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Analysis of Transom Stern Flows A. Reed, I. Telste (David Taylor Research Center, USA) C. Scragg (Science Applications International Corporation, USA) ABSTRACT The boundary value problem for a transom stern ship at moderate and high Froude numbers is formulated and solved. The solution is obtained using lifting potential flow techniques, and involves satisfying a Kutta condition at the after edge of the hull. The full problem is linearized about the free stream velocity, and this linearized problem is solved using two different approaches. One method uses Havelock singularities, and the other uses Rankine singular- ities. Both of these approaches are applied to the solution for the flow about a high-speed transom stern ship, with encouraging results. C cd CR Cw C' w Cwp Fn NOMENCLATURE Wave spectral function due to sources Wave spectral function due normal dipoles Residuary resistance coefficient, CR = RR/_PSU2 Wave resistance coefficient, Cw = Rw/-pSU2, computed from wave spectral energy Wave resistance coefficient, computed by inte grating predicted pressure over the surface of the hull Wave pattern resistance coefficient derived from measured wave pattern Froude number, Fn = U/~/~ g Gravitational acceleration G Green function i, j, k Unit vectors in the x, y, and z-directions, re spectively k Wavenumber ho Fundamental wavenumber, ho = g/U2 k=, k' Longitudinal wavenumber ky Lateral wavenumber Span Length of panel in the x-direction L Length of ship n Normal vector, taken into the fluid no, ny, nz Components of the normal vector, n, in the x-, y-, and z-directions Neumann-Kelvin Distance from singular point to field point, r = ,/(X _ ()2 + (A - ?1)2 + (A _ ()2 Distance from image of singular point to field - point, r' = ~/(x _ ()2 + (Y _ rl)2 + (Z + ()2 Surface of a panel on the hull, free surface, or wake N-K r r' si 207 U V An At x Sa Ship hull surface (zero sinkage and trim) SF Free surface (mean free-surface level) Sw Vortex wake surface a, v, a) Perturbation velocity components in the x-, y-, and z-directions Ship speed; magnitude of free stream velocity in ship fixed coordinate system Total velocity vector Component of the perturbation velocity in nor mal direction Component of the perturbation velocity in tan gential direction Vector coordinate of a field point, (x, y, z) x, y, z Coordinates of field point x in a right-handed ship fixed coordinate system, x-axis forward, y axis to port, z-axis upward x`, ye, z' Coordinates of the perimeter of the transom Z Wave elevation Dipole strength Vector coordinate of a singular point, ((, 71, () I, 7', ~Coordinates of a singular point, I, in the x-, y-, and z-directions Density of water Source strength Perturbation velocity potential p INTRODUCTION Naval architects have been familiar with the use of transom sterns on high-speed displacement ships for well over 50 years. The concept of a transom stern applied to displacement ships appears to be an outgrowth of its use on planing craft where it has been applied since around the turn of the century. An understanding of when the use of transom sterns is most appropriate was developed in the years around the Second World War, when extensive systematic-series experiments were performed. In the more recent past, this knowledge concerning the proper applica- tion and design of transom sterns seems to have been lost. Modern naval ship designs have used larger and larger tran- soms while the maximum speed of ships has decreased and the size of ships has increased. Both of these factors should weigh against the use of transom sterns on modern naval vessels. That the above statements are true is illustrated by the fact that model tests have shown that the resistance of modern naval ships such as the DD 963 is significantly

higher than it needs to be, due largely to the excessive tran- som area. In addition to the significantly elevated resistance of modern transom stern ships, these ships have significant wave breaking at the stern. This is manifest in two ways: first, by a large turbulent "rooster tail" aft of the transom, and second, by a large breaking transverse wave extend- ing approximately one ship beam off to each side of the transom. Both of these features have a significant negative impact on the wake signatures of modern high-speed naval displacement ships. While there has been substantial research in the field of wave resistance over the last twenty-five years, and while there has been substantial progress in the prediction of Kelvin wave flows (see for instance Lindenmuth et al. 1990), the treatment of transom stern flows has been largely ne- glected. As is obvious from the poor flow predictions in the region of the transom for Model 5415, given in Lindenmuth et al., the Neuman-Kelvin and Dawson method solvers need major improvements if the flow about a transom stern is to be predicted adequately. However, there have been a few encouraging developments in this area; one of these, which may show a proper approach to this problem, will be dis- cussed later. At moderate and high speeds, the flow about transom stern ships is characterized by smooth separation of the stream lines at the transom. The manner in which the flow separates from the hull at the transom immediately suggests an analogy to the flow at the trailing edge of a lifting surface (see for example Newman 1977~. In the extreme, for high Froude numbers, the problem becomes a planing problem. This limit provides some insight into the physics of the prob- lem, and indicates some physical phenomena which must be modeled in order that a correct solution to the prob- lem be obtained. At more moderate Froude numbers, the flow about a transom stern appears to have many analogies with the flow about a ventilated hydrofoil (again, see New- man 1977~. This observation also provides insight into the proper solution to the transom stern flow problem. Tulin and Hsu (1986) developed a model for high-speed slender ships with transom sterns. By making the assump- tion that both the beam and draft are small relative to the length, and examining the asymptotic limit as the Froude number approaches infinity, they were able to reduce the three-dimensional problem to a series of two-dimensional boundary value problems to be solved in the cross-flow plane. The problem solved at each cross section is similar to problems addressed in two-dimensional slender wing theory. On the free surface they employed a trailing vortex sheet in the wake of the hull to satisfy the free-surface boundary condition, which in their model was shown to be equivalent to the Kutta condition applied at the trailing edge of a wing. Perhaps the most significant result of their model was the existence of a drag force due to the presence of the trailing vortex sheet. This "stern-induced resistance" is equivalent to the induced drag associated with the shed vorticity in the analysis of three-dimensional lifting surfaces. Tulin and Hsu calculated the magnitude of the stern- induced resistance for models from Series 64 and made com- parisons with the residuary resistance measured at DTRC for a Froude number, En = U/~fi~;, of 1.49. The excellent agreement with the measured results suggests that, in the high speed limit where the wave resistance vanishes, the primary component of residuary resistance is induced drag due to vorticity shed from the transom. Furthermore, the magnitude of the stern-induced resistance is quite large, of the same order as the wave resistance which occurs at mod- erate Froude numbers, and consequently any attempt to model the flow about transom stern ships at finite Froude numbers must include both the generation of free-surface waves and the effects of vorticity shed from the hull. The trailing edge boundary condition which must be applied in the transom stern problem (actually the bound- ary condition must be applied on the hull surface just for- ward of the transom since the transom itself is assumed to be unwetted), is more restrictive than the Kutta condition applied in aerodynamics. In wing theory, we require that the flow at the trailing edge be tangent to the wing sur- face and that the pressure be continuous across the wake. For the transom stern problem, we require that the flow be tangent to the hull and that the pressure be equal to atmospheric pressure. This is equivalent to placing an ad- ditional boundary condition on the longitudinal component of the velocity, which can be satisfied by determining the appropriate longitudinal gradient of the dipole strength on the hull and/or by placing sources on the free surface aft of the transom. These sources aft of the transom would be equivalent to the sources which are used to model the cavity aft of a ventilated airfoil. Cheng (1989) provides a solution to the transom stern problem in which he satisfies a similar set of boudary condi- tions at the transom. However, Cheng's solution uses only sources on the body boundary to satisfy the boundary value problem. Obviously, with such an approach the trailing vor- tex wake can not be modeled, nor can the body boundary condition and the transom boundary condition be satisfied at all points on the body simultaneously. Furthermore, in attempting to satisfy both a normal and a tangential flow boundary condition by varying only the distributed source strengths, the solution will be very sensitive to the tangen- tial gradient of the source strengths, and consequently to panel size and location. In the analysis of the flow about a surface-piercing strut operating at a small angle of attack, a direct application of the linearized Bernoulli equation along corresponding free- surface streamlines on the high and low pressure sides of the strut will lead to the conclusion that there can never exist a discontinuity in the free-surface elevation at the trailing edge of the strut. This result is, of course, inconsistent with observation. However, if one applies the full nonlinear Bernoulli equation to the same problem, it can be shown that on the two streamlines there will exist a difference in free-surface elevation at the trailing edge which is propor- tional to the longitudinal vorticity shed by the strut at the free surface. Interestingly, near the strut the perturbation velocities are small and the linearized free-surface equations remain an adequate approximation for the calculation of the flow about the body, including the vorticity distribution. Apparently, this is a situation in which the vorticity gives rise to nonlinear free-surface effects, but the presence of these nonlinear effects does not significantly affect the vor- ticity distribution. Similarly, the nonlinear wavebreaking which occurs in the wake of a transom stern ship is strongly dependent upon the vorticity shed from the transom, but the vorticity distribution at the stern is not greatly affected by the presence of downstream wavebreaking. In this paper, we present a brief theoretical foundation for the transom stern boundary value problem. The full problem is linearized about the free stream velocity, and this 208

linearized problem is solved using two different approaches. One method uses Havelock singularities, and the other uses Rankine singularities. Both of these approaches are applied to the solution for the flow about a high-speed transom stern ship, at two Froude numbers. THEORETICAL FORMULATION Consider a transom stern ship traveling at a compar- atively high steady forward speed such that the flow sepa- rates cleanly from the hull at the transom. We assume that the fluid is inviscid and incompressible, and that the flow is irrotational everywhere except possibly along a sheet of trailing vorticity. We can then define a perturbation veloc- ity potential ¢' which satisfies Laplace's equation v2¢ = 0 (1) throughout the fluid domain. Using a ship-fixed coordinate system with the x-axis forward, the y-axis to port, and the z-axis upward, the velocity vector V is related to the po- tential by V = -pi + vim = ~-U+uji+vj+wk, (2) where U is the magnitude of the free-stream velocity, and a, v, and u' are the components of the perturbation velocity in the x-, y-, and z-directions, respectively. While the free surface generality exhibits energetic wave breaking at some distance aft of the transom, it is assumed here that the effects of these breaking waves do not propa- gate upsteam any significant distance. Therefore, there will exist a region aft of the ship and forward of the breaking stern waves over which the kinematic and dynamic free- surface boundary conditions are valid. Furthermore, if the effects of these breaking waves do not extend forward to the hull itself, then our solution in the region immediately surrounding the ship will be unaffected by an application of the kinematic and dynamic free-surface boundary condi- tions over the entire free surface. On the free surface Z. the non-linear kinematic boundary condition is written as ~-U + ¢~)Z~ + MAZY = oz. on z = Zip, y). (3) The Bernoulli equation is applied on the free surface to give us the dynamic free-surface boundary condition, gZ + ~ ~v~2 = ~ u2, on z = Zip, y). (4) It is assumed that both Equations (3) and (4) hold over the entire free surface, including the region directly astern of the transom. In addition, it is necessary to impose a radiation condition to ensure that the free-surface waves vanish upstream of the disturbance. The boundary condition to be applied on the hull sur- face SB, excluding the transom, is simply the zero normal flow condition V · n = 0, on y = SB(X, I; U), (5) where n is the hull normal vector (directed into the fluid domain), and we note that the sunk and trimmed position of the hull surface depends upon the forward speed. It is assumed that the flow separates cleanly at the transom and that the velocity vector at the trailing edge is tangent to the hull. Since there can exist no discontinuities in the pressure within the fluid domain, the pressure on the hull just for- ward of the transom must be equal to the pressure on the free surface just aft of the transom. Therefore, we have an additional boundary condition to be applied on the hull surface at the transom: 2 EVE = 2U2-gal, on y = SB((X!,ZI; U), (6) where x' and z' denote points at the intersection of the sunk and trimmed hull surface and the transom. Since Equa- tion (5) requires that the normal flow on the hull be zero, Equation (6) can be viewed as a restriction upon the magni- tude of the tangential component of the flow at the transom. We note since we are requiring that the pressure go to zero at the intersection of the hull surface with the transom, the present requirements are actually more restrictive than the Kutta condition applied at the trailing edge of an airfoil, for which we require only that the flow be finite and tangent to the foil, and that the pressure be continuous. E`ree-Stream Linearization If we assume that the potential and its derivatives are small relative to the free-stream velocity U. we can replace the non-linear free-surface boundary conditions with their linearized counterparts. Combining Equations (3) and (4) to remove the explicit variable Z. and retaining only the first order terms in ¢, we can write the linearized free-surface boundary condition as (7) where ho = g/U2 is the fundamenatal wavenumber. The boundary condition is now applied on the position of the undisturbed free surface. The hull boundary condition remains unchanged with the exception that it will now be applied at the position of the hull surface with zero sinkage and trim, SB(X,Z;U = O). Once the velocity potential has been determined, the sinkage force and the trim moment can be calculated and an improved estimate of the position on the hull surface SB(X,Z;U) can be used to solve the problem iteratively. However, for the validation cases presented in the following sections, the sinkage and trim were known from experimen- tal measurements and the hull boundary condition could be applied at the sunk and trimmed position of the hull surface on the first iteration. After dropping the non-linear terms, the pressure con- dition to be applied on the hull surface at the intersection with the transom, Equation (6), becomes simply ¢2~= + kook = 0, on z = 0, U¢2 = gal, on y = SB(Xt, Z[; U). (8) The linearized problem addressed here is very similar to the Neumann-Kelvin (N-K) problem: the potential must satisfy Laplace's equation, Equation (1); throughout the fluid domain subject to a zero normal flow boundary con- dition, Equation (5); on the hull surface and subject to a linearized free-surface boundary condition, Equation (7). However, in the present problem we have the additional re- quirement that the pressure on the hull must go to zero at the stern, Equation (8). This additional independent 209

boundary condition, a restriction on the longitudinal com ponent of the velocity which does not occur in the formula tion of the Neumann-Kelvin problem for ships which do not have immersed transoms, indicates the need for additional The function ~ is given by unknowns in our numerical approach to the problem. We solve this boundary-value problem by employing two different approaches, both of which are capable of mod eling vortex sheets in the presence of the free surface. One approach uses Havelock sources and dipoles which are dis tributed over the hull surface and in the wake. The lin earized free surface boundary condition is implicitly satis fied by the use of Havelock singularities, and therefore no singularities are required on the free surface. The other approach uses Rankine sources and dipoles which are dis tributed over the hull surface, the free surface, and in the wake. The details of the two approaches follow. Havelock Singularity Formulation Our first approach to the numerical solution to this problem distributes Havelock singularities over the hull sur face and along a trailing wake sheet. The hull surface and the wake sheet are divided into discrete panels comprised of coincident sources and normal dipoles of uniform singularity density. The Havelock singularity is a Green function for the problem which satisfies the linearized free-surface boundary condition on the mean free surface. The unknown singular ity densities are determined by imposing the hull boundary conditions at the centriods of each panel. There are several different methods for evaluating the potential ~ due to a distributed Havelock source of uniform density a, ~ = (T,/~; dSG(x,y,z;<,r1,()~ where G is the Havelock Green function and s' is the panel surface. We use a form of the Green function which allows us to analytically calculate a wave spectral function by in- terchanging the order of integration: G=-1+ 1 T T' ~ 2 Loo Loo ek(Z+~)+ik=(=-()+iky(y_~) + BRIM-ho J dry Jo dk~ kid-kok -i 2° /. day /3(ky) ek(Z+~)+ik' (~-~)+iky(y-77) ) -oo ~ where and r = /(x _ ()2 + (y _ 9)2 + (Z ~ ()2, (10) T = j(X _ ()2 + (y _ 9)2 + (Z + ()2 k = \/~. The variables kin and ky are the longitudinal and lateral wave numbers of the free-surface waves, respectively. In the single integral, the longitudinal wave number is not an independent variable, and is related to ky by 210 k' = [1 (ho + A)] ~/2 . 2 (ho + I/;) ~ = .. to + 4ky The first two terms in the Green function correspond to a Rankine source and its negative image above the free sur- face; their contributions to the potential are computed using standard techniques. The free-surface wave contributions to the potential are contained in the two integral terms of Equation (9~. To compute these terms we first define a wave spectral function C(k=, by ), C(k=,ky) = |/ dSek/:-it-inky (11) sit where the integration over the panel surface is performed analytically. The wave contribution to the potential can then be written as 2 ~ ~ ~ ~ ek3+ik~ =+iky y Flu, = a-ho J dLyt dkxC(k='kY) k2 _ kok -Hi ko / day is, C(k', ky)ekZ+iks2+ikYY, where it is understood that we are taking the real part of the expression. In practice, all of the boundary conditions involve derivatives of the potential rather than the potential itself, but the spatial derivatives can be calculated simply by multiplying the spectral function by the appropriate wave number prior to the integration. For a surface distribution of Havelock dipoles directed normal to the surface, the potential can be written as p = p J i dS On G(x y z; ( a/ () where ,u is the dipole density. When we substitute the def- inition of the Green function "Equation (9)], into this ex- pression, we find that the potential contains contributions from a Rankine dipole and its negative image above the free surface, and contributions from the wave terms. The wave spectral function for a Havelock dipole distribution, Cd(k2:, ky), can be analytically integrated over a flat panel, and is related to the wave spectral function for a Havelock (9) source, Equation (11), by Cd (k=, ky) = (nzk-invoke-inyLy)C(k=, By), where (ins, my, no) are the components of the unit vector n, normal to the dipole panel. The wave contributions to the potential can then be determined by integrating over kX and k Y I2 °° t°° ekz+ik=~+ikyy 4'w = ~-ho dry J dk=Cd(k='ky) k2 _ kok -Pi 2 | day /3, Cd(k',ky)ekz+ik'=+ikyy

Since both sources and dipoles are distributed over the panels on the hull surface, there exist two unknowns for each panel and we have some flexibility over the manner in which the singularity densities are to be determined. One could use a slender ship approximation to set the source densities a priori, a = Un~/4,r, and then determine the dipole densities by imposing the hull boundary conditions at the centroid of each panel. Alterna- tively, the source strengths could be determined initially by solving the Neumann-Kelvin problem in the absence of the pressure condition, and then solving for the dipole strengths which satisfy both the zero normal flow condition and the pressure condition. The zero pressure boundary condition, Equation (8), is highly dependent upon the tangential flow at the centroid of the last dipole panel on the hull surface. There exists a discontinuity in the tangential flow across a continuous distribution of surface dipoles which is proportional to the tangential gradient of the dipole strength. Unfortunately, this important term is lost when the continuous surface of the body is approximated by discrete panels of constant dipole strength. In order to include this term, we employ a finite differencing scheme for the panels immediately up- stream and downstream of the last panel on the hull surface. If the pressure boundary condition is to be applied at the ith panel, we denote the immediate upstream panel as i + 1 and the adjoining wake panel as i-1. Then the tangential gradient over the ith panel can be approximated by /,~ ~ (Ri+1-pi-! ) pan where Lpan is the length of the ith panel. The tangential component of the perturbation velocity Vt due to this dipole gradient is At = +2~pi, where the negative sign corresponds to the outboard side of the panel and the positive sign corresponds to the inboard side. This tangential velocity must be added to the veloc- ity calculated at the centroid of the uniform dipole panel only for those panels on which the zero pressure boundary condition is to be applied. If we panel the wake with M dipole panels, and the hull with N panels on which are distributed both sources and dipoles, then the zero normal flow boundary conditions applied at the centroids of the N panels on the hull surface can be written as ~ RjVni (pi) = Un ~`-~ ajVn; (*i), i = 1, . . ., N. ~ ~ . where IN is the normal component of the perturbation ve- locity. The additional M boundary conditions required are the zero pressure conditions applied at the centroids of the last M panels on the hull just forward of the trailing wake panels: -2,r^pi + ~ ~jU(Xi) = U at,-~ ajU(Xi), i = 1, . . ., M. The trailing wake sheets are extended straight aft from the transom for a distance of half a ship length. The effects of terminating the dipole sheets are therefore confined to a region well aft of the hull. Rankine Singularity Formulation The Rankine singularity approach involves seeking the perturbation potential 0 of Equation (2) as the solution to the integral equation 2~ = ~1 dS¢~3nt. r-//s~ dS 0nt r + //SF dS¢0n~ r I/SF ins r + //SW /1 [3nE r ' (12) obtained from Green's second identity. The field point may lie on the hull boundary SB or the mean free-surface level SF. The variable T. as it is in Equation (10), is the distance between the field point x and the singular point (, and no is the unit normal vector directed into the fluid domain. The hull surface is taken as the surface with either zero or known sinkage and trim. According to whether the field point x is on SB or SF, one of the integrals involving <~/0n: is a principal-value integral. The surface Sw represents a dipole sheet in the wake across which there is a jump ,~` in the potential. Using the integral equation as a basis of a solution tech- nique, we panel the hull, a portion of the mean free-surface level near the hull, and the wake. The panels in the wake are grouped in longitudinal strips within which all the panels have the same normal dipole moment. Each of the panels is flat and is assumed to have constant source and normal dipole distribution. These approximations allow one to dis- cretize the integral equation. The integrals over the hull surface and the mean free- surface level involve ~ and its normal derivative. On both surfaces, the boundary conditions can be used to reduce the number of unknowns at a boundary. On the hull surface the normal derivative is given by the zero normal flow condi- lion, Equation (5~. On the mean free-surface level, the lin- earized free-surface boundary condition, Equation (7), can be used to express the unknown normal derivative in terms of longitudinal derivatives of ¢, i.e. ¢~1:- An upsteam finite- differencing scheme involving the values of 4> at the centroids of free-surface panels is used to approximate <>X2; at a point x on the mean free-surface. Therefore, on the boundaries SB and SF only the function ~ at discrete points remains unknown. Since free-surface panels are arranged in longitudinal strips with their centroids lying on curved lines, the finite- differencing scheme for determining ¢~= must include panel centroids from several strips. In most cases an 11-point scheme is used: five points from the longitudinal strip of panels in which x lies and three points from each of two adjacent strips. At the upsteam ends of the strips of free- surface panels, we assume that ¢= and ¢~= are both zero (Sclavounos and Nakos 1988~. Aft of the transom, there are strips of free-surface pan- els that originate at the transom. On the panel closest to the transom, ¢~= can be approximated from the transom depth and the hull shape. At the next panel downstream, 211

a differencing scheme involving ~ on that panel and ¢, A=, and ¢~= on the panel nearest the transom is used. Approxi- mations for ¢= and ¢~= on the free-surface panel nearest the transom are obtained from the linearized Bernoulli equation and the fact that the position of the free surface at the tran- som is known. In particular, we use Equation (8) and the partial derivative with respect to x of Equation (8~. On the third panel downstream of the transom in these free-surface strips, a lower order upstream differencing scheme involv- ing only ¢' at panel centroids is used. For all other panels in these strips, the 11-point upstream differencing scheme is used. On the transom, although it is unwetted, a boundary condition must be specified because we have formulated the boundary value problem in terms of an integral equation. That integral equation involves an integral over a closed boundary which incudes the hull, the mean free-surface level, and possibly a vortex wake across which ~ may be discontinuous. Several treatments of the transom boundary are pos- sible. We may ignore the boundary which is equivalent to specifying that both ~ and ~= vanish on the transom, a possible overspecification of boundary conditions. However, if the perturbation potential and its gradient are small, this approximation may not be bad. Another option is to treat panels on the transom in the same manner as other panels on the hull are treated. In this case, ¢= on the transon is known and is set equal to Un=. Without a surface of dis- continuity extending downstream from the periphery of the transom, we should expect the flow to turn around the cor- ner of the transom as it streams past the hull. The resulting calculated flow field will not satisfy the criterion that the fluid leaves the hull tangentially, and it will violate our lin- earization assumptions. Treating transom panels like other hull panels thus seems to require a lifting surface extending downstream. A third option is to set ¢~ equal to a value determined by Equation (8), the linearized Bernoulli equa- tion. The three options are complicated by the fact that we have already decided to set ¢= and ¢~= at the free-surface panels immediately aft of the transom equal to those val- ues required by Bernoulli's equation. Thus, for the sake of continuity in the boundary conditions, we choose the third option and set ¢= equal to the value required by Equa- tion (8~. There remains the task of determining it, the strength of the normal dipole moment on the trailing wake. Two op- tions are readily apparent. We may, of course, set ~ to zero and thereby ignore the possibility or requirement of having a discontinuity of ¢' in the wake. Otherwise, we are required to determine ~ from Bernoulli's equation. This is done by considering the linearized form given by Equation (8~. For each wake strip, two points are chosen. One point is at the centroid of the hull panel nearest the transom. The other point is slightly aft of the transom on the side of the strip of dipole panels facing the fluid domain. A difference equation for ~ is obtained by discretizing Equation (8) using these two points. Equation (12) is not the only integral equation that can be used to obtain ¢. We could replace 1/r by 1/r + 1/r' and obtain a second integral equation. The main difference is that in the first equation ~ is expressed in terms of dipoles and sources on the mean free-surface level, whereas in the second case only sources appear on the mean free-surface level. We originally intended to use double-body linearization instead of free-stream linearization for the Rankine-source solution, but we encountered difficulties. In order to pre- vent the flow from turning the sharp corner formed by the transom, we used a wedge to extend the hull downstream for the double-body solution. There is no difficulty in obtain- ing such a solution. The non-zero Froude number solution, however, requires paneling on the entire free-surface near the hull, including the region aft of the transom. Normally, the double-body potential is itself differenced to build finite- difference coefficients so that the influence coefficients can be calculated. In order to proceed in this manner for the free-surface panels aft of the transom, the double-body po- tential was continued from the hull extension to the mean free-surface level by means of a Taylor series expansion. This was necessary because the fictitious potential inside the hull is identically zero. Once this was done, differenc- ing of the analytically contimled double-body potential was performed for first and second order derivatives in the mean free-surface level. Differencing for the first derivatives pro- duced reasonable approximations. However, differencing for the second derivative resulted in large numerical errors. A second and related issue with respect to linearization schemes is that the free-surface elevation computed from double-body linearization shows evidence of numerical in- stability, in that numerical noise seems to grow in the strips of free-surface panels in the downstream direction. This should be expected in view of the results of Sclavounos and Nakos {19881. The fact that this numerical instability is not so pronounced in the free-surface linearization scheme was not expected. The differencing schemes we used for double-body and free-stream linearization are not the same. The double- body di~erencing scheme involves centroids on the same free-surface strip because the free-surface paneling is deter- mined so that the strips of free-surface panels are bounded by double-body streamlines. For free-stream linearization, the free-surface paneling cannot line up with the stream- lines of the free-stream flow because free-surface paneling must conform to the shape of the hull and not penetrate into the hull. Thus the required differencing scheme is more complicated because it must include centroids of panels in several strips of panels. However, once it has been decided which centroids to include in the finite difference stencils, the finite-difference coefficients for approximating ¢~= can be determined rather easily by eliminating truncation er- rors. A linear system of equations is set up to eliminate these errors up to fourth order. The method of singular value decomposition is used to obtain a set of differencing coefficients among the possible sets of differencing coeffi- cients. For free-stream linearization at a particular Froude number, the coefficients of the derivatives of 0 in the free- surface boundary condition are known constants. This is not the case in double-body linearization, in which we are required to numerically approximate the coefficients of the derivatives of ¢, which introduces another source of error. PREDICTIONS In order to validate the computational methods which have been developed based on the theory which was just dis- cussed, predictions have been made for a high-speed tran- som stern model at two Froude numbers, 0.25 and 0.4136. The hull form examined for this study is that of DTRC Model 5415, which was studied as part of the Compara 212

Fig 1 Panelization of high speed transom stern ship (DTRC Model 5415~. tive Study of Numerical Kelvin Wake Code Predictions, re- ported by Lindenmuth et al. (1990~. A sample paneling of Model 5415 is shown in Fig. 1. On this model, Lindenmuth et al. reported a wave trough at the transom for Fn = 0.25 and a crest at the transom for Fn = 0.4136. Because of concerns about violating linearization cri- teria, it was thought best to concentrate developmental ef- forts on the Havelock singularity code at the lower Froude number. The Rankine-source code requires paneling the mean free-surface level near the hull. Because the wave- lengths associated with a lower Froude number are smaller, more paneling is required to resolve the same expanse of the free surface. More paneling requires more computer time and slower development time. Therefore, for the sake of efficiency, the development and modification of the Rank- ine singularity code were concentrated at the higher Froude number. For these reasons, the results in this section are presented in a somewhat inconsistent order. For the Have- lock singularity method, results are presented first for the low Froude number and the major part of the results corre- spond to this Froude number. For the Rankine singularity method, the opposite is true; results are presented first for the higher Froude number, and then for the lower Froude number. Havelock Singularity Results For the Havelock singularity calculations, Model 5415 was paneled at the measured sinkage and trim, with 320 panels on the hull and 8 trailing dipole panels in the wale. For the first test case, we set the source strengths using the slender-ship approximation, and solved for the dipole strengths. We found that the dipole strengths required to satisfy the zero normal flow condition on the hull were un- realistically large, and this approach has been abandoned. By starting with source strengths obtained from a N-K solu- tion without any Kutta condition, we are in effect, using the sources to satisfy the normal flow condition, and using the dipoles to satisfy the tangential flow condition. This leads to a better behaved solution. The source strengths on the hull, from the N-K solution without a Kutta condition, for Fn = 0.25, are presented in Fig. 2a. Each curve in the fig- ure corresponds to a row of panels on the hull (although the panels on the bulb are included in the solution, their source strengths are not plotted on this graph for reasons of clar- ity). These source strengths account for the interactions between the hull and the free-surface waves, but not for the zero pressure condition at the transom stern. The dipole tar o O \ o o a. E to rat .~ In ~1 0.5 0.4 0.3 0.2 0.1 0.0 -0. 1 -0.2 -0.3 -0.4 -0.5 Conget~udunaL posutcon / shop length (a) Source strengths on the hull. . , . , . , . , . -, . . . . . . . . n.s 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0 5 borage tud~r~aL pose talon / shop Length (b) Dipole strengths on the hull. ., ., ., ., ., .,,,,,,,, 1 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 l~at~eraL pose talon / transom beam / o o o~ ~D o . =) o E o o o o JO o ,0~` \_ \ (c) Dipole strengths across the wake. Fig 2 Singularity strengths on the hull, En = 0.25. Each curve represents a row of panels at a different depth. strengths required to satisfy the Kutta condition, Fig. 2b, are driven entirely by the zero pressure condition, in the sense that the dipole strengths would go to zero in the ab 213

sence of this boundary condition. Apparently the ejects of imposing the Kutta condition on the hull at the transom, as represented by the dipole distribution, do not extend very far forward of the transom. The dipole strengths are negligible over most of the length of the hull, but as the flow accelerates to satisfy the zero pressure condition, the dipole strengths drop quickly and smoothly to a minimum value at the transom. The dipole strengths are held con- stant on the trailing wake panels. The dipole distribution across the wake is shown in Fig. 2c. The similarities be- tween this distribution and the spanwise distribution on an airfoil are striking. The significance of this trailing vortic- ity can be appreciated by noting that the magnitude of the dipole strengths is comparable to the strengths calculated on an airfoil of equal span, operating at a negative angle of attack of approximately five degrees. The sense of the vorticity will result in an upwelling flow on the centerline of the wake, and a diverging free-surface current across the wake, and may well be the physical mechanism responsi- ble for such flows observed in the centerline wake region of transom stern ships. Contour plots of the contributions to the near-field wave elevations due to the sources alone and due to the dipoles alone are presented in Figs. 3a and 3b (note that the contour interval is reduced in Fig. 3b). The elevations have been non-dimensionalized by the fundamental wavenumber ho. The changes in the wave field due to the dipoles are lim- ited to a small region directly aft of the transom and a group of diverging waves which occur along a cusp line which orig- inates near the stern. The total near-field wave elevations which result from the combined distribution of sources and dipoles on the hull are presented in Fig. 3c. The correspond- ing experimental results are shown in Fig. 3d. Including the Kutta condition at the transom leads to a solution which exhibits a very steep rise in the wave elevation aft of the transom, which will likely lead to wave breaking. Wavecuts along y/L = 0.324 are presented in Fig. 4. The measured wavecut is presented in Fig. 4a, while that predicted from a N-K solution without a Kutta condition is shown in Fig. 4b. In Fig. 4c we note that the transverse waves due to the dipoles are very small relative to those generated by the sources. The most significant far-field ef- fect of the Kutta condition is the same group of diverging waves noted in the near-field contour plots. The combined wave fields are given in Fig. 4d. The inclusion of the dipoles results in a free-wave amplitude spectrum which is in good agreement with that obtained from the experimental wave- cuts, Fig. 5. The predicted wave resistance, Cw, of 0.00042 compares well with the experimental wave pattern resis- tance, Cwp, of 0.00037. Fig. 6 presents contour plots of wave elevation for Fit = 0.4136. Fig. 6a presents the results for the combined dis- tribution of sources and dipoles. The near-field waves gen- erated by the dipoles alone are shown in Fig. fib. The cor- responding experimental results are shown in Fig. 6c. As before, the effects on the wave field of imposing the Kutta condition at the transom are relatively limited. The dipole distribution across the wake is shown in Fig. 7. The vor- ticity is somewhat stronger and has the same sense as be- fore, leading to an upwelling along the centerline and a di- verging flow across the wake. The measured wavecut along y/L = 0.324 is presented in Fig. 8a. The predicted wavecut due to the combination of source and dipole panels is shown in Fig. 8b. Although the predictions are slightly higher than /`koZ = 0.02 . ~ (a) Due to Havelock sources alone. `koZ = 0.01 `koZ = 0.02 (b) Due to Havelock dipoles alone. ~'1 ,~ ~7~.5~ (c) Due to combination of Havelock sources and dipoles. I ~ 1 /`koZ = 0.02 /`koZ = 0.02 ,,. :~- O. (d) Measured. (from Lindenmuth et al. 1990) O rat ~ U1 - o - 11~\ (:) to ~,,,. ,,,~,,' '>?>', ~ :~)'J~ , ~ -o.s o.o o.s 1.0 1.s 2.0 2.s 3.0 3.s ·.o (e) Due to Rankine singularies. (Transom panels, no wake panels) Fig 3 Non-dimensional near-field wave elevation contours for Model 5415 at F., = 0.25. 214

lo - o.o to.o 20.0 30.0 40.0 so.o 60.0 70.0 80.0 90.0 ~oo.o X ~ HOVENUMBER (a) Measured. (from Lindenmuth et al. 1990) ........ . .o 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 X ~ W8VENUMBER (b) Due to Havelock sources alone. O T o.o 10.0 20.0 30.0 40.0 so.o 60.0 70.0 80.0 90.0 100.0 X ~ WRVENUMBER (c) Due to Havelock dipoles alone. Fig 4 Wavecut at y/L = 0.324 for Model 5415 at F,l 0.25. the data, the qualitative agreement between the measured and predicted far-field wave cuts is very good. The free- wave amplitude spectra are compared in Fig. 9. 215 . . . . . . . . . . . .o 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 X 34 WRVENUMBER (d) Due to combination of Havelock sources & dipoles. Fig 4 (Cont.) Wavecut at y/L = 0.324 for Model 5415 at Fn = 0.25 . Preducted li1ec~sured o.o t.o 2.0 3.0 4.0 5.0 6.0 i.o 8.0 9.0 10.0 LateroL Wave Number ( Ky ) Fig 5 Comparison of measured (dashed line) and pre- dicted (solid line) free-wave spectra for Fn = 0.25. (measured from Lindenmuth et al. 1990) Rankine Singularity Results Since the flow configuration is assumed to be symmetric about the center plane of the ship, only half of the hull, free- surface, and wake are paneled. In either case the hull was paneled with 324 panels. The transom was paneled with 8 panels and the wake with 8 strips of panels. The free surface was paneled with 780 panels for the high Froude number case: 10 strips of 62 panels to the side and 8 strips of 20 panels aft of the stern. For the low Froude number case, more panels were deemed necessary because of the smaller wavelengths. Therefore the same expanse of the mean free- surface level was paneled in 10 strips of 80 panels to the side and with 8 strips of 23 panels aft of the stern. The paneling for the high Froude number case was allowed to be finer near the bow and stern whereas the paneling for the low Froude number case was nearly uniform in the longitudinal direction. Fig. 10 depicts the free-surface paneling for Fn = 0.4136; the hull paneling is similar to that shown in Fig. 1. Figs. 6d and be present contour plots of the wave ele- vation predicted using the Rankine source method for Fn = 0.4136. Fig. 6d corresponds to the case where wake panels

/`koZ = 0.02 O ~ (a) Due to combination of Havelock sources and dipoles. AknZ = 0.01 (b) Due to Havelock dipoles alone. lo AkoZ = 0.02 i=-~o~(: (c)-Measured. (from Lindenmuth et al. 1990) lo ,~, . _ . AkoZ = 0.02 °mL~1-~ (d) Due to Rankine singularies. (No transom panels, wake panels) AkoZ = 0.02 of_: -0.5 o.o 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (e) Due to Rankine singularies. (Transom panels' no wake panels) Fig 6 Non-dimensional near-field wave elevation contours for Model 5415 at En = 0.4136. -0.3 -0.2 -0. 1 0.0 0.1 0.2 0.3 0.4 0.5 Lateral posctcon / transom beam Fig 7 Dipole strengths across the wake for o Fir = 0.4136. ~ . . . · - . D.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 X ~ hlRVENUMBER (a) Measured. (from Lindenmuth et al. 1990) i 0 ., ., ., ., ., ., . ',,,, 1, _ 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 10 .0 X ~ HRVENUMBER (b) Predicted due to combination of sources and dipoles. Fig 8- Wavecut at y/L = 0.324 for Model 5415 at En = 0.4136. are present and the panels on the transom are neglected. The normal dipole strengths of the wake panels were de- termined from the linearized Bernoulli equation. Fig. be corresponds to the case in which transom panels were in- cluded, but the normal dipole strengths in the wake were set to zero. There is very little difference between these two sets of results. A third attempt at finding a solution was made for which both the wake and the transom panels were 216

-O .8 o.o 1.0 2.0 3.0 to 5.0 6.0 7.0 8.0 9.0 10.0 Lateral Wave Number 1 Ky ) -Comparison of measured (dashed line) and pre dicted (solid line) free-wave spectra for Fn = 0.4136. (measured from Lindenmuth et al. 1990) 0.75 - Hull Depth | Wave Elevation x/L 1.0 (Transom) 1.25 (a) Fn = 0.4136. 1.5 Fig 10-Free-surface panelization for Fn = 0.4136. present. The third attempt failed because the resulting lin- ear system of equations was too ill-conditioned to obtain a solution in 32-bit arithmetic. We are using direct solvers here because the iterative schemes we have been using do not converge for the problems being considered here. Because of the presence of dipoles on the free surface, we are not able to separate the trailing vorticity due to the presence of the transom from that normally on the free surface. Therefore, there is no figure corresponding to Figs. 2c or 7. Comparing the predicted wave elevations in Figs. 6d and be with the wave height obtained from experiments, Fig. 6c shows that the bow wave and the mid-ship trough are underpredicted. The discrepancy at the bow may be due to the paneling not being fine enough. In the stern area the predicted wave height does match the hull depth at the transom, as is indicated in Fig. lla. This figure shows hull depth versus longitudinal position in the left half and predicted wave height versus longitudinal position in the right half. The hull depth and wave height are plotted for the eight longitudinal strips of panels on the main hull and the eight longitudinal strips of free-surface panels extending from the stern downstream. The wave height and the hull depth match at the stern. We may thus conclude that the pressure condition, Equation (8), is indeed satisfied. The predicted wave height rises from the transom depth to a peak of koZ = 0.14 before falling again. There are no ex- perimental data in this area to determine the accuracy of these predictions. To the side and aft of the stern both the experimental data and the predictions show a wave height Outer Transverse Location Hull Depth Wave Elevation 0.75 x/L 1.0 (liallsom) 1.25 (b) Fn = 0.25. Fig 11 Hull depth and predicted wave height verses longi- tudinal position near the stern at eight transverse locations. koZ = 0.08. A pressure integration over the portion of the hull beneath the mean free-surface level except for the transom was performed to obtain the wave resistance and induced drag as the force in the longitudinal direction, Cw. The resulting value for Cw was 0.00243, which is close to the Cwp of 0.0024. Fig. 3e shows the predicted wave height contours for Fn = 0.25; these results should be compared with the ex- perimental measurements of Fig. ad. The computations cor- respond to the case in which the transom has panels and there are no dipole panels in the wake. In this case the predicted results are not as good as at the higher Froude number because of the difficulty in resolving the finer de- tails of the flow even with the finer free-surface paneling. In the stern area, Fig. fib shows that the fluid leaves the hull tangentially just as it did for the higher Froude num- ber case. This is a good indication that the atmospheric- pressure boundary condition on the hull at the transom is being satisfied. There are more data points from measure- ments for this case. The experimental data indicate a max- imum wave height in the stern area of about koZ = 0.10 217

and a minimum wave height of about koZ = -0.06. The predictions have a peak value of about koZ = 0.14 and a trough with depth of about koZ = - 0.10. In this case the wave resistance, Cw, is predicted to be 0.00053, while the value from wave pattern analysis, Cwp, is 0.00037. CONCLUSIONS The authors have developed two methods which are capable of modeling the flow about a ship with a transom stern, using vortex sheets to model the lift effects. Both methods implement a Kutta condition based on the fact that the pressure on the hull at the transom must be zero. This Kutta condition is used to determine the strength of trailing vorticity or the strengths of dipoles on the tran- som. The first method employs Havelock singularities which can be distributed over the hull surface and in the wake. The linearized free-surface boundary condition (obtained by free-stream linearization) is implicitly satisfied by the use of Havelock singularities, and therefore no singularities are re- quired on the free surface. The second method employs Rankine singularities which can be distributed over the hull surface and in the wake. The technique requires that Rank- ine sources also be distributed over the free surface, and in this way a linearized free-surface boundary condition (lin- earization here is also about the uniform free stream) is satisfied. Computations based on the two methods have been compared with each other for flow about a high-speed transom stern ship (DTRC Model 54154. The results show encouraging agreement with one another and with experi- ments for the flow configurations considered. The Havelock singularity method uses Havelock sources to satisfy the normal flow boundary conditions on the hull, and Havelock dipoles to satisfy tangential flow conditions. This leads to well-behaved solutions to the boundary-value problem. Imposing a zero pressure condition on the hull at the transom appears to have little effect upon the predicted far-field Kelvin wave. The effects in the near field are much more significant. The present results show a wave trough just behind the ship, and the wave elevation now matches the depth of the transom. There is a rapid rise to a wave peak immediately downstream of this trough. The waves generated by the Havelock dipole exhibit a distinct cusp line emanating from the corners of the transom. The pre- dictions are consistent with observed wave fields generated by transom stern ships. Perhaps more significant than the predicted wave field is the prediction of the vorticity shed by a transom stern ship. The flow is shown to accelerate toward the transom to satisfy the zero pressure condition; the result is an increase in the downward dynamic force on the hull. The shed vor- ticity is roughly equivalent to that shed by a hydrofoil with a span equal to the transom beam, operating as a negative angle of attack of approximately 5 degrees. The sense of the vorticity is such that there will be an upwelling flow on the centerline of the wake, and a diverging free surface current across the wake. Similar flow fields are frequently observed behind transom stern ships. To our knowledge, this is the first numerical result which offers an explanation for the source of this vorticity. The Rankine singularity approach shows little differ- ence in predicted wave resistance for two forms of problem formulation: first, when the transom is paneled and wake panels are absent and, secondly, when the wake is present and the transom is neglected. A third alternative corre sponding to paneling both the wake and the transom proved to be numerically ill conditioned. The ill conditioning very likely arises from the fact that the equations required to determine the strengths of the trailing vorticity essentially duplicate those set up to determine dipole strengths on the transom. At any rate, from the results we have seen, it seems best to neglect the wake panels altogether and to place dipole panels on the transom for this type of lineariza- tion. When we attempted to formulate a Rankine singularity method based on a Dawson type double-body linearization, we had difficulties. To solve the double body problem, we extended the hull with a solid surface aft of the transom in order to obtain a double-body flow that did not turn a corner at the transom. We then attempted to apply a Dawson type free-surface condition on this extension of the transom which appeared not to work. Perhaps the solid hull extension should be replaced with a wake composed of a sheet of dipoles extending downstream from the perimeter of the transom to infinity. We have found linear solutions to the problem of the steady flow about a transom stern hull form at realistic Froude numbers, using models which are capable of prop- erly including the effects of shed vorticity. Obviously linear solutions are valid only forward of the region of energetic wave breaking which occurs in the wake downstream of the transom. However, this approach allows us to accurately calculate the flow field near the hull, including the free- surface wave elevations, the pressure on the hull, the shed vorticity and the stern-induced resistance. This transom stern analysis method should allow the design of transom sterns with lower resistance and lower wale signatures. ACKNOWLED GMENTS This work was supported by the Applied Hydrome- chanics Research program of the Applied Research Division of the Office of Naval Research, and administered by the David Taylor Research Center. The efforts of Suzanne Reed who edited the text and assembled the paper are greatly ap- preciated. REFERENCES Cheng. B. H. 1989. Computations of 3D Transom Stern Flows. Proc. Fifth International Conference on Nu- merical Ship Hydrodynamics, National Academy Press: Washington, DC, pp. 581-92. Lindenmuth, W. T., T. J. Ratcliffe and A. M. Reed. 1990. Comparative Accuracy of Numerical Kelvin Wake Code Predictions "Wake-Off." DTRC Ship Hydromechan- ics Dept. R & D Report DTRC-90/010, 234+ix p. Sclavounos, P. D. and D. E. Nakos. 1988. Stability Analysis of Panel Methods for Free-Surface Flows with Forward Speed. Proc. Seventeenth Symposium on Naval Hydro- dynamics, National Academy Press: Washington, DC, pp. 173-93. Newman, J. N. 1977. Marine Hydrodynamics. The MIT Press: Cambridge, MA, 402+xiii p. Tulin, M. P. and C. C. Hsu. 1986. Theory of High-Speed Displacement Ships with Transom Sterns. J. Ship Res., 30~3~:186-93. 218

DISCUSSION Hoyte Raven Maritime Research Institute Netherlands,~e Netherlands I have a few questions on this very interesting paper. 1. If no special treatment of the transom stern is applied, a free surface is predicted that intersects the transom at some point. Since one expects it to flow off the edge of the transom, one looks for modifications. But in a linearized method we generally do not care about the precise location of the intersection as its influence is of higher order. Is not, then, this method (though a very successful one) to incorporate in a linearized method something that in principle prohibits the linearization, vis. the presence of a sharp corner close to the free surface? 2. You point out the analogy with a Nutta condition. There is perhaps another analogy with free-streamline theory. This, however, predicts an infinite curvature of the separating streamline at the transom edge. Could such a behaviour fit into your method? 3. I was impressed by the fact that the Rankine singularity method with transom panels but without trailing wake panels was as good as that which incorporates the value. Does not this contradict the importance of trailing vorticity? AUTHORS' REPLY i. This meeting is indeed an attempt to circumvent a difficulty in devising a linearization scheme. The objective is to find a basic flow from which the true flow deviates little. With a dipole sheet extending to downstream infinity from the edges of the transom, we are building into the basis flow the fact that fluid moves smoothly past the transom. The perturbation from the basis flow potential should then be small. 2. The method of free streamlines might be used to find a hull extension about which a double-body flow can be calculated. Flows corresponding to nonzero Froude numbers could be calculated based on linearizing the free-surface boundary conditions on the mean free- surface level outside the hull with its extension and on the surface of the extension. 3. We do not understand this seeming contradiction, but it may be that we can carry and are in fact carrying vorticity on the free surface. This matter needs to be studied more. DISCUSSION Kazu-hiro Mori Hiroshima University, Japan Although the linearized pressure condition (8) is consistent in your framework, it may not always provide nonjump in pressure. How was the resulted pressure there? Is the pressure condition satisfied? AUTHORS' REPLY For the Havelock singularity method, the dipole strength in the wake is determined in such a way that the linearized zero-pressure condition (8) is satisfied. For the Rankine singularity method, the differencing scheme immediately aft of the stern is set up so that this condition is satisfied. Consequently, for either method, the pressure is zero at the intersection of the free surface with the hull. For the Rankine singularity method, this can be seen in Figs. Ha and fib where the wave elevation aft of the stern has been calculated based on the linearized zero-pressure condition (8). The pressure based on the fully nonlinear Bernoulli equation has not been calculated. DISCUSSION Dimitris Nakos Massachusetts Institute of Technology, USA The approach followed in the paper appears to be able to w tune. linear and/or quasilinear numerical solutions of the steady ship wave problem behind a transom stern, so that they are in better correlation with reality. My question pertains to one of the most critical, and potentially most troublesome, assumptions behind the technique described in the paper...that is the positioning of the line across which separation occurs. In the case investigated by the authors, the loading condition of the vessel is such that part of the transom stern is submerged even in calm water, and the sharp edge of the transom appears as the safest alternative for Separation line.. In many cases, however, the transom-like flow may be anticipated. Do the authors have any suggestions about the positioning of the separation line in such cases? AUTHORS' REPLY The cases in question are nonlinear flows which we are not ready to handle. As a first approximation, one would not concern himself with the hull configuration above the level of the undisturbed free surface. When sinkage and trim is accounted for in subsequent calculations, it might be appropriate to also take into consideration this transom-like flow. 219