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Computations of 3D Transom Stern Flows Bill H. Cheng David Taylor Research Center Bethesda, USA Abstract A practical computational method for 3D transom stern flows is presented. The theory and numerics of computing transom stern flows are described in detail. The special treatment of the linearized free surface boundary condition is included. In particular, the boundary condition for a dry transom is derived within the framework of a free surface potential flow. The transom is treated as an inflow boundary and the transom boundary condition is then used to specify the starting values of a linearized free surface calculation. This computational method has been incorporated into a Rankine source panel method, the XYZ Free Surface (XYZF,S) program version 2.0. The XYZES program has been used to predict the wave resistance for a large number of transom stern ships. The compute d wave resistance has com pare d favorably with measured wave pattern resistance, correctly predicting the relative merit of competing hulls. The agreement with experimental measurements is remarkably good for Froude numbers between 0.35 and 0.50, corresponding to the normal speed range of high-speed transom stern ships. 1. Introduction A practical computational method for ship by the the transom stern flows is of special interest to hydrodynamicists. Such interest is intensified the peculiar property of the flow pattern. If t ship speed is high enough, the transom clears surrounding water and the entire transom area is exposed to the air. The transom flow detaches smoothly fro no the underside of the transom, and a depression is created on the free surface behind the transom. Saunders (14 described this flow pattern in his book on hydrodynamics and his description of a dry transom is reproduced in Fig. 1. This flow pattern has been credited with the reduced wave resistance for high-speed transom stern ships, as compared to their cruiser stern equivalents. 581 Early numerical studies encountered problems in modeling transom stern flows. Rankine source and Havelock source methods were hampered by the difficulty of treating the intersection curve between the transom and the free surface. Many investigators replaced the fluid domain behind the transc)'n faith solid surfaces. For example, Gadd t2] and Ghang [3] simulated the surface depression by adding a tapering extension to close the body behind the stern. Realistic flow patterns cannot be obtained in this way and more accurate computations of transom stern flows are needed. This paper describes an in pro ed method for com puting transom ste r n flows. The theory and numerical method for computing 3D transom stern flows are presented step by step. The special treatment of the linearized free surface boundary condition is given in detail. The boundary condition for a dry transom is derived within the framework of a free surface potential how. The physical constraints imposed by this transom boundary condition require that the static.pressure be atmospheric and that the flow leave tangentially at the transom. This computational technique has been successfully incorporated into a Ranltine source panel method originally developed by D awson t4,5~ and further developed as the XYZ Fre e Su rface ~ XYZFS ~ program Ve rsio n 2.0. The XYZFS program has been used to analyze the wave resistance and local flowfield for a large number of transom stern ships. Comparisons of wave resistance prediction and experimental measurements have been made and have shown general agreement. Cheng et al. t6: described the hydrodynamic analyses of DTRC Model 5403 and Model 5404, which represent typical transom stern ships with bow domes. A parametric study on stern wedges, as an example of transom stern variations, was described by Cheng et al. t74. An FFG-7 hull with a 15-degree stern wedge was

TRANSOM MEAN WATER LEVEL ~ ~ ---I in_ DEPRESSION IN WATER BEHIND HANSOM STERN UNDISTURBED WA1 - SURFACE Fig. 1. Flow pattern behind a transom stern (sketch from Saunders t 1 ] ). analyzed by Hoyle et al. [8~. Cheng et al. t94 presented a comparison of the compute d prope lie r inflow with the corresponding data from wake survey experiments for a variant of the research vessel, R/V Athena hull. Transom stern studies prior to 1983 were summarized by Cheng et al. A. Subsequently, Wilson and Thomason [10] published a parametric study on transom sterns. The effects of transom depth and transom width variations were identified using a combined numerical and experimental approach. More recent studies on transom sterns included residual resistance computations for Series 64 by Tulin and Hsu [11~. These computations used a strip theory which is applicable in the limit of an infinite F'-oude Amber. In this paper, the computed flow pattern behind the transom is presented and compared with experimental measurements by Jenkins t12] for the R/V Athena hull. In addition, a comparison of wave resistance prediction and experimental measurements is presented for D TRC Model 5416. 2. Physical problem Consider a steady flow directed from bow to stern of a ship. Attention is focused on the region Just forward and aft of the stern as shown in Fig. 1. Figure 2 is a schematic diagram representing a sideview of a transom stern flow in a centerplane. The keel line typically slopes upward relative to the mean water level, which is designated by a horizontal dashed line in Fig. 2. The wavy solid line represents a streamline which passes under the transom, detaches from the stern, and forms part of the free surface. Immediately downstream of the transom, the water rises rapidly toward the mean water level, overshoots, and then reaches a maximum elevation before descending under the - KEEL LINE Fig. 2. Schematic of transom stern flow for the centerplane shown in a sideview. influence of gravity. Such ascending and descending motion is repeated further downstream. The wave motion described here can be seen in the wave pattern computed by Coleman t 134 using a finite difference method. 3. Numerical method 3.1 Panel method for cruiser stern flows Before the computational method for 3D transom stern flows is introduced, a panel method for cruiser stern flows is described. The usual treatment of cruiser sterns is presented since the special treatment for transom sterns is similar and is an extension of the cruiser stern method. In the Rankine source panel method, a hull surface is m athem atically subdivided into hundreds of small source panels. Each panel is characterized by its centroid, normal unit vector, and area. The velocity at the control point (i.e., centroid) of a given panel i induced by another panel j is a function of the geometry of panel j and of the distance between panel i and panel j . For example, the x-component of the velocity induced by panel j on panel i is equal to the induced velocity (per unit source strength), denoted as the influence coefficient CXi;, multiplied by the source strength Sj, which is a constant for each panel. The sum of the free stream velocity Uoo and the velocity induced by all other panels j gives the velocity at panel i: N Ui = Uoo + ~ CXi,jSj · (1) j =1 In Eq. 1, the summation is carried out for all the panels on the hull surface, and the quantity N denotes the total number of panels. The unknowns on each panel are the source strength to be determined from a solution of the boundary value problem subject to the Neumann boundary condition on the hull surface. This panel method scheme was developed by Hess and Smith [14~ for the flow of an infinite fluid past a ship-like body. 582

Another way of modeling a ship is to combine the submerged ship hull with its mirror image above a plane of symmetry at the mean water level. This combination is referred to as the double model in the literature. The flow past a double model is obtained by a numerical solution of the boundary value problem subject to the Neumann boundary condition on the double model hull. This double model solution is the approximate solution to the free surface problem in the limiting ease of zero Froude number when the free surface is approximated by a rigid wall. After the double model solution is obtained, the double model streamlines are traced on the mean water level. These streamlines will not penetrate the hull surface and are used to set up a free surface grid. Constant source panels are placed on this free surface grid. Then, the free surface boundary condition is linearized using the double model solution as the basic solution in the sense that the deviation from the double model flow is considered small: `f = ~ + ¢', (2) where ~ denotes the velocity potential for the free surface flow, ~ the velocity potential for the double model, and ¢' the perturbation velocity potential. D awson t4] gave the resulting free surface boundary condition with the double model linearization as ~ ~ 2¢ ~ + go = 2 ~ l 2~ ~ I ~ 3 ~ SPECIAL SECTION At', ~~ =~\ BEHIND THE@ 3 ~ , ~ ~ TRANSOM fit _ : ~ == ~ ~ ~ .~ ,~77> at the mean water level z O . The free surface boundary condition involves the gradient of the velocity potential along a streamwise direction designated by I, and differentiation is carried out along the corresponding double model streamlines. For example, the streamwise velocity on the free surface is computed by ~l= x ox + Y by . (4) j~X2 + ~y2- j~X2 + ty2 Note that this differentiation scheme approximates the free surface flow direction by the double model flow direction. This appre~im ation is a crucial ste p to be used later in the derivation of the transom boundary condition. In the numerical calculation of the convective term, finite differencing is used to calculate derivatives between adjacent panels along a double model streamline. A four-point upstream finite differencing scheme is used to eliminate upstream propagating disturbances as recommended by Dawson ~4~. For the foremost upstream point, a two-point upstream finite difference operator is used. The starting values are specified by a uniform flow and a corresponding wave elevation of zero. In the panel method for free surface flows, the source strength distribution must satisfy simultaneously the linearized free surface boundary condition and the Neumann boundary condition on the hull. For a case with N panels representing the total number of the hull panels and free surface panels combined, the resulting system of N by N equations is full, nonsymmetrieal, and not diagonally dominant. A Gaussian elimination scheme is used to solve this system of equations. For the ease of a transom stern, the main section of the free surface (as shown in Fig. 3) can be handled in the same way as for cruiser sterns. The double model linearization and the upstream finite difference operator are also used. However, the flow domain behind the transom presents a new problem and needs special treatment. In particular, a new section of free surface panels is introduced, the transom is treated as an inflow boundary, and the starting values of a free surface calculation must be specified at the transom. This problem is addressed in the remainder of this paper. MAIN SECTION7 FREE SURFACE Fig. 3. An aerial view of the free surface paneling for a transom stern hull. 3.2 Theory of the transom boundary condition As for a cruiser stern ease, a transom stern solution begins with a double model computation. The geometric model for a transom stern hull is shown in Fig. 4. Note that the transom is intentionally left open and there are no panels on the transom. The hull surface for the double model serves as a stream surface separating the external flow about the hull from a fictitious internal flow. The double model flow computed in this way is appropriate for a day transom with the exit flow detaching at the transom tangential to the hull surface. 583

Fig. 4. Perspective view for the numerical model of a transom stern hull. Note the open transom. For the free surface solution, the free surface boundary conditions must be adapted to the iansom stern geometry. To simplify the analysis of transom flows in practical applications, several assumptions are made in the theory for transom stern flows. It is assumed that the transom is left open as in the double model flow computations and that the potential flow detaches from the transom at well defined locations. These locations are identified as the points on a curve which has the same shape as the transom bottom. The shape of the transom bottom may be described by specifying the transom depth ZT as a function of the transve'rse coordinate y: ZT = f ( Y ) at x = XT, (5) where the subscript T denotes the transom and XT the longitudinal coordinate of the transom. Along the intersection curve between the transom and the free surface, the static pressure is equated to the atmospheric pressure pOO since the transom clears the surrounding water and is exposed to the air: Pr = pOO at x = XT and z = ZT for a given y In Eq. 6, the effect of an air wake behind the transom is neglected and the atmospheric pressure is considered a global constant. With the dependence on pressure removed, Bernoulli's equation describes a steady-state balance of kinetic energy and potential energy: ~ 1 AX + by + ~Z2 ~ + gZT = 1 U2 (7; where ~ ~x, by, Ha ~ denotes the gradient of the velocity potential in ~ x, y, z ~ directions. Eq. 7 may be rearranged as ~ 2 + ~ 2 + ~ 2 ~ 2& ZT (8) Thus, the kinetic energy for the water at the transom can be determined by the right hand side of Eq. 8, where ZT lies below the mean water level and takes on a negative value. This equation is the first constraint for transom stern flows. Since a flow cannot penetrate an impermeable surface, another constraint to be satisfied at the transom is that the exit flow must be tangential to the hull surface. The magnitude of the velocity at this point is given by the square root of the right hand side in Eq. 8. The flow direction is part of the free surface solution and is determined by the transom geometry, which can be spe cified by a ( local) tangential unit vector: X j¢2 + ¢2 + ¢2 ( ~ he ~ 10) tax + by + Adz = go (11) j¢X2 + ~y2 + ~Z2 A first approximation of the tangential unit vector is obtained by replacing the potential in Eqs. 9 through 11 bar the double model potential to give the tangential unit vector in the direction of the double model flow: Tx = six ( 12) j~X2 + ~y2 + ~Z2 Ty = i t 13) j~X2 + ~y2 + ~Z2 Tz = Z , ( 14) j~X2 + ~y2 + ~Z2 584

FREE SURFACE BOUNDARY CONDITION ( x, y, z ) by 1/2 L . Equations 15 through 17 can then be rewritten as follows: i + 1 ~:A UT = ~ Tx ( 18) TRANSOM BOUNDARY CONDITION \' NEUMANN I BOUNDARY AT = ~ / 1 _ ZTS Ty ( 19 ) CONDITION v n Flux Fig. 5. Description of boundary conditions for WT = N/1~ T.:, transom, free surface, and hull. where the vector ~ fix, Eye By ~ represents the velocity of the double model flow for a hull panel whose centroid lies just forward of the sharp corner at the transom. This hull panel has the same y- value as the transom and is denoted by the index NQ as shown in Fig. 5. The approximation of the free surface flow direction by the double model flow direction is consistent with applying the free surface boundary condition along double model streamlines on the main section of the free surface, as spe cifie d by Eq. 4. The validity of this approximation has been verified by comparing computed and measured wave profiles behind the transom. When the velocity magnitude from Ea. 8 is (20) where Fn2=U2 /gL and ZT= ~zT/L . Equations 18, 19, and 20 are applied as the transom boundary conditions for the free surface calculations in the case of a dry transom. In so doing, the static pressure is forced to be atmospheric and the free surface flow is forced to leave the transom tangentially in a direction specified by the double model flow. The transom boundary conditions show that the velocity at the transom is directly related to transom depth and the Froude number and is indirectly related to other transom characteristics (i.e., the buttock angle, the deadrise angle, and the run angle) through the direction of the double model flow. 3.3 Implementation of the free surface boundary condition combined with the flow direction from Eqs. 12 through 14. the three velocity components at the In this section, the special treatment for the transom are approximated by: UT = N/~TX (15) VT = A/W,~ Ty (16) WT = N/~ Tz , (17) where the vector ~ UT, VT' WT) has been nondimensionalized by ~ US and denotes the velocity at the transom in the ~ x, y, z directions, respectively. The sign of the vector ~ UT, VT, WT ~ IS determined by the tangential unit vector. The quantity °° represents the square of gZT the Froude number based on the transom depth. In usual numerical calculations, it is more common to use the Froude number based on the waterline length L . This convention is achieved by scaling free surface boundary condition behind a transom stern is described. The free surface boundary condition in this region uses the conventional linearization about a uniform flow: 2Fn2 ux + ·v = 0, (21) where the length is nondimensionalized by 1/2 L and the vector ~ u, v, w ~ represents the perturbation velocity in a dimensionless form. The free surface boundary condition is applied at the mean water level, on which flat panels are placed. Computations are performed for the centroid of a panel, as indicated by "bullets" in Fig. 5. In the discretization of the convective term in Eq. 21, an upstream finite difference operator is used to eliminate upstream propagating waves. In particular, a two-point upstream finite difference operator must be used to start the computations for transom stern flows: UX; = CAi Ui + CB; us, (22) where CBi = x _ = - CAi . ~ 23) 585

In Eqs. 22 and 23, the index i denotes the foremost point behind the transom and the index t the upstream point at the transom. The quantity ui is an unknown to be determined and us, the velocity at the upstream point, must be specified. The value us is taken from the transom boundary condition Eq. 18. However, a minor modification is required here since UT represents the total velocity and us represents the perturbation velocity superimposed on the uniform flow. Thus Ut = ~ + UT . (24) Substituting Eqs. 22 and 24 into Eq. 21 and . ~ rearranging gives, 2Fn2 CA; u; + wi = - 2Fn2 CB; [ 1 + UT] . (25) This equation is the discretized form of the free surface boundary condition at a particular y-value for the foremost free surface panel with index i next to the transom. The unknowns reside on the left hand side and the knowns on the right hand side. For the the downstream panel at the same y-value with index i + 1, the following equation is used: 2Fn2 ~ CA+ up + CBi+~ ui ~ + win = 0, (26) where CB = =- CA 27 i+l Xi - Xi+1 i+} Equations 26 and 27 are applied to succeeding panels with indices i+ 2, i+ 3, etc. until the downstream boundary is reached. Equations 22 through 27 are applied for other y-values to cover the special section of free surface (Fig. 3) downstream from the transom. The resulting system of difference equations is then solved using the Rankine source panel method. Substituting Eq. 1 into Eqs. 25 and 26 gives 2F`n2 ~ CAj CXj j Sj - firs; = - 2Fn2 CB; ~ 1 + UT] and 2Fn2~ [ CAj+1 CXi+l,j + CBi+l C~i,j ]Sj - 2~Si+1 = 0 The index N denotes the total number of panels for the hull surface, the main section of free surface, Add the special oil behind the Tom. The influence coefficient CXj j denotes the x- component of velocity induced on panel i by panel j per unit source strength. In the computations of influence coefficient for each hull panel, the contribution from its mirror image must be added. The source sty engths s; are the unknowns to be determined through the solution of a boundary value problem, and a Gaussian elimination scheme is used in the numerical solution. The distribution of source strength must satisfy simultaneously the free surface boundary condition and the Neumann boundary condition on the hull, as represented in Fig. 5. A two-point upstream finite difference operator has been used in the special section behind the transom for wave resistance computations. These calculations are applied to a short computational domain covering one half of a ship length downstream from the transom. The four-point operator will be introduced for future calculations, which extend farther downstream, to improve the numerical accuracy of transom stern computations. 4. Comparison with experiments This section presents a comparison of numerical predictions with the corresponding experimental measurements for high-speed transom stern ships. Such a comparison shows the extent to which the numerical model of transom stern flows can predict the hydrodynamics of the physical model. Another objective is to identify opportunities for future research in transom stern flows. Two examples are presented here: the local flow field around a ship and the force on a ship. The local flow field is represented by the wave pattern behind the transom for R/V Athena. The force on a ship is represented by the wave resistance for D TRC Model 5410. 4.1 Wave profiles Figure 6 gives a sideview of the computed wave profile behind the transom for R/\ Athena. The vertical and horizontal axes are nondimensionalized by 1/2 L and are plotted to (28) the same scale. The numerical model of the transom stern is shown on the right. The dashed line represents the mean water level and the solid wavy line represents the free surface computed on pane is closest to the ce nte rplane. The computations were performed for a Froude number of 0.48. The free surface has a small slope, which t29y is appropriate for this Froude number, and the flow 586

0.4875 2 z/L O _ COMPUTED WAVE PROFILE _ _ . at . ~ . ~ _ MEAI`J WATER LEVEL t TRANSOM U X ~ ~0.4875 . . . . . ~ ~ l ' _ -2.05 - 1 .9S -1.85 -1 .75 -1 .65 - 1.55 -1.45 - 1.35 -1 .25 -1.15 -1 .05 ~0.95 -0.8S -0.75 2 x/L Fig. 6. Computed wave profile in the centerplane behind the transom for R/V Athena at a Froude numbe r of 0.48. angle can be measured graphically. Notice the ascending motion followed by the descending motion as expected. A gap is left between the transom and the computed free surface to indicate the centroid location of the foremost panel, where the free surface calculations are started. An upstream extrapolation of the free surface shows that the free surface intersects the transom at the transom depth ZT. Figure 7 gives a close-up view of Fig. 6 near the transom. Again the dashed line and solid wavy line represent the mean water line and the computed free surface closest to the centerplane, respectively. The corresponding wave profile measured by Jenkins t12] is represented by isolated points and the rearmost point indicates the extent of experimental data. Thus, the experiment indicates that the transom clears the water. and the assumption of a dry transom is valid for a Froude number of 0.48. The numerical prediction and experimental measurements of wave profiles behind the transom show close agreement, espe cially just downstre am of the transom. The double model approximation of the flow direction seems Justine d in this case. 4.2 Wave resistance . D TRC Model 5416 represents a typical high-speed transom stern ship without a bulbous bow. Dr. Michael Wilson of DTRC designed this hull form as a candidate for a low resistance ship. The computations for Model 5416 were performed at least one year prior to model construction and tank testing at DISC. The model tests involved a longitudinal wave cut experiment using capacitance wave probes to measure wave profiles. A wave spectrum analysis of the measured wave profiles gave the measured wave pattern resistance. The 0.1875 2 z/L o ~0.1875 l l l -1.25 - 1.15 -1.05 - 0.95 2 x/L COMPUTED WAVE PROFILE (PRES - T STUDS ° MEASURED WAVE PROFILE (JENKINS 111]) -O 85 ~0.75 Fig. 7. Comparison of the computed and measured wave profiles for R/V Athena at a Froude number of 0.48. measured wave pattern resistance was then compared to the wave resistance predicted by integrating the pressure on the hull using the XYZFS program version 2.0. - .v 2.5 . Is ° 2.0 _ z - - o by - Al: 1.5 _ 1.0 _ 0.5 _ _~ '~ _ I ~ ~ ~ ~ 0 0.1 0.2 0.3 0.4 0.5 0.6 CALCULATED (XYZFS) TV \~ MEASURED / Cw(p) f5 (WILSON) '8~ FROUDE NUMBER, Fn Fig. 8. Comparison between calculated wave resistance and me asure d wave pattern resistance versus Froude number for Model 5416. This comparison of computed wave resistance and measured wave pattern resistance is presented in Fig. 8. The computations were performed for Froude numbers from 0.25 to 0.50 at increments of 0.05. The hull form was held fixed 587

at the even keel position for these computations and for the experiments. The calculated wave resistance and measured wave pattern resistance show general agreement. The agreement is rem arkably good for Froude numbers from 0.35 to 0.50, corresponding to the operating speeds of high-speed transom stern ships and corresponding to the case of a dry transom. For Froude numbers below 0.35, the slope of the wave resistance curves is in agreement. However, computations predict a hump around a Froude number of 0.3, which is absent from the measured wave pattern resistance. This discrepancy is partially attributed to the breakdown of the dry transom assumption at lower Froude numbers. Transom stern flow at lower Froude numbers is the subject of a future study, and further numerical and experimental research is needed. 5. Conclusion A practical computational method for 3D transom stern flows has been developed. A dry transom boundary condition has been derived for linearized free surface potential flows. The physical constraints imposed by the transom boundary condition require that the static pressure be atmospheric and that the flow leave tangentially at the transom. The pressure constraint has been applied to Bernoulli's equation to determine the magnitude of the velocity at the transom as a function of transom depth and Froude number. The velocity tangency constraint has been enforced by requiring the free surface flow to leave tile transom tangentially in a direction specified by the double model flow. The nurrnerical implementation of the free surface boundary condition involves the use of an upstream finite difference operator to eliminate upstream waves. The transom boundary condition has been used to specify the starting values for the free surface calculations in a special section behind the transom. This computational method has been incorporated into a Rankine source panel method, the XYZFS program version 2.0. The computed flow pattern behind the transom has been presented and compared with experimental measurements for the R/V Athena hull. Comparison of wave resistance prediction and experimental measurements have been presented for D TRC Model 5416. The results show that the wave resistance predictions are remarkably good for Froude numbers between 0.35 and 0.50, corresponding to the normal speed range of high- spe e d transo m ste rn ships. 588 Acknowledgments This study was supported by the Surface Ship Technology Exploratory Development Program and managed by the Ship Hydromechanics Department (SHD ~ of the D avid Taylor Research Center (i;~'l'fCCj. The author is indebted to Mrs. Joanna W. Schot of D IRC for her encouragement and support during the course of this study. The author wishes to thank Dr. Henry Haussling of DTRC for his advice and his constructive comments on the manuscript. D r. Michael Wilson of SHD kindly furnished the experimental results for Model 5416. Mrs. Janet Dean of DTRC has contributed helpful suggestions and discussions. References . Saunders, H.E., "lIiydrodynamics an Ship Design, " SNAME, Vol. 1 i i pp.326-327 ( 1957) . 2. Gadd, G.E., "A Method of Computing the Flow and Surface Wave Pattern Around Full Forms, " Trans. RICE. Vol. 118, p. 207 ~ 1976~. 3. Chang, M.S., "Wave Resistance Predictions by Using a Singularity Method," Proc. of He Workshop on Ship Wave-it esistance Computations, D TNSRD C, Bethesda' MD (Nov 1979~. 4. D arson, C.VV., "A Practical Computer Method for Solving Ship Wave Problems," in Proc. Second International Conference on Numerical Ship Hydrodynamics, University of California, Berkeley ~ Sep 1977~. D awson, C.W., " Calculations with the XYZ Free Surface Program for Five Ship Models," Proc. of the Workshop on Ship Wave- Resistance Computations, D TNSRD C, Bethesda, Maryland, (Nov 1979~. 6. Cheng, B.H., J.S. Dean and J.L. Jayne, "The XYZ Free Surface Program and Its Application to Transom Stern Ships with Bow Domes," in Proc. Second Workshop on Ship Wave-Resistance Computations, D TNSRD C, Bethesda, MD (Nov 1983~. 7. Cheng, B.H., G.G. Borda, J.S. Dean and S.C. Fisher, "A Numerical/ Experimental Approach to Wave Resistance Predictions," in Computer Aided Design, Manufacture and Operation in the Marine and Offshore Industries, Washington, D .C. ~ Sep 1986~. 8. Hoyle, J.W., B.H. Cheng, B. Hays, B. Johnson, and B. Nehrling, "A Bow Bulb Design Methodology for High Speed Ships," Transactions SHAME) Vol. 94 (Nov 1986).

9. Cheng, B.H., J.S. Dean, R.W. Miller, and W. L. Cave III, "Hydrodynamic Evaluation of Hull Forms with Podded Propulsors," Naval Engineers Journal, Vol. 101, (May 1989~. 10. Wilson, M.B. and T.P. Thomason, "Study of Transom Stern Ship Hull Form and Resistance," D TNSRD C report 85/072, D TNSRD C, Bethesda, MD ~ Apr 108Gi). 11. Tulin, M.P. and C.C. Hsu, "Theory of High- Speed D isplacement Ships with Transom Sterns," Journal of Ship Research, Vol. 30, No. 3 ~ Sep 1986~. 12. Jenkins, D.S., "Resistance Characteristics of the High Speed Transom Stern Ship R/V ATHENA in the Bare Hull Condition, Represented by O TNSRD C Model 5365," D TNSRD C report 84/024, D TNSRD C, Bethesda, MD ~ June 1984~. 13. Coleman, R.M., "Nonlinear Flow about a 3- D Transom Stern, " Fourth International Conference on Numerical Ship Hydrodynamics, Washington, D.C. (Sep 1985~. 14. Hess J.L. and A.M.O. Smith, "Calculation of Potential Flow About Arbitrary Bodies, " Pergamon Press Series, Progress in Aeronautical Science, Vol. 8 ( 1966~. 589

DISCUSSION by K. Nakatake I appreciate your paper treating the transom stern flow. Fig.8 shows a good agreement of calculated and measured Cw. But, in this high Fn range, the effects of trim and sinkage become important. Did your calculation include such effect? If possible, please show the wave height contour around the model. Author's Reply Dr.Nakatake asked about the effects of sinkage and trim in the computations. The results presented in Fig.8 of my paper correspond to the fixed case for both the computations and experiments. To include sinkage and trim effect, we can reposition the hull according to sinkage and trim predictions from fixed case calculations. We did not analyze and plot wave height contours for a sunk and trimmed case and is presented in Fig.A1. The corresponding wave pattern resistance curve, as measured by Dr. Michael Wilson of DTRC, is presented for comparison. The computed results could be further improved by using the experimentally measured sinkage and trim to reposition the hull. at, 0.6 dl Fig.A1 Comparison between calculated wave resistance and measured wave pattern resistance versus Froude number for Model 5416 (sunk and trimmed case) DISCUSSION by J. Ando I'd like to congratulate for your good results for 3D transom stern flow, and I'd like to discuss on the radiation condition. When we try to satisfy the radiation condition, if we use finite difference operator, we have some troubles. In my experience, point-to-point oscillation of the source strength occurred near the downstream boundary. Sometimes, the source strength oscillates so large that the wave pattern is affected. And the results change due to the kind of finite difference operator which uses number of points. Did you have such experience? So I'd like to present a method which does not use any finite difference operator in order to satisfy the radiation condition. In this method, the radiation condition is satisfied automatically by shifting the source panel in the downstream direction by one panel length. We call it Kyushu University method which is abbreviated as KU method. ~- CALCULATED // (mFS) MA ~ I REV I r~ /~` MEASURED / cw(p) / (WILSON) Next, I show some results for Wigley hull. Fig.A2 shows panel arrangement on the still water surface. By KU method, we don't use any finite difference operator along the stream line of the double model flow. Fig.A3 shows comparison of wave patterns. This pattern by KU method looks like the experimental result. Dawson's method gives wider propagation of f rue waves; 0~ . ~ -1.0 0.0 1.0 2x/L 2.0 KU Method 2z/L -Q8- . llllll!llllL-~, , , - ,P -1.0 0.0 J I -- 01 02 0.3 04 0.6 06 ;7 Dawson's Method FROUDE NUMBER, Fn -Q4 O Fig.A2 Comparison of panel arrangement 590 I, ., ~ I, . . . Cj ~ I · I I 11 I j ~ I al jll 1 T rTrrTr~ red I I a, T. j ~ jI I I I jl ~ ~ jl jt I jI ret r r I I I r I r , T [,, T r] ~ r~ _ ~ CLI i I I ~ I ~ i jl jr ~ jI Trot I r r rT I I I I ~ I 1 ~ r1 1 r I I, ...... ~ ~ j ~ ~ j j i ~ j ~ ~ j j I ~ r ~ ~ T ~ I I T ~ r ~ . T ~ ~ I I . 1 . .1 . .~ ~ ~ 11,,,, 111 ~ 2x/L 2.0

1 Fn=0.289 - ~.t'~-"'t"~ ~ 'I I. ~ I ~~< Measurement (by SRI) KU Method Dawson's Method Fig.A3 Comparison of wave pattern Author's Reply I would like to thank Mr. Ando for his interest in my paper. It is true that the XYZ Free Surface problem uses upstream finite difference operators to eliminate upstream propagating waves, thus satisfying the radiation condition numerically. However, we have not experienced the point-to-point oscillations of the source strength near the down stream boundary as referred to by Mr. Ando. The reason is that we handle the downstream boundary in the following manner. As the downstream boundary is approached, a four-point operator is switched to a three- point operator and then to a two-point operator for the rearmost panel. The two-point operator gives considerable numerical damping when the rearmost panel is relatively large. Mr.Ando's "Dawson method" calculation seems to include a reflection from the side boundary. Such a reflection can be avoided by extending the side boundary of the computational domain to one ship length, measured from the ship's centerline. The Kyushu University (KU) method for satisfying the radiation condition sounds interesting and seems to be similar to the method by Jensen[A1]. It would be helpful for the research community to know more about the KU method than has been presented in Mr. Ando's discussion. The results of the Wigley hull look promising and I encourage Mr. Ando to continue his studies on the radiation condition. [Al] Jensen, P.S.: On the Numerical Radiation Condition in the Steady State Ship Wave Problem, J. of Ship Research, 1988. ,` _ ~ ~ _ %. . ~ _ . 591