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Computation of a Free Surface Flow around an Advancing Ship by the Navier-Stokes Equations T. Hino Ship Research Institute Tokyo, Japan Abstract The finite-difference solution method for the Navier-Stokes equations with nonlinear free surface con- dition is applied to the simulation of flow field with a free surface around an advancing ship. The body-fitted coordinates system is used in order to cope with a ship of an arbitrary hull form. The coordinates system does not fit to the free surface geometry which must be de- termined as the part of solution in the time marching procedure. The nonlinear free surface condition is im- plemented in the numerical scheme. The algebraic tur- bulence model is used together with the wall function on the body boundary condition to simulate high Reynolds number flow. The numerical results are compared with the experimental data. 1. Introduction A viscous flow field around a ship is strongly non- linear even when the free surface effects are neglected. When a free surface deformation is taken into account, the geometry of free surface boundary should be deter- mined as a part of solution by the nonlinear free sur- face condition and flow field becomes more complicated in both physical and numerical aspects. A number of efforts have been made to solve this nonlinear prob- lem. Among them, the finite-difference solution meth- ods for the Navier-Stokes equations with free surface effects t1,2] seem to be most promising because of their generality. However, even the Navier-Stokes solvers for double-model flow around a ship [3,4] have not been established well in the case of turbulent flow simula- tion because of various problems. There are much more difficulties to be overcome in the development of free surface flow solvers, such as the treatment of free sur- face boundary condition, grid generation strategy and turbulence model. In the present paper, the finite-diRerence method for the Navier-Stokes equations with nonlinear free sur- face condition developed in Reference t13 is extended to high Reynolds number flow simulation around an ad- vancing ship. Nonlinear free surface condition is imple- mented in the scheme. The algebraic turbulence model 103 is used together with the wall function method for the body boundary condition. The outline of numerical scheme is described in Section 2. The numerical results for Wigley's parabolic hull and Todd's Series 60, Cb=0.6 are shown and com- pared with measured data in Section 3 and 4, respec- tively. The concluding remarks is given in Section 5. 2. Numerical Scheme 2.1 Governing Equations The governing equations are the Reynolds aver- aged Navier-Stokes equations and the continuity equa- tion for mean velocity of unsteady three-dimensional incompressible fluid. They are written in dimensionless form as follows; at ~ mu + vuy + wuz =p-+(l/Re+z~t)(u-2+uy,~uzz) (la) Vt + us + vvy + wvz =PI + (1/Re + lJt`)(~V~ + VYY ~ VZZ) ( OCR for page 103
In Eqs. (1)-(4), subscripts, x, y, z and t mean the partial differential. The body-fitted curvilinear coordinates system (if, 71, ) is introduced to cope with the body boundary of an arbitrary form, where ~ is the direction from fore to aft, ~ the direction from a ship or a center plane to the side outer boundary and , the girth direction from keel to deck. As same as the previous method [1], the computational coordinates do not fit to the free surface shape, so they are not time-dependent. The coordinates transformation is given as follows; = ((a, y, a), 7~ = 71(z, y, Z), = (z, y, Z), t = t (3) The momentum equations (1) and the continuity equation (2) are transformed through Eqs. (3) as up + Us + Vu + We = - (~` + rears + ~) + (l/Re + z~t)(V2u) (4a) vet + Us + Vv,7'+ We ((y~+0y~7+y~)+(l/Re+L/t)(V2v) (4b) we+ Us +Vw77+ Ww: = - ((z< + r~z,7 + Czar) + (1/Re + z~t)(V2w) (4c) BUD + n,,~u,7 + ,ru~ + (yin + ~yV77 + yVc +(zw~ + ~zw,7+zw: = 0 (5) where where (U. V, W) are the unscaled contravariant velocity components and defined U = flu + (yv + Saw (6a) V = YOU + 71yV + 77zW (fib) W = Mu + yV + zw (6c) ~ is pressure from which hydrostatic component is ex- tracted; ~ = p + z/Fn2 (7) V2 is the transformed Laplacian operator and defined as V2q = (~2 + f2 + f2)q66 + (~2 + 02 + 02)q +(+ + y + Z )q(< +2( + By fly + (zNz)q6n +2(r~+ + pyy + Hzz)quc +2(2~+y~y+zfz)q(E +(~m + (9 + (zz)q~ + (em + H99 + ~zz) +(~= + 99 + zz)q~ where q is arbitrary scalar quantity. A, by and so on appeared in Eqs. (4)-(8) are the metrics of the grid. 2.2 Basic Algorithm The basic algorithm is same as that of the MAC method [5~. The discretization is made in the non- staggered grid, that is, all variables are defined in the intersections of grid lines. The present method is based on the time marching procedure and is divided into two stages. On the first stage, velocity is updated by the mo- mentum equations (6). The forward difference is used in time. The spatial differences are the third-order up- stream difference by Kawamura and Kuwahara t6] for the convection terms, the second-order central differ- ence for the pressure gradient terms and for the diffu- sion terms and the fourth-order central difference for the grid metrics terms. On the second stage, pressure on the next time step is computed so that the velocity field on the next time step may satisfy the continuity condition. By taking di- vergence of the momentum equations (6), the following Poisson equation for pressure is derived. v2 = IKE~KuDIE LyleKyleyLc t9 (zM~~zMnzM~ Dt K = US + Van + Wuc(l/Re + ut)(V2u) L = US + Vv77 + Wvc(l/Re + L/t)(V2v) M = US + Vw,'+ Ww:(l/Re + z~t)(V2w) and D = Cup + mud + -u; + love + ~yv,7 + yvc +(zw~ + Hzwn + zw: The right-hand-side of Eq.~9) is evaluated by the values at the present time step. The spatial differences for K, L and M are same as that for Eqs.~6). The time differential appeared in the last term is expressed by the forward difference. Then D, divergence of velocity, on the next time step is set zero from the continuity condition, while D on the present time step which is not necessarily zero is evaluated by the second-order central difference to avoid accumulation of numerical errort53. The left-hand-side of Eq.~9) is evaluated by the second- order central difference and solved iteratively by the Jacobi method. The initial condition is still state, that is, velocity and wave elevation are zero and pressure is hydrostatic (8) in the whole domain of computation. The constant ac- celeration is made by adding the inertia force to the momentum equation in x-direction, Eq.~4a), until the inflow velocity becomes unity. 04

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2.3 Free Surface Conditions When the effects of viscosity and free surface ten- sion are neglected, the free surface conditions consist of the following two conditions. One is the pressure con- dition that means that pressure on the free surface is equal to atmospheric one. The other is the kinematic condition that tells the fluid particles on the free surface keep staying on it. Because the grid points are not on the free surface in the present grid system, it is not easy to satisfy the free surface conditions on the exact location of the free surface. The pressure condition is implemented in the solution process of the Poisson equation for pressure. To give the boundary condition at the intermediate point between grid points, the 'irregular stars method' used in the SUMMAC method [73 is extended to the curvilinear coordinates system. The kinematic condition is used to determine the free surface shape in the time marching process. The wave elevation is defined in the computational coordi- nates as ~ = h((, 0, t) (10) The kinematic condition is written as 2.6 Turbulence Model ht + US + VhnW = 0 on = h (11) Eq.~11) is transformed into the finite-difference form in the same manner as that for the momentum equations (6). Velocity (U. V, W) on the free surface is extrapo- lated equally from the value at the lower grid points. 2.4 Body Surface Conditions For the body surface condition, the wall function approach is used to reduce the computation time. With the no-slip condition, the minimum grid spacing in the direction normal to the body surface should be small enough to resolve the viscous sublayer of the boundary layer on the body. Because the explicit scheme in time is used, the time increment is limited by the CFL condi- tion and should be also small in proportion to the grid spacing. The total computational time to convergence would be too large for the present computer power. The wall function used here is the general logarith- mic law, that is, q/ur = 1/,c In y+ ~ B (12) where q is velocity magnitude, u, the friction velocity and ye the normal distance from the wall normalized by I//UT. The constants tic and B are set 0.4 and 5.5, respectively. Following Chen and Patel [8i, the two velocity points ~ j = 2 and 3, where j = 1 is the wall ~ are assumed to be located in the logarithmic region. From q and the normal distance from the wall at j = 3, UT is determined from Eq. (12). Then q at j = 2 can be calculated from UT and the normal distance of the point. Velocity at j = 2 is treated as the boundary value in the velocity updating process. The direction of velocity at j = 2 is assumed parallel to the wall. In the accelerating period, the no-slip condition is imposed for velocity on the body. Pressure and wave elevation on the wall (j = 1) is set equal to those at j = 2. By the use of the wall function, the minimum grid spacing can be more than ten times as large as that in the case of the no-slip condition. In the present compu- tations, typical value of y+ at j = 3 is taken about from 20 to 100. The computing time is reduced drastically by this procedure. 2.5 Other Boundary Conditions On the inflow boundary, velocity is uniform flow in x-direction after the acceleration and pressure is hy- drostatic with zero wave elevation. On the outflow and side boundaries, pressure, velocity and wave elevation are extrapolated with zero gradient from the inside. Turbulence model used is the two-layer algebraic model by Baldwin and Lomax t93. It is widely used in the aerodynamic computations and also in the incom- pressible flow computation around a ship by Kodama t33. In the present study, flow is enforced to be turbulent from the fore end of a ship. The free surface effect on turbulence is not included in the model. There has not been any turbulence model that can be applied to the boundary layer and wake of a surface-piercing body like a ship. Further investigation in both computation and experiment is required to establish a turbulence model under the free surface effects. 3. Computation for Wigley's Hull 3.1 Computational Condition The first computational results are for Wigley's parabolic hull. The waterlines and the frame lines of the hull geometry are defined by the parabolic lines. The computations are made with two, coarse and fine, grids. The grid generation scheme based on the geo- metrical method is used. The coarse grid is shown in Fig.la. The number of grid points is 51, 20 and 18 in the ((, A, A) directions, respectively. The H-O grid topology is adopted. The grid points are clustered near the body and near the still water surface. The computational domain in the dimensionless coordinates ( where x = 0 is the midship, y = 0 the center plane and z = 0 the still water level) IS 105

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Fig. la Coarse computational grid for Wigley's hull. Fig. lb Fine computational grid for Wigley's hull. -1 < x < 1, 0 < y < 0.5, -0.5 < z < 0.0555 It should be noted that the domain includes the region above the undisturbed free surface, that is, z ~ 0. The grid points below the still water surface is 51 x 20 x 10. The number of grid points inside the fluid varies as the wave field develops. The minimum grid spacing in direction is 0.001. The fine grid shown in Fig.lb has 100 x 20 x 38 grid points in ((, ~1, ) directions, respectively. The computa- tional domain is same as that for the coarse grid, except that0.5 < z < 0.036. The grid points under the free surface is 100 x 20 x 30 and the minimum grid spacing in redirection is 0.0008 in this case. The Froude number is 0.25 and the Reynolds num- 106 her is 106 in both computations. The acceleration is made in the first 500 time steps. The dimensionless time increment is 0.0005 for the coarse grid computa- tion. In the fine grid case, the dimensionless time incre- ment is 0.0005 from 1 to 2000-th time step and 0.0003 from 2001-th time step for stabilization of computation. 3.2 Accuracy Analysis - The time derivatives in the momentum equations (6) and in the free surface kinematic condition (11) are replaced by the forward one-sided difference, that means accuracy in time of the present method is the first-order. For the spatial differences, the pressure gra- dient terms and the diffusion terms have the second- order accuracy. The third-order difference is used for the convection terms, in which the leading error is the fourth-derivative term and does not affect the diffusion

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/: '\\ ~~ : ~ ~ ~ ~ ~ :~ ~ ~ ...- i,:.:::''' i,\~/'~ 1 Fig. 2a Time evolution of computed wave pattern around Wigley's hull with the coarse grid. Contour interval is 0.02 x 2gh/UO. Dotted lines show negative values. Top; 6000-th step ~ t = 3.0 ), middle; 8000-th step ~ t = 4.0 and bottom; 10000-th step ~ t = 5.0 ). - -'' """" .'~).J'. J Fig. 2b Time evolution of computed wave pattern around Wigley's hull with the fine grid. Contour interval is 0.02 x 2gh/UO. Dotted lines show neg- ative values. Top; 6000-th step ~ t = 2.2 ), middle; 8000-th step ~ t = 2.8 and bottom; 11000-th step ~ t = 3.7 ). 107

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1 an' W Fig. 3a Pressure distribution on hull surface, center plane and free sur- face around Wigley's hull computed with the coarse grid. Contour interval is 0.02Cp. Dotted lines show negative values. J ~2' - ~ ' l'' W<.~=''2''''\ Fig. 3b Pressure distribution on hull surface, center plane and free surface around Wigley's hull computed with the fine grid. Contour interval is 0.02Cp. Dotted lines show negative values. terms of the momentum equations. The grid metrics are evaluated by the fourth-order difference. Numerical errors due to these finite differencing are the function of the time increment and the grid spacing. Other factor that determines accuracy is conver- gence. As for convergence of the Poisson solution for pressure, the residual is typically 0~10-4) after 20 iter- ations. In the time integration process of the present method, the quantity that converges most slowly is the wave elevation. Therefore, convergence of the solution is examined by steadiness of the wave patterns. The comparison of the numerical results with the fine and coarse grids provides information concerning grid density effect. Fig. 2a shows the time sequence of the wave patterns around Wigley's hull computed with the coarse grid. The wave pattern has not reached the steady state at 10000-th time step when the dimen- 108 sionless time is 5.0. The grid resolution seem to be not sufficient to get convergence. In the case of the fine grid shown in Fig. 2b, on the other hand, the wave pattern has become almost steady at 1100~th time step (the dimensionless time is 3.49~. The waves far from the body in the coarse grid case are less steep than those in the fine grid case. The numerical dissipation due to the finite differencing error decreases the wave amplitude when the grid spacing is large. Fig.3a and 3b show the pressure distribution on body surface, center plane and free surface in the coarse and fine grid cases, respectively. Pressure value on the free surface is identical to the wave elevation, because hydrostatic component is extracted from static pres- sure. Strong wiggles of pressure can be seen on the body surface in the coarse grid case. That may be one reason why the solution has not converged. In the fine grid

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Fig. 4 Perspective view of computed waves around Wigley's hull. Wave height is three times magnified. 2*9*h/(U*U) 0.40 0.20 0~-~ -0.20 Computed Measu red J J ~.5 x/L Fig. 5 Comparison of computed and measured wave profiles on ship sur- face of Wigley's hull. En = 0.25, Re = 106 in computation and Re = 5 X 106 in measurement. case, however, the wiggles are limited in the restricted regions such as near the fore end and the aft end or near the free surface and their magnitude is small. 3.3 Results Hereafter, only the fine grid results are shown. Fig.4 shows the perspective view of the free surface configuration, where the wave amplitude is magnified by three times. Waves far from the ship hull cannot be seen clearly. The grid spacing is still too large, partic- ularly far from the ship hull, to diminish the numerical dissipation effects. Fig.S is the comparison of the com- puted and the measured [10] wave profiles on the body surface. Agreement is very well in wave amplitude and in wave length. The orgin of wave making, apart from the propagation of waves which is affected by the nu- merical dissipation, is simulated properly. 109 In Figs.3b and 5, stern wave generation which is not clear in the previous computation for the low Reynolds number laminar flow [1] is simulated in the present results. Stern waves are related to pressure recovery at the stern region and this becomes higher as the Reynolds number increases. Computed pressure distribution on body surface is shown in Fig.6 together with the measured one [103. Pressure patterns are in good accordance with each other, except for the region of wiggles described above. These wiggles seem to come from inappropriate treat- ment of the boundary condition. Pressure recovery at the stern in the high Reynolds number flow described above is simulated well. Figs.7 show the wake (u velocity) contours and the cross flow vectors (v and w velocity) at various stations. The vertical velocity component appears beneath the

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Computed 0.1 0 -0.1 0 _., . . o AP Fig. 6 Comparison of computed and measured pressure distribution on ship surface of Wigley's hull. Contour interval is 0.02Cp. Dotted lines show negative values. En = 0.25, Re = 106 in computation and Re = 3.4 x 106 in measurement. :,\~\ x=0 .5 (AP) Fig. 7 Computed wake contours and cross flow vectors at various stations of Wigley's hull. Contour interval is 0.1 x u. Top left; x = - 0.5 (F.P.), top right; x = 0 (midship), middle left; x = 0.3, middle right; x = 0.4, bottom left; x = 0.5 (A.P.), bottom right x = 0.6. 110

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x=0.4 x~0.5 (AP) x=0.6 Fig. 8 Computed eddy viscosity distribution at various stations of Wigley's hull. Contour interval is 4.0 x 10-6ut. Top left; x = 0.3, Top right; ~ = 0.4, bottom left; x = 0.5 (A.P.), bottom right x = 0.6. free surface. In particular, large upward velocity is seen at F.P. station, which corresponds with the generation of bow waves. The wake contours at A.P. station seems to be too thick though the corresponding measured data is not presented. The improvement of the turbulence model and/or the wall function approach is required. In Figs.8, the eddy viscosity distributions at var- ious stations are presented. The discontinuity in the distribution comes from the fact that the eddy viscos- ity is determined line by line. The strong eddy viscosity regions spread widely near the free surface at x = 0.4 and 0.5 stations. This may be because the turbulence model is affected by the free surface motion and it does not have physical meaning. 4. Computation for Series 60, Cb=0.6 4.1 Computational Condition Fig. 9 Computational Grid for Series 60, Cb=0.60 The second result is for the practical ship hull form, Series 60, Cb=0.6. The computational domain is -1 < x < 1, 0 < y < 0.5, -0.5 < z < 0.0384 and the number of grid points is 100 x 20 x 38 which is same as the fine grid case for Wigley's hull. The com- putational grid is shown in Fig. 9. The minimum grid spacing in direction is 0.0008. The Froude number is 0.22 and the Reynolds number is 106 in this case. The acceleration is made in the first 500 time steps. The di- mensionless time increment is made smaller gradually as the time step increases, otherwise the computation cannot be stable. From 1st to 1200-th time step, the time increment is set 0.0005, then 0.0003 from 1201-th to 2000-th time step, 0.00025 from 2001-th to 3000-th time step and 0.0002 after that. 111 4.2 Results

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t=7 .09 t=7.49 I . . . . . / ~~ :~ G ,.~ :~.~1 .''~, t=789 ~ )J{~ Fig. 10 Time evolution of computed wave pattern around Series 60 Con- tour interval is 0.02 x 2gh/UO. Dotted lines show negative values. Top; 31000- th step ~ t = 7.09 ), middle; 3300~th step ~ t = 7.49 ~ and bottom; 35000-th step ~ t = 7.89 ). The time evolution of wave patterns are shown in Figs.10. The wave field has not reached the steady state at 35000-th time step (the dimensionless time is 7.89~. It takes very large time to get converged solution for free surface problems by the time marching proce- dure, though the use of the wall function approach con- tributes to time saving to some extent. Improvement of the numerical scheme is required to get faster con- vergence. One reason why it takes longer to get con- vergence in the case of Series 60 than in the case of Wigley's hull may be the difference of flow complexity. The hull geometry of a practical ship, such as Series 60, is more complicated than that of Wigley's hull and flow around the complicated geometry does not become steady in a short time. Hereafter, the numerical results at 3500~th time step are shown, because the flow field near the ship can be considered as almost steady. Fig. 11 shows the pressure distribution on body sur- face, center plane and free surface. Wiggles of pressure can be seen on the body surface near the fore and aft ends and near the free surface. Wiggles beneath the free surface is partly due to the discontinuity of grid spacings near the free surface which comes from the constraint of the grid generation scheme. Fig.12 shows the perspective view of the free sur- face configuration, where wave amplitude is magnified by three times. Fig.13 is the comparison of the com- puted and measured wave profiles t11] on the body sur- face. The slight unphysical oscillation of wave elevation is found in the aft part. The hull geometry of Series 60 is more complicated, particularly in the stern part, than that of Wigley's hull and the grid lines around the stern are more distorted. Numerical error due to the distorted grid causes the oscillation of waves. Except that, the computed result agrees well with measured data in wave amplitude and in wave length. In Fig.14, the computed pressure distribution on body surface is compared with the measured one t124. Pressure patterns are in good accordance with each other except for the slight wiggles of computed results. Free surface effects on pressure distribution beneath the free surface, that is, the low pressure zone below the wave trough and the high pressure zone below the wave crest, can be seen in both computation and measure- ments. Figs.15 show the wake (u velocity) contours and the cross flow vectors (v and w velocity) at various sta- tions together with the measured data t123. At the F.P. station, large upward velocity appears as well as in the case of Wigley's hull. The wake contours at the midship station becomes thin around the bilge circle in the mea- surement and this is well simulated in the computation. At the A.P. station, longitudinal vortices can bee seen

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Fig. 11 Pressure distribution on hull surface, center plane and free sur- face around Series 60. Contour interval is 0.02Cp. Dotted lines show negative values. Fig. 12 Perspective view of computed waves around Series 60. Wave height is three times magnified. f: 0.20 nit ~ - ~ .vv 0.5 - 0.4 ~O. 1 ~ -0.20- 2*9*h/(U*U) 0.4OI Computed M ensured 04 0.5 x/L Fig. 13 Comparison of computed and measured wave profiles on ship sur- face of Series 60. En = 0.22, Re = 106 in computation and Re = 1.39 x 107 in measurement. 113

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Computed ~ ;g' it' ~ . . -0.1 ~ ~ Fig. 14 Comparison of computed and measured pressure distribution on ship surface of Series 60. Contour interval is shown in figure. Dotted lines show negative values. Fn = 0.22, Re = 106 in computation and Re = 7.7 X 106 in measurement. Computed rid rid / Measured Computed ,Ii,,~m!, j 1 ~ 1, '\ x=0.25 Measured / Computed 0.9 / 0.9 I (~'\'''"N'\~.~:,_= \ a\\\'\\\\ 1111111\ \ \ x=0.5 (AP) Measured Computed x=0 (midship) - Measured Computed / 0.9 /0.9 (go\ 1 ~ ,.-lU Measured Computed O .9 1 /: / f ' Fig. 15 Computed and measured wake contours and cross flow vectors at various stations of Series 60. Contour interval is 0.1 x u. En = 0.22, Re = 106 in computation and Re = 7.7 X 106 in measurement. Top left; x = - 0.5 (F.P.), top right; x = 0 (midship), middle left; x = 0.25, middle right; x = 0.4, bot- tom left; x = 0.5 (A.P.), bottom right x = 0.6. 114

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/ / 1 west 7- 1_~ Fig. 16 Computed eddy viscosity distribution at various stations of Series 60. Contour interval is 4.0 x 10-6L,t. Top left; ~ = 0.25, top right; x = 0.4, bottom left; x = 0.5 (A.P.), bottom right x = 0.6. in both measurement and computation at the almost same position. However, the computed wake contour is thicker than measured one. At the section of x=0.6, the positions of the longitudinal vortices are different from each other. Figs.16 show the eddy viscosity contours at various stations. Non-physical eddy viscosity beneath the free surface appeared in the case of Wigley's hull cannot be seen in this case. 5. Concluding Remarks The finite-diderence solution method for the Reynolds averaged Navier-Stokes equations is applied to the simulation of high Reynolds number flow with a free surface around an advancing ship. The numerical results for Wigley's hull and Series 60, Cb=0.6 show good agreement with the experimental data in wave profiles and pressure distributions on the hull surface. The further efforts are required in the turbulence mod- eling under the free surface effect to obtain quantitative agreement for the wake distribution. The improvement of the numerical scheme is also needed to get converged solution faster. Acknowledgment The author would like to express his sincere grati- tude to the members of the CFD group at Ship Research Institute for their valuable discussions. In particular, he is grateful to Dr. Yoshiaki Kodama, Ship Research In- stitute, for his suggestions and for providing the author with the graphic output code. The computations were carried out by ACOS 910 with Scientific Attached Processor at the Computer Center, Ship Research Institute. The CPU time was 115 about two hours per 1000 time steps in the fine grid case. References [1] T. Hino, "Numerical Simulation of a Viscous Flow with a Free Surface around a Ship Model", Jour- nal of The Society of Naval Architects of Japan, Vol.161, ~ 1987 ). t2] T. Sato, H. Miyata, N. Baba and H. Kajitani, "Finite-Difference Simulation Method for Waves and Viscous Flows about a Ship", Journal of The Society of Naval Architects of Japan, Vol.160, (1986~. t3] Y. Kodama, "Computation of High Reynolds Num- ber Flows Past a Ship Hull Using the IAF Scheme", Journal of The Society of Naval Architects of Japan, Vol. 161, ( 1987~. [4] A. Masuko, Y. Shirose, Y. Ando and M. Kawai, "Numerical Simulation of Viscous Flow around a Series of Mathematical Ship Models", Journal of The Society of Naval Architects of Japan, Vol.162, (1987~. F.H. Harlow and J.E. Welch, "Numerical Calcu- lation of Time-Dependent Viscous Flow of Fluid with Free Surface", The Physics of Fluids, Vol.8, (1965~. t6] T. Kawamura and K. Kuwahara, "Computation of High Reynolds Number Flow around a Circular Cylinder with Surface Roughness", AIAA paper, 84-0340, (1984~.

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[7] R.K.C. Chan and R.L. Street, "A Computer Study of Finite-Amplitude Water Waves", Journal of Computational Physics, Vol.6, ( 1970~. [8] H.C.Chen and V.C. Patel, "Calculation of Trailing- Edge, Stern and Walce Flows by a Time-Marching Solution of the Partially-Parabolic Equations", lIHR Report, No.285, (1985~. [9] B. Baldwin and H. Lomax, "Thin-Layer Approxi- mation and Algebraic Model for Separated Turbu- lent Flows", AIAA paper, 78-257, (1978~. [10] "Cooperative Experiments on Wigley Parabolic Models in Japan", 1 7th ITTC Resistance Commit- tee Report,(1983~. [11] H. Takeshi, T.Hino, M. Hinatsu, Y. Tsukada and J. Fujisawa, "ITTC Cooperative Experiments on a Series 60 Model at Ship Research Institute", Pa- peTs of Ship Research Institute, Supplement No.9, (1987~. [12] F. Mewis and H. J. Heinke, " Untersuchungen der Umstromung eines Modells der Serie 60 mit Cb=0,60", Sciffbaujorschung, Vol.23, No.3, (1984~. 6

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DISCUSSION by S. Ju 1. In the wall function approach, y+ values of third grid points from the body between 20 and 100 look too small, since y+ values of 2nd grid points from the body will be less than 10 and in the region of laminar sublayer. 2. How the transition from the laminar to turbulent flow is treated? Author's Reply 1. Yes. The grid spacing near the wall in this calculation is a little too small. This should be improved in the future calculation. 2.Though the original Baldwin-Lomax turbulence model can cope with the transition, flow is assumed to be turbulent from the fore- end of a ship in this computation. DISCUSSION by R.C. Ertekin I would like to ask two questions. 1) Because you are solving a symmetric problem, and therefore, using the symmetry condition on the center plane, the normal vector at the point where the center plane meets the bow or stern is double valued or it has discontinuous first derivatives. As a result, the Jacobian of the transformation matrix vanishes there. How did you cope with this problem? Where were the stagnation points on the bow and stern? Have you had any difficulty with the no-slip boundary condition at these points? 2) I am surprised not to see any comparisons with the care in which the linear free surface boundary condition is supposed. Why can you solve the nonlinear problem but not the linear one? I don't also understand why you can not calculate the resistance experienced by the hull. After all, shouldn't the objective of such calculations be the determination of power requirements? By the way when I say the linear problem I mean the linear (viscous) governing equations and boundary conditions! Author's Reply 1. Along the line of mapping singularity, flow quantities, are not computed but set the average of values of the adjacent grid points. The present scheme does not use the no-slip boundary condition but use the wall-function approach. Anyway, I don't have any difficulty on the bow or stern. 2. The implementation of the linearized free surface condition in the present scheme is not impossible, but apparently the solution becomes less accurate. We cannot compute the wave-making resistance separately from the numerical results but, of course, we can compute total resistance. Please see the reply to Dr. Musker. DISCUSSION - by A.J. Musker I should be interested to know whether you have made any efforts to calculate total resistance. The pressure contours appear to be well predicted and you presumably also have calculated data on wall shear stress. Perhaps you could comment. Author's Reply Pressure resistance and friction resistance are computed by the integration of numerical results as follows. . Ship Fn Re rp rF rT _ Wigley O .25 101 ~ 1. 44xlO-d 3.73x103 5.17xlO-: Series O .22 106 0.41x10-3 4.13x10-3 4.54x10-3 where rp = (Pressure Resist. ) / 2 ,oU2S rF = (Friction Resist. ) / 12 ,oU2S rT = rp + rF

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