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Numerical Solution of the "Dawson" Frec-Surface Problem Using Havelock Singularities C. Scragg, I. Talcott (Science Applications International Corporation, USA) ABSTR ACT A method of solving the steady free-surface ship wave prob- lem satisfying "Dawson's" double-body linearization of the free- surface boundary condition, which employs distributed Havelock singularities on both the hull surface and on the free surface is pre- sented. The use of Havelock singularities, as opposed to Rankine singularities, allows the solution to be extended into the far-field without difficulties. The present technique combines the superior aspects of Rankine/Dawson methods in the calculation of near- field waves and the far-field superiority of the Havelock methods. ~,n~r`~,rn'.~in~ results are presented for two simple hull forms, a (JUAN ~` ~6. 1445 , ~ in. vat `__ ~ r submerged body of revolution and a Wigley hull. NOMENCLATURE B C(k=, ky) Cw g G k$ k y ho L n p Rw S(z, y) U V (x, y, a) Z(x, y) To INTRODUCTION Beam Wave spectral function Wave resistance coefficient Gravitational constant Green function Draft Wave number Longitudinal wave number Lateral wave number Characteristic wave number = g/U2 Ship length Unit normal vector into the fluid Pressure Wave resistance Hull surface Ship speed Fluid velocity vector Ship-fixed coordinate system, with x forward, y to port, and z upward Free-surface elevation Double-body wave elevation Velocity potential Double-body velocity potential Perturbation potential Fluid density Havelock source density In 1977, Dawson [1], introduced a method of linearizing the free-surface boundary condition using a perturbation about the zero-Froude number potential. Since then, there has been signif- icant interest in utilizing zero-Froude number or "double-body" linearization schemes in the field of wave resistance and in the prediction of Kelvin waves. Although there are several different methods of linearizing the free-surface boundary conditions (see Raven, Hi), we refer to this basic approach as Dawson's method even though we do not actually use the same version of the lin- earized free-surface equations given by Dawson in his pioneer- ing work. Several researchers have developed computer codes which satisfy the exact hull boundary condition and Dawson's free-surface condition by distributing Rankine singularities over the ship's hull and on the free surface. During the 1988 Workshop on Kelvin Wake Computations (Lindenmuth, et al. [33), it be- came apparent that the best of these Rankine/Dawson codes were capable of predicting quite accurately the wave elevations in the near-field region directly around the ship. However, these codes encountered difficulties in the prediction of the freely-radiating far-field Kelvin waves. The solutions exhibited excessive numeri- cal wave damping and/or wave reflections off the computational boundaries. Although Rankine singularities provide a convenient and effi- cient method for the calculation of the zero-Froude number prob- lem, they actually introduce some numerical difficulties into the calculation of the Kelvin wave field. These difficulties are avoided by solving Dawson's problem with distributed Havelock singular- ities. Since Rankine sources are symmetrical, it is necessary to impose some sort of numerical radiation condition to prevent up- stream radiating waves. This difficulty is not encountered when Havelock singularities are used since the Havelock singularity in- herently satisfies the radiation condition. The use of Havelock singularities also eliminates problems associated with wave re- flections off the computational boundaries. Since neither Rankine singularities nor Havelock singularities distributed over the hull surface alone can satisfy Dawson's free-surface boundary condi- tion, it is also necessary to panelize some region of the free surface surrounding the hull. With Rankine singularities, wave reflec- tions at the edge of the computational domain can create serious problems, usually solved by the introduction of some numerical damping scheme. But with Havelock singularities, there is no dif- ficulty at the edge of the panelized region since Havelock singu- larities always satisfy the linearized free-surface boundary condi- tion, and consequently the far-field waves always propagate away from the hull as linear Kelvin waves. At moderate distances from the hull, the zero-Froude number potential approaches the undis- turbed free-stream potential, and consequently, Dawson's free- surface boundary condition limits to the linearized free-surface boundary condition satisfied by Havelock singularities. There- fore, with Havelock singularities distributed on the free surface, the singularity strength necessary to satisfy the Dawson free- surface boundary condition will smoothly approach zero as the double-body flow approaches the free-stream. The computational domain is defined quite naturally as the limited region of non- zero singularity strength directly around the hull. Furthermore, this computational domain in which free-surface panels are re- quired, is determined by examining the zero-Froude number so- lution, eliminating the need for elaborate free-surface panelization schemes. 259

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This approach leads to a solution which satisfies the same field equations and the same boundary conditions as the Rank- ine/Dawson codes, and therefore, the near-field solutions are the same. But since the use of Havelock singularities eliminates the numerical problems associated with Rankine singularities, this method leads to solutions which are also valid in the far-field. THEORY Consider a ship moving with steady forward speed U in the presence of a free surface. We define a ship-fixed coordinate sys- tem with the positive x-axis in the direction of travel, the y-axis directed to port, and the z-axis vertically upward. The origin is located on the mean free surface. We assume that the fluid is incompressible and inviscid and that the flow is irrotational. Consequently, we can define a velocity potential ~ which satisfies the Laplace equation throughout the fluid domain, V2~ = 0, (1) and is related to the fluid velocity vector V by V = Vie. (2) On the surface of the body S(x, z), we require that the flow be tangential to the hull surface, Vat n = 0, on y = S(x, z), (3) where n is a unit normal vector directed out of the hull. On the free surface Zip, y), the velocity potential must sat- isfy the kinematic free-surface boundary condition, ~2 = USED + Gym, on z = Z~x,y), (4) and the dynamic free-surface boundary condition, gZ + 2vq} Vie = 2U2, on z = Ztx,y), (5) where 9 is the gravitational constant. In addition, we require that the disturbance created by the body must vanish at points infinitely far away, and we require that the far-field free-surface waves generated by the body may not radiate upstream of the ship. The free-surface gradients Zen and Zy can be written in terms of the velocity potential by differentiating equation (5) with re- spect to x and y. Then by substituting the gradients into equa- tion (4), we can write a single free-surface boundary condition which must be satisfied by the potential: 2(V~.V~+2(V~.V.)y~y+g~z = 0, on z = Z(z,y). (6) The manner in which this non-linear free-surface boundary con- dition is linearized is what distinguishes the Dawson problem from the Neumann-Kelvin problem. In both problems, we seek a solution to the Laplace equation (1) which satisfies an exact hull boundary condition (2~. In the Neumann-Kelvin problem we rewrite the potential as the sum of a free-stream potential and a perturbation potential id', and we assume that the perturba- tion potential is, in some sense, small relative to the free-stream potential, 4? = -Up + up'. (7) If we substitute equation (7) into the free-surface boundary con- dition (6), and retain only terms which are linear in A', then we obtain the linearized Kelvin free-surface boundary condition ~+ko~' =0, on z=0, (8) where ho is the characteristic wave number defined by o u2 (9) To show that the boundary condition can be applied at the po- sition of the mean free surface, one can expand the potential in a Taylor series about z = 0, and assume that the free-surface elevation Z is of the same order as the perturbation potential. In Dawson's approach to the problem, the potential is di- vided into a double-body potential ~ and a perturbation potential in, it = ,+~p (10) and it is assumed that the perturbation potential is small rela- tive to the double-body potential . The double-body potential corresponds to the limiting solution as the Froude number goes to zero (i.e. g >> U ), for which case the free surface acts as a reflection plane. The double-body potential is a solution to the Laplace equation at all points outside of the body, V2 = 0, (11) and satisfies the exact hull boundary condition, n V) = 0, on y = S(x, z), (12) and a reflection boundary condition applied on the position of the undisturbed free surface, (: = 0, on z = 0. (13) The double-body solution can be readily obtained by well-known panelization methods utilizing Rankine (1/R) singularities, and it will be assumed throughout the remainder of this discussion that the double-body potential and its derivatives can be treated as known quantities. The perturbation potential must be a solution to the Laplace equation throughout the fluid domain, and must satisfy the same hull boundary condition, n Via = 0, on y = Sly, z). (14) To obtain the linearized free-surface boundary condition which must be satisfied by the perturbation potential, we substitute equation (10) into the free-surface boundary condition, equa- tion (6), and retain only first order terms in A, 2(V~ V)sy$ + (V~ Versus + 2(V~ V)yyy + (V~ Vy~y~y + gyz = -9z - 2 (V) verse= (15) -2 (V~ V)y by, on z = Z(z, y). In order to apply the free-surface boundary condition at the po- sition of the undisturbed free surface, it is necessary to expand equation (15) in a Taylor series about z = 0. By defining a wave elevation Z0 which depends only upon the double-body potential Ze = 2} (u2 - V) V), (16) and assuming that the wave elevation Zip, y) is composed of Z0 plus additional terms which are of the order A, we can obtain the linearized boundary condition to be satisfied on the mean free surface: (s)2Yss + 2~Sy~ysy + (y,2yyy + DOSES + (y~xy)Ys + 2~s~xy + Amp Fly, + gyp: = - 9Zo~zz -Obsess + ~y~xy)s ~ hasty + ~y~yy)y, (17) where we have used the reflection condition, As = 0, to remove terms involving the vertical component of the double-body flow on z-0. It is important to note that since the double-body potential tends to the free-stream potential as we move away from the body, the Dawson free-surface boundary condition limits to 260

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source panels over the hull surface and M panels over the undis- turbed free surface, the pertubation potential can be written as U2~cc+gyz = 0, as ~ ~ oo. (18) This is identical to the linear Kelvin free-surface boundary con dition written in equation (8~. Consequently, the differences be tween the Dawson solution and the Neumann-Kelvin solution must be due to the differences between their respective free surface boundary conditions, equations (17) and (8), which occur in a relatively limited region around the hull. We propose to represent the double-body perturbation po tential y by a distribution of Havelock singularities on the hull surface and on the free surface in a region immediately surround ing the hull. At points farther away from the hull, where the Daw son free-surface condition tends to the Kelvin condition, the Have lock source strength necessary to satisfy the free-surface boundary condition goes to zero. In this approach it will be necessary to add the double-body potential , which is represented by a dim tribution of Rankine singularities, to the perturbation potential A, represented by Havelock singularities, in order to calculate the total flow field around the hull. Alternatively, one could represent both the double-body po tential and the perturbation potential by distributions of Have- where lock singularities on the hull and the free surface. In this ap proach, we seek the total potential A, which must be a solution to the Laplace equation subject to the hull boundary condition (3~. To obtain the linearized boundary condition which must be sat isfied by ~ on the mean free surface, we substitute equation (10) and into equation (17~: (<;5 )2/(x_ t.)2+(y_~)2+(Z+~)2, k= I, where kit and by are the longitudinal and lateral wave numbers of the free-surface waves. In the single integral, the longitudinal wave number is not an independent variable, but is related to by by [2 ( ~)]1/2 . 2 (ho + ~ ) - ~/ko + 4ky The unknown source densities are determined by applying the hull boundary condition, equation (14), at the centroid of each hull surface panel and applying the Dawson free-surface bound- ary condition, equation (17), at the centroid of each free-surface panel. Since the Havelock singularity automatically satisfies the Kelvin free-surface boundary condition, the source density re- quired to satisfy the Dawson condition should smoothly tend to- ward zero at points away from the hull where the Dawson con- dition limits to the Kelvin condition. To evaluate the total po- tential A, it is necessary to include a term corresponding to the free-stream velocity, N =-US + Eli | dsG(x,y,z;t,,rl,,) i si (22) M + En, Hi | ds G(x, y, z; (, 71, () ~ i Hi The source densities ~` in equations (20) and (22) will of course be different due to the fact that they will be determined using different boundary conditions on the free surface, equations (17) and (19). We note the absence of the waterline integral which is necessary in the typical formulation of the Neumann-Kelvin problem where Havelock singularities are distributed only over the hull surface and along the intersection of the hull and the free surface. The waterline integral arises out of an application of Stokes Theorem to the integration over the free surface, re- (lucing the surface integral to a contour integral. Since we have 261

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retained the free-surface integration, no contour integral around the waterline occurs. The first two terms in the Green function correspond to a Rankine source and its negative image above the free surface. These terms can be evaluated using standard techniques. The contribution to the potential due to free-surface waves is con- tained within the two integral terms in equation (21~. To calcu- late this wave potential we first define a wave spectral function C(k=, by) which can be evaluated analytically for a flat panel of uniform source density: C(k=,ky) = ~ deem-ik=6-ityn 3i (23) Then the contribution to the potential from the two integral terms in the Green function is 2 l /~ ek=+ik=~+ikuy =a-ho J dLyt dk~ C(k='ky) k2 _ kok (24) -i,, ,/ day /3 C(k', ky~etz+ik=~+ilyy, -oo ~ The boundary conditions on the hull and the free surface involve first and second derivatives of the potential which can be cal- culated by multiplying the spectral function by the appropriate wave number prior to performing the integration. The calculation of the Kelvin waves generated by a discrete Havelock singularity at zero depth can be particularly trouble- some. However, in the present formulation of the problem, the singularities are distributed uniformly over flat free-surface panels of finite dimensions. By first performing the spatial integration over the panel, we effectively filter out much of the high wave number content of the Havelock source and we can obtain well behaved spectral functions for an arbitrary panel, even one at zero depth. Some of the higher order derivatives of the potential which occur in the Dawson free-surface boundary condition can lead to an integrand which contains sufficient high wave-number content to present numerical difficulties. However, these high wave num- bers correspond to waves which are not properly resolved by the free-surface panelization and which should be removed from the spectral function to prevent aliasing. In the results presented in the following sections, a cosine squared filter has been applied to the spectral functions with the filtering length set at twice the length of the free-surface panels. The spectral functions have been cut-off at half the panel dimension. It is interesting to examine the limiting behavior of a distri- bution of Havelock singularities as both the depth of the panel and its collocation point go to zero. If the panel is located at an infinitesimal depth ~ < 0, then the collocation point should be interpreted as being locating at the limit as the field point approaches the panel from below, z = c~. The linearized free- surface boundary condition is actually satisfied on the other side of the panel, at z = 0, and there will exist a 4'T discontinuity in At across the panel due to the first Rankine term in equation (21~. Consequently, a free-surface distribution of Havelock singulari- ties will not satisfy the Kelvin free-surface boundary condition at points located on the panel itself, although it will satisfy the boundary condition at all other points on the free surface. There- fore, on panels located at points away from the hull, where the Dawson free-surface condition approaches the Kelvin free-surface condition, the Havelock source density must go to zero, for oth- erwise the free-surface boundary condition could not be satisfied at the collocation point. The N unknown source strengths on the hull surface panels and the M unknown source strengths on the free-surface pan- els are obtained by solving a set of independent linear equations composed of the N hull boundary conditions and the M free- surface boundary condition. Consequently, we have eliminated the finite differencing schemes usually employed in the solution to the Dawson problem. RESULTS - SUBMERGED BODY OF REVOLUTION To investigate our numerical approach, we initially examined a fully submerged body, since this would avoid any difficulties associated with the intersection of the hull with the free surface. We chose the submerged prolate spheroid for which Neumann- Kelvin results have been presented by Doctors and Beck [53. The ellipsoid of revolution can be defined by r = ~0 L [1 - (2x/L)2] (25) where r is the radius of the body and L is the length. The hull cen- terline was located at a depth of z = 0.16L. The body was pan- elized with 240 panels on the half-body (symmetry about y = 0 is assumed), using 8 rows of 30 panels with cosine spacing in the longitudinal direction. For this body of revolution, the hull sur- face can be panelized with flat quadrilaterals, and panel warpage is not an issue. The free surface was panelized using 240 square panels (on the half-space y > 0) with dimensions of L/10 on each side. The free-surface panels were arranged on a rectangular grid centered over the body, with 30 panels longitudinally and 8 pan- els laterally. The panelization of both the free surface and the submerged spheroid is shown in Figure 1. This panelization was chosen to cover the entire region of the free surface over which the double-body velocity magnitude differed from the free-stream by more than 1.0%. For this submerged body, the maximum dif- ference between the double-body velocity magnitude and the free stream velocity is ~ 9%. The magnitude of the double-body velocity on z = 0 is plotted in Figure 2, where the solid lines represent 1% contours for which the magnitude is greater than the free-stream and the dashed lines represent contours less than the free-stream velocity. We expect that the Dawson free-surface condition differs from the Kelvin condition only over this limited region, and consequently, the Havelock singularity density should go to zero on the panels near the edges of our panelized domain. Approach I- Perturbation Potential At a Froude number of 0.4, the solution of the perturbation potential, id, resulted in the distribution of Havelock singulari- ties on the free surface shown in Figure 3 (positive contours are shown as solid lines and negative contours are dashed). As ex- pected, the source density is greatest near the body and goes smoothly to zero at the edges of the domain. The greatest source density which occurs on any edge panel is less than 2% of the peak value (-0.012), which occurs directly over the body. The corresponding Havelock singularity densities on the hull surface are shown in Figure 4. Each curve represents a row of panels at one circumferential angle on the body. The peak singularity densities found on the hull occur on the row of panels nearest the free surface, and are about twice the value of the peak free- surface sources. Contrast this singularity distribution with that obtained using our Neumann-Kelvin solution technique, Figure 5, which of course, does not have any singularities on the free sur- face. The Havelock/Dawson solution has peak singularity densi- ties which are about one fourth of the peak values obtained in the Neumann-Kelvin solution, but more importantly, the Have- lock/Dawson solution is significantly smoother, indicating that the geometric approximation of flat panels of constant source strength is more accurate for an equal number of hull surface panels. We also note that the Havelock/Dawson solution yields singularity densities which go smoothly to zero at both the bow and stern. The near-field free-surface elevations are calculated from the pertubation potential using a linearized version of the Bernoulli equation which is consistent with the linearized Dawson free- surface boundary condition: p = 2pU2 - pgz - 2pV) Vie - pV) Vie, (26) 262

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where p is the pressure and p is the density of the fluid. The free-surface elevation calculated over the panelized domain is pre- sented in Figure 6 (non-dimensionalized by the characteristic wave number ko). As expected, the edges of the computational domain do not create any wave attenuation or reflections. The extension of this solution to the far-field is accomplished by us- ing equation (23) to calculate the free-wave spectrum C(k', by) associated with the Havelock singularities on both the hull and the free surface, and then calculating the far-field potential from the far-field limit of equation (24~: he ~-~ ho ~ day t~ C(k' ~ by jEkZ+ik=~+ityy, (27) -Go ~ In Figure 7 is shown a comparison of the far-field Kelvin waves calculated from the Havelock/Dawson solution (solid line) and from the Neumann-Kelvin solution (dotted line). The similarity is remarkable. The components of the far-field waves generated by the singularities on the hull and on the free surface can be calculated separately, and this result is shown in Figure 8. Al- though the singularity strengths on the hull are twice as large as those on the free surface, the far-field waves generated by these singularities (solid line in Figure 8) are significantly smaller due to the exponential attenuation with depth. Approach II- Total Potential The second numerical approach we examined was the solu- tion of the total potential ~ by distributed Havelock singularities on the hull and free surface. The free-surface singularities ob- tained are shown in Figure 9. Again we see the rapid reduction in the singularity density as we move away from the body. At the outermost panels, the densities (less than 3% of the peak values) are similar in absolute value to those obtained in the previous ap- proach, and may represent some measure of the numerical noise. The peak values (-0.007) are less and the distribution of Have- lock singularities seems to be limited to an even smaller region than was observed in the solution to the perturbation potential. Unlike the pertubation potential, the singularities do not seem to go monotonically to zero as we move laterally away from the body, but exhibit a slight oscillatory behavior, suggesting that this approach may require a finer free-surface panelization. The singularity densities distributed over the hull surface are shown in Figure 10. Both the qualitative and quantitative similarities between these source densities and the corresponding Neumann- Kelvin results (Figure 5) are striking. The strong singularities at the bow and stern are an order of magnitude greater than the peak values found on the free-surface panels. As before, the near-field free-surface elevations are calculated from a consistent linearized version of the Bernoulli equation. To obtain the pressure equation in terms of the total potential ~ rather then the perturbation potential id, we substitute equa- tion (10) into equation (26), which leads to p = 2pU2 - pgz + 2pV) V) - pV) Vat . (28) The non-dimensional free-surface wave elevations are given in Fig- ure 11. Although the wave heights obtained here are slightly higher than those obtained from the perturbation potential (Fig- ure 6), the overall agreement is quite good. The far-field wave elevations obtained from the free-wave spectrum are compared to the Neumann-Kelvin results in Figure 12. The present results (solid line in Figure 12) are somewhat higher than those obtained from the perturbation solution (Figure 7) and consequently, the quantitative agreement with the Neumann-Kelvin results (dotted line) is not quite as good, although qualitatively, the results are similar. The contribution to the far-field waves from the hull sur- face singularities dominates the solution, as shown in Figure 13. The waves generated by singularities on the hull are shown by the solid line and those generated by free-surface singularities are shown by the dotted line. Wave Resistance Calculation The wave resistance can be calculated either by integrating the linearized Bernoulli pressure, equation (26) or (28), over the hull surface, or by calculating the energy in the far-field Kelvin waves. Pressure integration is the standard technique employed in Dawson's method, while most Neumann-Kelvin solvers calculate the wave resistance from the energy in the free-wave spectrum. Using equation (23) to calculate the free-wave spectrum from the distribution of Havelock singularities on both the hull and free surface, we can obtain the wave resistance from RW = 27rpho Idly ~ Ok', ky)~2 (29) The non-dimensional wave resistance coefficient Cw is defined as C RW -DSU2 ' (30) cat where S is the total wetted surface area of the hull. We will use Cd to designate the comparable resistance coefficient ob- tained from pressure integration. It is probably more reasonable to compare C,l, with a drag coefficient obtained by integrating the full non-linear Bernoulli pressure over the hull surface rather than the linearized version contained in equations (26) and (28~. The results obtained from the free-wave spectrum and from pres- sure integration over the hull are presented in Table I for both Havelock/Dawson approaches and compared with the Neumann- Kelvin result. The Neumann-Kelvin solution given by Doctors and Beck [5; is within 1% of the comparable result presented here. Table I. Wave Resistance Coefficient for Submerged Body C(~) c(2) Coo H/D Soln: ~ 0.0132 0.0143 0.0150 H/D Soln: ~ 0.0148 0.0162 0.0190 N-K Soln. 0.0124 0.0124 C(~): Obtained by integrating the linearized Bernoulli pressure. C(2~: Obtained by integrating the non-linear Bernoulli pressure. C'`,: Obtained from free-wave spectrum. The steep gradients found in the source densities directly over the body suggest that the solutions would benefit from the use of smaller panels in this region. The calculations were re- peated using quarter size panels on an inboard region defined by -0.6 < z/L < 0.6 and-0.2 < y/L < 0.2, with the outboard region using the larger panels shown in Figure 1. The solutions had more well deRned singularity densities but the resulting wave- fields and wave resistances did not change significantly. RESULTS- WIGLEY HULL Our initial attempts to solve the Dawson problem using dis- tributed Havelock singularities employed the submerged body de- scribed above. In this section we report on the results obtained when the same numerical approach was applied to a surface pierc- ing body, specifically the Wigley hull form defined by Y = 2B [1-(2X/L) ] [1-(Z/H) ~ , (31) where B = beam = L/10, and H = draft = L/16. The pan- elized free surface, Figure 14, contains 300 square panels (L/10 on each side) in the half-space y > 0. The hull surface paneliza- tion contained only 50 quadrilateral panels, each panel having a 263

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length of L/10 and a depth of H/5. This panelization is much coarser than we would usually use for a comparable Neumann- Kelvin solution, but we wanted to match panel size on the hull with adjoining free-surface panels. The fact that the hull surface panels are flat quadrilaterals means that there will exist some gaps between the panel edges. The double body velocity magnitudes are shown in Figure 15. Due to the high length/beam ratio of the Wigley hull and the rel- atively large free-surface panels, the double-body velocity magni- tudes calculated at the centers of the panels never differ from the free stream velocity magnitude by more than 3%. On the out- ermost panels, the double-body velocities are reduced by more than one order of magnitude. Consequently, we would not ex- pect the Havelock/Dawson solution to differ significantly from the Neumann-Kelvin solution. The slight fore/aft asymmetry which can be seen in the contours is one effect of the gaps in the hull surface created by the use of flat quadrilateral panels. The free-surface singularity distribution corresponding to the solution of the perturbation potential at a Froude number of 0.40 is shown in Figure 16. The singularity densities go smoothly to zero at a very short distance from the hull. Since the free-surface panelization does not seem to adequately resolve the distribution of singularities near the hull, it would appear that smaller panels distributed over a more limited free-surface domain would yield a better result for the same number of panels. The corresponding hull surface singularity distribution is given in Figure 17. Each of the 5 curves in the figure represents a row of panels at the same depth on the hull. The singularity densities are comparable in peak values to the densities on the free surface, although adjoin- ing hull and free-surface panels can have quite different densities, as evidenced on panels near both the bow and stern. The den- sities on the free surface reach their peak values near the ends of the hull while the hull-surface singularities seem to be tending smoothly to zero at the ends. Comparing these singularities with the Neumann-Kelvin solution shown in Figure 18, we note that (like the similar comparison for the submerged hull) the Have- lock/Dawson solution results in much smaller peak values and much smoother trends. Calculated non-dimensionalized near-field free-surface eleva- tions are given in Figure 19. As before, the computations are not affected by the edge of the panelized domain, and can be readily extended into the far-field. The far-field wave elevations are com- pared to the Neumann-Kelvin result in Figure 20, where the solid line represents the Havelock/Dawson result and the dotted line represents the Neumann-Kelvin result. There is more high wave number energy in the Neumann-Kelvin result, as evidenced by the steep diverging waves which occur at non-dimensional distances of-10 to-12, but otherwise the results are quite similar. The generation of the far-field waves is dominated by the free-surface singularities, as can be seen in Figure 21. For this particular case, the wave resistance coefficient ob- tained from the Havelock/Dawson solution Cw = 0.00204 com- pares very well with the value obtained from the Neumann-Kelvin solution (Cw = 0.00212~. The drag coefficient obtained by in- tegrating the linearized Bernoulli pressure is somewhat higher ICE = 0.00258) while the drag coefficient obtained from the full non-linear Bernoulli pressure is 0.00248. However, these results may not be too meaningful given the coarseness of the paneliza- tion on the hull surface. The second approach to the Havelock/Dawson problem, solv- ing for the total potential rather than the perturbation potential yielded a disappointing result. The free-surface singularities, Fig- ure 22, did not tend smoothly to zero at the edges of the panelized domain, but exhibited regular oscillations as we moved laterally away from the stern. The singularity densities shown in the fig- ure are quite small, one tenth the magnitude of the hull-surface singularities (which are quite similar to the Neumann-Kelvin sin- gularities), but they generate a comparable wave field due to their free-surface location. In fact, the far-field wave elevations shown in Figure 23 show that a remarkable degree of cancellation will occur between the waves generated by the free-surface sing~lar- ities and the hull-surface singularities. We speculate that the solution is suffering from a loss in numerical accuracy caused by this cancellation. It would appear that this problem is unique to this particular approach. In the first approach, the solution for the perturbation potential, the free-surface waves are dominated by the singularities distributed over the free-surface panels, and the loss of accuracy does not occur. . . . SUMMARY AND CONCLUSIONS Our original objective in this effort was to develop a method ot combining the superior near-field predictions of the Rank- ine/Dawson codes with the superior far-field predictions of the Havelock codes. We set out to demonstrate that the use of Have- lock singularities distributed over the free surface as well as on the hull surface, rather than Rankine singularities, could result in a solution to the Dawson problem which was free of the wave reflections often caused by the boundaries of the computational domain and free of the wave attenuations introduced by numer- ical damping schemes. This Havelock/Dawson solution could be extended to an arbitrary distance in the far-field. By examining two simple geometries, a submerged spheroid and a Wigley hull, we have demonstrated the feasibility of the method. The free-surface singularity densities go to zero at the outer edges of the panelized domain and both the near-field and far-field wave elevations are well behaved, with no evidence of wave reflections or artificial damping. The far-field wave eleva- tions have been shown to be comparable to those obtained from a solution to the Neumann-Kelvin problem. We have presented results obtained from two different nu- merical approaches. In the first approach, we solve for a pertur- bation potential represented by distributions of Havelock singu- larities, to which must be added a double-body potential which is obtained in the usual manner from distributions of Rankine singularities on the hull. In the second approach, we solve for the total potential represented by distributions of Havelock sin- gularities. The two approaches should yield comparable results, but we found that with the first approach we did not experience the degree of numerical difficulties encountered with the second approach. The singularity distributions obtained when solving for the perturbation potential were exceptionally well behaved. The peak - values of the singularity densities were comparable on the hull and on the free surface, and were significantly smaller in peak value and noticeably smoother than the singularity distributions obtained from the Neumann-Kelvin solution. For the two simple geometries examined in this study, the hull-surface singularities tended toward zero at both the bow and the stern. The wave fields were dominated by the contributions from the free-surface singularities. The second approach, solving for the total potential, yielded acceptable results only for the submerged body. For both the sub- merged spheroid and the Wigley hull, the hull-surface singularity distributions obtained were remarkably similar to those obtained with our Neumann-Kelvin solver. The free-surface singularity distributions were significantly smaller than the hull-surface dis- tributions, but their location on the free surface resulted in wave fields which were comparable to those generated by the singular- ities on the hull. For the Wigley hull, it appeared that cancel- lation between the waves generated by the hull singularities and the free-surface singularities resulted in a loss of accuracy. It has yet to be demonstrated that the Havelock/Dawson approach can reproduce, for realistic hull forms, the superior near-field results of the Rankine/Dawson methods which were reported by Lindenmuth et al. [33. The only geometries investi- gated to date have been so simple that one would not expect sig- nificant differences between Neumann-Kelvin results and Rank- ine/Dawson results. However, these encouraging results indicate that the Havelock/Dawson method may well give us the ability 264

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to combine the best aspects of both the Rankine/Dawson codes and the Havelock codes. ACKNOWLEDGMENTS This work was supported by the Applied Hydromechanics Research program of the Applied Research Division of the Of- fice of Naval Reseacrh, and administered by the David Taylor Research Center. REFERENCES [1] Dawson, C.W. "A Practical Computer Method for Solving Ship-Wave Problems". The Proceedings of the Second Interna- tional Conference on Numerical Ship Hydrodynamics, Berkeley, California, 1977. [2] Raven, H.C. "Variations on a Theme by Dawson". Seven- teenth Symposium on Naval Hydrodynamics, The Hague, The Netherlands, 1988. [3] Lindenmuth, W.T., T.J. Ratcliffe and A.M. Reed. Com- parative Accuracy of Numerical Kelvin Wake Code Predictions - "Wake-Off". David Taylor Research Center Report DTRC/SHD- 1260-01, May 1988. [4] Wehausen, J.V. and Laitone, E.V. Surface Waves. "Encyclo- pedia of Physics," Vol. IX. Springer-Verlag, Berlin, 1960. t5] Doctors, L.J. and Beck, R.F. "Convergence Properties of the Neumann-Kelvin Problem for a Submerged Body". Journal of Ship Research, Vol. 31, No. 4, December 1987. 265

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PRNEL I ZRT I ON SOURCE DENSITY Contour cntervaL - O. 0005 o.s o.o o.s '.o 1.5 X / ShLp Length Figure 1. Hull panels on a submerged body of revolution and corresponding free-surface panels. DOUBLE BODY TOTRL VELOCITY - In o 1 1.~5 o.oo Figure 2. Double-body velocity magnitude on the free surface. Solid lines are contours greater than 1, dashed lines are less than 1. Figure 3. Source density on the free-surface resulting from a solution to the perturbation potential at Fn=0.40. Solid lines are positive contours, dashed lines are negative. SOURCE STRENGTHS Hull SLnguLarLtLes ~ Perturbation Potential ~_.. 0.5 0.4 0.3 0.2 o. I o.o -o. ~ -0.2 -0.3 -0.4 -0.5 tongutudLnat posLtLon / shLp Length Figure 4. Source strengths on hull surface panels resulting from a solution to the perturbation potential. Each curve represents panels along one circumferential angle. 266

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u] o o lo - ~ o- o i lo lo o o cn (n SOURCE STRENGTHS Neumann-KeLv~n Sol utcon 0 . ~ l 0 , ., ..., ..... .5 0.4 0.3 0.2 0.1 O.o -0. 1 -0.2 -0.3 -0.4 - .s CongutudenaL positron / strep length Figure 5. Source strengths on hull surface panels resulting from a solution to the Neumann-Kelvin problem at Fn=0.40. NEVE ELEVRT I ON Contour Interval ~ 0. 050 o lo u, 1.45 O.OC Figure 6. Free-surface elevations resulting from a solution to the perturbation potential at Fn=0.40. Solid lines are positive con- tours, dashed lines are negative. _ -2.0 -3.0 WRVE ELEVRT I ON a, Froude Number ~ O .40 YO ~ 2. 00 lo cot O At: cat _ 0 o is lo I_ G ~ _ ha r~ o -4.0 -5.0 -6.0 -7.0 -8.0 -9.0 -10.0 -li .o -12.0 LONG I TUD I NRL D I STRNCE / SH I P LENGTH Figure 7. Comparison of far-field wave elevations obtained from the solution to the perturbation potential Dawson problem (solid line) and the Neumann-Kelvin problem (dashed line). WRVE ELEVRT I ON Froude Number - 0.40 YO - 2. 00 ., Figure 8. Contributions to the far-field wave elevations from the hull-surface panels (solid line) and from the free-surface panels (dashed line), corresponding to the solution to the perturbation potential. 267

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SOURCE DENS I TY Contour Intervals - O. COOS WRVE ELEVRT I ON Contour Intervals - O. 050 - 1.45 0.00 Figure 9. Source density on the free-surface resulting from a solution to the total potential at Fn=0.40. SOURCE STRENGTHS HuLl' SunguLarctces ~ Totals Potentcal ' - r.l ~ " -a 0.5 o.4 0.3 0.2 o. 1 o.o -o. t -0.2 -0.3 -0.4 -o.s l~ongctudenal" positron / shop Length Figure 10. Source strengths on hull surface panels resulting from a solution to the perturbation potential at Fn=0.40. Figure 11. Free-surface elevations resulting from a solution to the total potential at Fn=0.40. WRVE ELEV8T I ON Froude Number - O.40 YO - 2.00 -2.0 -3.0 -i.o -s.o -6.0 9-i.o -8.0 -9.0 -to.o -~i.o -~2.0 LONG I TUD I NRL D I STANCE / SH I P LENGTH Figure 12. Comparison of far-field wave elevations obtained from the solution to the total potential Dawson problem (solid line) and the Neumann-Kelvin problem (dashed line). 268

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WRVE ELEVRT I ON N Froude Number ~ 0.40 YO ~ 2. 00 0 WOK o, ~ . , . , . , , , . , , -2.0 -3.0 -4.0 -s.o -6.0 -7.0 -8.0 -9.0 -10.0 -1 1 .o -12.0 LONG I TUD I NRL D I STRNCE / SH I P LENGTH Figure 13. Contributions to the far-field wave elevations from the hull-surface panels (solid line) and from the free-surface panels (dashed line), corresponding to the solution to the total potential. PRNEL I ZRT I ON . DOUBLE BODY TOTRL VELOC I TY Contour Interpol ~ 0. 005 o~ 8 a o 1.45 - 0.00 -1 .45 Figure 15. Double-body velocity magnitude on the free-surface for the Wigley hull. SOURCE DENS I TY . . . . . -1.5 -1.0 -0.s o.o o.s X / Shep Length 1.0 ~.5 Figure 14. Hull panels on a Wigley hull and corresponding free- surface panels. Figure 16. Source density on the free-surface resulting from a solution to the perturbation potential for the Wigley hull at Fn=0.40. 269

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SOURCE STRENGTHS Hulls Sungul~aruti~es - Perturbatton PotentcaL - r~ A a - o o 2 0 cn . ~ an - o . o N o o o A _~ 0.5 0.4 0.3 0.2 0.1 0.0 -0. 1 -0.2 -0.3 tong~tudunal~ positron / shop Length ^o. ~-o.s Figure 17. Source strengths on the hull-surface panels of Wigley hull resulting from a solution to the perturbation potential. Each curve represents a row of panels at a constant depth. SOURCE STRENGTHS ~Neumann-Kel~v~n Solution o o on o - Gl N~ WN . . . 0.5 0.4 0.3 0.2 0.1 0.0 -0. 1 -0.2 -0.3 -0.4 - . LongetudunaL position / sheep Length Figure 18. Source strengths on hull-surface panels of Wigley hull resulting from a solution to the Neumann-Kelvin problem at Fn=0.40. ARVE ELEVRT I ON Figure 19. Free-surface elevations resulting from a solution to the perturbation potential for Wigley hull at Fn=0.40. URVE ELEVRT I ON Froude Number ~ O .40 TO - 2.00 . , . , . , . , . i!. ,'. , . , , , , , . ,., . 1 . -2.0 -3.0 -~.0 -5.0 -6.0 -7.0 -B.0 -9.0 -10.0 -11 .0 -12.0 -13.0 -14.0 -15.0 LONG I TUD I NRL D I STRNCE / SH I P LENGTH Figure 20. Comparison of far-field wave elevations obtained from the solution to the perturbation potential Dawson problem (solid line) and the Neumann-Kelvin problem (dashed line). 270

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WOVL LLEV8TION ~ ~ - O ~ - 2~ WHVE ELEVOTION ~ ~ - ~ - 2~ -2.0 -3.0 -4.0 -5.0 -6.0 -7.0 -8.0 -9.0 -10.0 -11.0 -12.0 -13.0 -14.0 -15.0 LONG I TUQ I NAL O I STONCE / SH I F LENOTH Fi~re 21. Contributions to tbe ~-Reld w~e elev~ions hom tbe bulLsur~ce p~ek tsobd Une) ~d hom tbe he~sur~ce paneb Q^d 1~), ~d~g ~ t~ ~^ ~ t~ pet~b~- pote~i~. SOURCE DENSITY ~- ~1 1.45 Fi~re 22. Source denshy on tbe he~sur~ce resuldug hom ~lution to tbe tots1 p~enti~ ~r tbe Wig~y bull ~ Fn=0.40. .o -i.o -~.o -s.o -d.o -) o -a.o -g.o -lo.o -ll.o -12.0 -13.0 -14.0 -15.0 LONG I IUD I NAL D I STONCE / SH I F LENGTH Fi~re 23. Conthbutions to tbe ~-Reld w~e elev~ions Q~ tbe . bulLsur~ce p~eL (scud bue) ~d hom tbe he~sur~ce p~s (d~bed line), corresponding to the solution to tbe tota1 potential. 271

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