particle, n is the unit normal vector of the body surface pointing into the fluid. din is the normal derivative on So, and V is the velocity of the body surface, which we assume is given. The quantities are nondi- mensionalized by setting the gravitational acceleration, the fluid density and an appropriate length scale equal to unity. For the unsteady problem, initial conditions are required. Here, we study the flows generated by disturbances starting from rest. Therefore, at t = 0, we require that ~ _ O and the free surface elevation ~-O. The coordinate system is shown in Fig. 1. Free Stream _ ~ _ H D'C=FF I ~ -I l 1~1i~' 1 ~L Fig. 1 Problem definition and coordinate system 4 Solution Procedure The initial boundary value problem (1-5) and the asso- ciated initial conditions are solved by the mixed Eulerian- Lagrangian method. In this method, the following boundary value problem with a Dirichlet condition on the free surface and a Neumann condition on the body surface is solved in the Eulerian frame at each time step: /~= 0 (in Qj, A= To ~ =V.n {fin Vet ~ O (on Sf), (8) (9) where ¢0 and Sf are known from the previous time step. After solving the boundary value problem, the velocities of the fluid particles constructing the free surface can be cal- culated and the free surface conditions (2) and (3) can be integrated with respect to time following the fluid particles to update their potentials and positions which serve as the boundary conditions at the next time step. This procedure is repeated as time goes. 418

There are many methods to solve (6-9~. The method we use is the desingularized boundary integral method. Simi lar to conventional boundary integral methods, it reformu lates the boundary value problem into a boundary integral equation. The difference is that the desingularized method separates the integration and control surfaces, resulting in nonsingular integrals. There are two versions of the method: direct and indirect. In the direct method, the integral equa tion is obtained from Green's second identity evaluated on a surface (control surface) somewhere outside the problem do main and the integration surface is the problem boundary. In the indirect method, the solution is constructed by in tegrating a distribution of some fundamental solutions over a surface (integration surface) outside the problem domain. The integral equation for the distribution is obtained by sat isfying the boundary conditions on the problem boundary and (control surface). The effectiveness and accuracy of desingularized bound ary integral methods have been examined by Schultz and Hongi3 for two-dimensional problems, Websteri6 for three dimensional steady flows, and Cao, Schultz and Beck7~7 for three-dimensional unsteady waves caused by simple under water disturbances. The following are advantages of the desingularized boundary integral method: More accurate solutions may be obtained by a desin gularized boundary integral method for a given trun cation. The kernels are nonsingular, so special care is not re quired to integrate the singular contribution. Simple numerical quadrature greatly reduces the computa tional effort by avoiding transcendental functions. Fewer nodes may be required since simple quadrature eases the restrictions of a flat panel. There is more flexibility since higher-order Green func tions or fundamental solutions can be more easily in corporated. The indirect desingularized boundary integral method has two more advantages when compared to the direct one: · Integrals can be replaced by a summation if the desin gularization distance is sufficiently large. This makes the computation even simpler. · The indirect method may result in smaller errors due to truncation of an infinite boundary. Desingularization also causes some difficulties associated with uniqueness and completeness. However, if the singular point is located away from the boundary a distance pro portional to the local mesh size, the singular point will get closer to the the surface as the mesh becomes finer. In the limit, the desingularized formulation becomes identical to the singular formulation.7 Because of its advantages, we use the indirect desingu larized method. We construct the solution using a source distribution on a surface (Sf) above the free surface and a source distribution on a surface (Sb ~ inside the body surface: 419 ¢(X)=~S'af(Xa)~_~ ids ~ISb ~za~ (10) By applying the boundary conditions, (7) and (8), we obtain a boundary integral equation for the unknown strength of the singularities, Afros) and ab~x2,j, + ,|,| ab(Xs) ~~ i ~dS = 550(Xf) (on Sf) (11) S' a/(~'~S);3n (a D _ _ At) dS + ,| ,|, at ~3 (I ~ _ _ if) dS = V · n (on Sb), (12) where Us is the integration point on surfaces Sf and Sb, Of is the control point on Sf, and xb is the control point on Sa. After af~x8) and abbot) are determined, the fluid parti- cle velocities on the free surface can be calculated. Then the time marching procedure integrates the free surface bound- ary conditions. The pressure on the body surface is evaluated using the Bernoulli equation: _p = Bi + 2 ~V¢~2 + Z = d'+ + ~ 2 V ~ - V) · V) + Z. (13) where ~ = (~93' + V V)¢ is the substantial derivative of the potential at fixed points on the body surface. The second form is more useful when following points fixed on the body moving with velocity V. The forces and the moments on the body are calculated by integrating the pressure over the body surface: J. J/Sb (14) and M = ~ Is-per x node (15) where r is the position vector of the body surface point to a reference point (usually the center of the body). 5 Numerical Implementation In the results presented in this paper, the submerged disturbance (either a simple source-sink pair or a spheroid) moves in the-x direction smoothly starting from rest to a final speed. The strength of the source-sink pair is also smoothly increased from zero to a final value. The free surface conditions (2) and (3) are in the fixed coordinate system. This has an advantage since no spa- tial derivatives are required on the free surface which helps reduce numerical reflection from the truncated boundary.

For large time simulations, the computational window moves with the disturbance. At certain time steps, some fluid par- ticles are ignored downstream and new particles with zero values of potential and elevation are axlded upstream. The initial free surface grid (at t _ 0) is equally spaced in the x direction and the spacing is increased algebraically in the y direction. The moving computational window makes it difficult to have a non-uniform grid in the ~ direction with finer spacing near the disturbance. Collocation is used to satisfy the boundary conditions on the surface grid. The solutions are constructed by replac- ing the integrals over surfaces Sf and SO in (10) by isolated sources on the surfaces. The sources are placed approxi- mately perpendicular from the node points on the bound- aries at a distance Lo determined by Lot = Iy(Dm) (16) where lo is a parameter that reflects how far the integral equation is desingularized, Dm is the nondimensional local mesh size (we choose Dm as the square root of the average of the areas of the four elements around the node point) and ~ is a parameter associated with the convergence of the mesh refinement. An appropriate ct lies between O and 1 to ensure the convergence of the mesh refinement and the uniqueness and completeness properties of the solution of the integral equation. We have examined the influence of la for two problems: 1) a simple potential problem in which a dipole is below a ~ = 0 infinite flat plane and 2) a prelim- inary study of the nonlinear waves by a simple source-sink disturbance. It was found that good solutions could be ob- tained for 1 < la < 3 and the solutions are not sensitive to the variation of lo in this region. More detailed discussion on the selection of cat and lo can be found in Cao, Schultz and Beck7. We found that lo = 1 is "optimal" in considera- tion of the condition of the resulting algebraic system. This value is therefore used in the present computations for the free surface desingularization. For the example with the submerged body, the mesh size on the free surface is usually larger than that on the body by about 10 times and the differences among the influence matrix coefficients are large, so that the resulting system for of and ab is likely to be poorly conditioned. To avoid this, we split the system into two, one for of and the other for at which are alternately solved using LU decomposition for each subsystem. Each set of equations is much better behaved and more accurate solutions can be expected. An- other advantage of splitting is that the coefficient matrix for ab does not change with time and needs only to be inverted once for the entire time simulation. The matrix for of does not change during the iteration between the body and the free surface and only needs to be inverted once for the cur- rent instant of time. Of course, the matrix for of changes at next instant of time. In contrast, the source-sink pair distur- bance has fewer unknowns and can be solved very efficiently with a GMRES minimization procedure.7 The pressure on the body is evaluated at the node points. The substantial derivative of the potential ~ in (13) is calcu- lated using a four-point forward difference scheme. A fourth- order Runge-Kutta-Fehlberg method is used in the nonlinea* free surface integration. An initial time increment is set, but is modified by the Runge-Kutta-Fehlberg subroutine where appropriate. 6 Results 6.1 Numerical aspects In the two examples presented in this section, the dis- turbance velocity is given by V(t) = Fr(1-ems) in the-x direction, where Fr is the Froude number. For the source- sink disturbance, the strength of the source and sink is given by arty = aO(1-en. The problems are assumed to pos- sess symmetry about the xz plane. The free surface is discretized using Nf nodes and Nf nodes in the x and y directions, respectively, to form Nf = Nf x Nf free surface nodes. For the body problem, the spheroid used by Doctors and Becker is chosen. The diameter- t~length ratio D/L is 0.2. The basic grid on the surface is shown in Fig. 1. The grid lines are spaced uniformly in the circumferential direction. The grid lines have a cosine spac- ing in the longitudinal direction. The body has a grid with Nb elements in x direction and Nb elements in the circum- ferential direction, resulting in Nb = (Nb _ 1) x (Nb + 1) + 2 body nodes including the two end points. To improve the computational far-field behavior, we add negative images of the disturbance singularities.7 The pressure is integrated over the body in (14) and (15) using Simpson's rule first in the circumferential direction and then the longitudinal direction. The usual hydrodynamic coefficients (CD, CL, and CM) are obtained by multiplying F · l; F · k and M ·.1 by 2/(SFr2), where S is the area of the spheroid surface. The pressure coefficient, Cp, is defined as 2p/Fr2. We require the ratio of the element size in :e direction to the wave length to be less than 1/10 to resolve the waves. The nondimensional wave length ~ is estimated by ~ = 2,rFr2 using two-dimensional linear theory. 6.2 Waves generated by a source-sink pair mov- ing below a free surface In this example, the length scale is chosen to make the depth of the submerged disturbance unity. The distance between the source and sink is chosen to be 0.1. The Froude number based on depth is unity. The midpoint between the source and sink is initially located at point (5,0, - 1~. The grid on Sf at t = 0 has 41 x 16 node points within 0 ~ x < 20 and O < y < 7.5. The spacing increases by 10 percent in the y direction. The initial time increment in the time marching is 0.2. The potential ~ is expressed as a sum of 1) the source- sink disturbance pair at the distance h below the undis- turbed free surface, 2) the image disturbance above the undisturbed free surface, and 3) a sum of Nf sources of unknown strength in an array a distance La above the dis- turbed free surface using (16~. 420

The method is first applied to the waves generated by a sufficiently small disturbance such that linear wave theory is a good approximation. The results of the present method using fully nonlinear free surface conditions are compared to an "exact" solution computed from a time-dependent Green function for a Kelvin wave source that satisfies the linearized free surface condition.~9 Fig. 2 shows the comparison of the wave elevation along the symmetry plane at t = 10 com- puted by the present method to that computed by the linear calculation for a weak disturbance (aO = 0.05~. The nonlin- ear and linear results agree very well. Independent compu 0.02 0.01 o.oo -O . 0 1 -o . 02 1 l ------- 1~1onlinear \/ ~ Line" 0 5 10 x 15 20 Fig. 2 Wave profiles along y = 0 (weak disturbance) 0.3 0.2 i: 0. 1 ._ o.o -O. 1 i, -o . 2 --^ Nonlinear 0 5 10 x 15 zo Fig. 3 Wave profiles along y = 0 (strong disturbance) tations using: a) a smaller computational domain (with the same mesh spacing within 0 < x < 15 and 0 ~ y < 7.5), b) finer mesh grids (81 x 16 and 41 x 31 with the same compu- tational domains, and c) doubling the time increment, result in negligible difference for the nonlinear calculation. This in- dicates that even for the small disturbance example studied here, the differences in Fig. 2 are primarily due to nonlinear effects. Fig. 3 shows the results for a stronger disturbance COO = 0.75), showing the larger nonlinear effects of the free surface conditions, especially at the troughs. 6.3 Waves generated by a spheroid moving be- low a free surface In this example, the length scale is chosen to make the length of the spheroid l unity. The center of the spheroid is initially located at (2,0,-h), again with a moving compu- tational window. On the free surface, we use 61 x 16 nodes within 0 < x < 7.5 and 0 < y < 1.875. The spacing in the y direction increases by 10 percent for each row of nodes further from the centerline. For comparisons to the results presented in Doctors and Beck, we use the same submer- gence depths of the spheroid (h/L = 0.16 and 0.245~. The potential ~ is expressed as 1) a sum of NO sources of unknown strength inside the body, 2) the image of 1) above the undisturbed free surface, and 3) a sum of Nf sources of unknown strength above the disturbed free surface. The desingularization distances of the sources above Sf are given by (16~. To represent the body, the singularities (except at the bow and stern) were distributed on a spheroid of smaller minor axis inside the body. After some preliminary calculations, the ratio of the minor axes of the two spheriods was fixed at 0.3 in our calculations. Fig. 4 shows a three-dimensional view and contour lines of the wave pattern caused by the spheroid for h/L = 0.245 and Fr = 0.6 at t = 25. A smooth startup (,ll = 2) and Nb = 2N6' = 16 are used. We compare the waves pro- duced by the spheroid and those made by the relevant sim- ple source-sink pair disturbance in Fig. 4 and Fig. 5. The strength of the pair and the distance between the source and sink are determined to give a Rankine oval having the same length and midsectional area as the spheroid moving in an infinite fluid. The comparison becomes meaningless if the disturbance is too close to the free surface because the simple source-sink no longer represents the body well. The same depth of submergence, location of the center and mo- tion of the disturbance as those for the spheroid are used for a direct comparison. Comparison of Fig. 4 and Fig. 5 shows that the wave patterns of the spheroid and the rel- evant source-sink are very similar except that the spheroid generates steeper waves near the stern. Fig. 6 shows the drag, lift and moment acting on the spheroid as a function of time. As seen, the solution is close to the steady state after the body has moved 10 to 15 body lengths. Fig. 7 shows an influence of the two different startups of the body (p = 2 and oo) on the hydrodynamic forces. Although both eventually merge to the same values, the body experiences very different forces during the transi- tion. We notice that the body experiences a negative drag for a short time soon after an impulsive startup. 421

1 875, ---~-~~:, ! / ~ / '' Ir t ~ \ ~ -1 8751 _ 1_ O.0000 _ _ ~ 1 _ _ _ _ _ _ _ _ _ I J _ _ _ 1.875 3.750 Fig. 4 Wave pattern (by spheroid) 5.625 7 500 0.008 (elevation contours are 0.02 apart) ~ .~, 0 006 0. 004 0. 002 ° ~ °°° Fig. 9 shows the convergence of the drag, lift and moment on the body as a function of node number Ni using Nb = 2N~ = 8,12,16 and 20. For all these cases, the free surface grid (61 x 16) adequately resolves the waves. The comparison of the hydrodynamic coefficients to lin- ear theories at different Froude numbers is shown in Fig. 10. Our results for h/L = 0.245 (solid triangles) compare well with linear calculations. Our computations used a finer free surface grid (71 x 16) for the smaller Froude numbers to re- solve the waves. For h/L = 0.16, the body is too close to the free surface. For all attempted Froude numbers, the free sur- face is sucked down and touches the body surface which in turn stops the computation. The linear calculations are not affected by this because the free surface boundary conditions o.o~o -0.002 -o .004 -o .006 \\\/ - "-_... 0 5 CM 0 15 20 25 t Fig. 6 Hydrodynamic coefficients vs. time 0.020 Fig. 5 Wave pattern (by source-sink pair) 0.015 (elevation contours are 0.02 apart) Fig. 8 compares the pressure on the spheroid using the present method and the Neumann-Kelvin calculation at t = 25 for the same conditions as those for Fig. 4. The compari son is made for the pressure along the body centerlines of the top, bottom and side. In the Neumann-Kelvin calculation, the body surface is divided into flat panels with distributions of constant source strength. The strengths are determined by satisfying the body boundary condition at the centers of the panels. The pressure is calculated at the center points. The pressure elsewhere is obtained by interpolation. The differences between the linear and the nonlinear results are noticeable. a ,~ 0. 010 o ~ o . 005 ;^ o :r -o .005 -0.010 0 1 Gradual start-up - Imoulsive start-up 2 3 t 4 5 Fig. 7 Influence of the start-up of the spheroid 422

are satisfied at z = 0. (Note: To compare to Doctors and Becki8, the moments for this figure are taken about the ver- tical projection of the spheroid center onto the undisturbed free surface. The moments shown in the other figures are taken about the centroid of the spheroid). The computations-were carried out on a Cray Y-MP. The results shown in Fig. 4 took approximately 4.3 CPU seconds to solve the boundary value problem (6-9) which required ~ .o 0.8 ~ 0.4 0.2 o.o -o . 2 Fr=0.6 t = 25 | Nonlinear 0.6 L I ~ Linear Bottom center line ~1 1 ~: -0.4 l l l l Bow X Stern Fig. 8 Comparison of the pressure on the spheroid .° 8.0 B.0 . 4.0 _ 2.0 _ 0 50 100 150 Ni 200 250 Fig. 9 Sensitivity of body mesh Slender body Haveloclc Doctors and Beck x x x x Farell ~ ~ ~ ~present method (h/L = 0.245) 0.02 0.016 0.012 CD BOOR 0.004 O _ _ ~ 1 i///'--~16 0.2 0.3 on 05 l 0.6 0.7 0.8 Fr 003 0.025 0.02] 0.015 - CL 0.01 0.005 01 -0.005 -0.01 0.002 01 -0.002 -0.004 C,~ -0.006 -O.OQ8 -0.01 4 -0.012 -0.014 ~. 0.2 0.3 16 ~ \ ,= 02450 ~16 / H/L = 0.2450 OR l 02 0.3 0.4 0.5 0.6 Q7 0.8 Fr Fig. 10 Comparison of hydrodynamic coefficients (modified from Doctors & Beck) 423

solving each subsystem approximately 5 times. About 10 percent of this time was required for matrix setup. The CPU for the entire time simulation was 45 minutes. The computations for the simple disturbance in Fig. 5 took ap- proximately 3.9 CPU seconds to solve (6-9) and 40 minutes for the entire time simulation. The difference in the CPU time for the two computations is small because the splitting procedure for the body problem oniv r~r~llir-~ the ~'l~l;+;~-l evaluation of multiple right-hand sides. 0.02 c 0.01 ._ 0.00 -0.01 .J I.. I 3~"ll~;~ b11= "UUlblOll~ -0.02 02 4 X ,' Id = 1.0, 2.0, 3.0~4.0 Id = 0.5 ~ 8 Fig. 11 Effect of disingularization factor Id ° 30 . 0 25.0 20.0 5.0 0.0 s.o o.o 1 _ l ~ Work // ray , . . , o 2 4 ~8 10 t Fig. 12 Balance of wave energy and work done by spheroid 424 A faster iterative matrix solver can be used effectively in the simple disturbance example.7 About one-fifth of the CPU time was required using the General Minimal Residual Algorithm (GMRES) as compared to using LU decompos- titon. Although GMRES could be used for the spheroid example, it would require special preconditioning. We tried solving the entire system for the spheroid example using both LU decomposition and GMRES without precondition- ing. The LU algorithm gave inaccurate solutions while GM- RES did not converge. Fig. 11 shows the eRect on the wave computations of five different values of desingularization Old = 0.5, 1.0, 2.0, 3.0 and 4.03. The results are not significantly affected except for la = 0.5, which is too small for the integration by the summation (or equivalently one point Gauss quadrature). We also performed an energy conservation check for a fixed control volume bounded by the free surface, the body surface, a horizontal bottom and four vertical surfaces repre- senting the truncated far-field boundaries. The body starts to move from rest at the center of the control volume. The energy of the fluid in the control volume and the work done by the body are shown in Fig. 12. The energy and the work balance each other well for t < 6 before the waves and the body reach the truncated boundary. Since the energy flux is neglected, agreement is not to be expected after the body or the waves reach the boundary. 7 Conclusions Desingularization performs well for fully nonlinear free surface problems without surface piercing bodies. Desingu- larization is not dispersive nor dissipative. Similar to those methods requiring free surface discretization, our method fa- vors high Froude numbers since fewer nodes are required to resolve the waves. Iteration between the free surface and the body surface conditions is required. The waves produced by the spheroid and the relevant source-sink disturbance are similar if they are sufficiently submerged. This indicates that a simple disturbance (with a much simpler iterative procedure and fewer unknowns) can be used if the surface waves are the main interest. Acknowledgment This work is supported under the Program in Ship Hy- drodynamics at The University of Michigan, funded by The University Research Initiative of the Office of Naval Re- search, Contract Number N000184-86-K-0684. Computa- tions were made in part using a CRAY Grant at the Univer- sity Research and Development Program of the San Diego Supercomputer Center. We acknowledge A. Magee for the linear calculation of pressure shown in Fig. 8. References 1. Longuet-Higgins M.S., and Cokelet,C.D., "The Defor- mation of Steep Surface Waves on Water: I. A Numer- ical Method of Computation," Proc. R. Soc. London A350, 1-26 (19763.

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