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Finite-Difference Simulation of a Viscous Flow about a Ship of Arbitrary Configuration M. Zhu, H. Miyata and H. Kajitani University of Tokyo Tokyo, Japan Abstract The improved version of the WISDAM-II method, a finite-difference solution method for a three-dimensional viscous flow about a ship of arbitrary configuration, is described. A zonal method is used for the boundary-fitted coordinate system so that the boundary layer is sufficiently resolved with proper boundary conditions on the body surface. The robustness and the accuracy are improved by the introduction of a new difference scheme. Computations are performed for a flow about a Wigley hull at Re=108 and the appropriateness of the zero-equation and the SGS turbulence models are examined. l. In troduction A great deal of efforts have been focused for the development of theoretical or numerical methods of solving the whole features of the flow about a ship advancing steadily in the deep water t5] [6] [8]. Since the difficulties arise from the high Reynolds number viscous flow, its separation, its interaction with the free-surface waves and so forth, we are still far away from the completion of the method. However, methods so far developed have already turned out to be useful for the partial explanation of the flow about a ship. One example is the TUMMAC-IV method by Miyata et al., which is currently used for the design of the fore-part of the hull, see Miyata et al tl] [2] [3] [4]. The success of this method is mostly due to the fact that the wave length of the ship waves is sufficiently long and can be resolved by the available grid system. For the numerical solution of a viscous flow about a ship, many research activities are known. Larsson and his coworkers are developing a method of designing hull forms by use of the integral method for the boundary layer, see Kim and Larsson t5]. Chen and Patel have developed a partial-parabolic and a fully-elliptic method for the viscous flow about the after-body of a ship t7] t8]. However, it is still difficult to have satisfactory solution of the separated flow, the streamwise vortices and the viscous flow 119 under the influence of the free-surface. For the elucidation of the details of a turbulent flow a numerical approach by so- called large eddy simulation (LES) technique is often employed, see Moin and Kim, et al [11] [12] [13]. Since a turbulent flow at high Reynolds number is composed of vertical motions of wide-ranged spectrum and small- soaled motions may also play an important role, the resolution of viscous motions of high frequency is very important. They must be directly solved or appropriately approximated in the numerical solution method. With the aim of developing a LES-like technique for a flow about a body of complex geometry with free-surface, a new finite- difference method called WISDAM-II is developed, see Miyata et al t6]. A boundary- f itted coordinate system, which moves at each time step owing to the deformation of the free-surface caused by waves, is employed and the subgrid-scale (SOS) turbulence model is incorporated following Deardorff ill]. This method seems to be very promising since it is very close to the direct solution of the Navier-Stokes equation, and both the viscous motion and free-surface motion are simultaneously solved. However. the improvement of the robustness and is postponed to the f uture study. The objective of this paper is is to show the improved version of II method and the other is to appropriateness of the turbulence order to obtain sufficiently fine the accuracy twofold, one the WISDAM- examine the models. In spacing in the boundary layer a zonal method is employed. To attain sufficient robustness as well as accuracy the fourth-order accurate differencing scheme combined with the artificial dissipation of the fourth- derivative term is used. Both the zero- equation model and the SGS model are used and compared. The movement of the free-surface is not considered in this paper. 2. Grid system for a zonal method Elliptic partial differential grid generation system proposed by Thompson et al. [9] is adopted to construct a boundary-

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fitted curvilinear coordinate system similar to the previous study t6] both in topology and in numerical process. T he three- dimensional grid system has a H-H type topology. Figure 2.1 illustrates the transformation from the physical region D (X ~ ,X2,X3 ) to the imaginary transformed region R (E ~ ,$ 2,$ 3 ), where the streamwise direction is approximately parallel to the ~ ~ direction, the lateral grid lines are approximately perpendicular to the ship hull surface are in the ~ 2 direction, and the grid lines parallel to the girth line of ship hull is the ~ 3 direction. The grid generation is conducted in a well-documented grid generation procedure t6] by solving the transformed Poisson equation with Richardson relaxation method, that is gij ;) r + pk fir = 0 where for convenience, the notation of the geometric coefficients is defined as gij = TkT58k' g = det~gij) gij= 1 -leitil,feJpy 2 g gap go where T'i is the transformation matrix, giJ the covariant metric tensor, and g t j the contravariant metric tensor. ~ is the Kronecker delta and e the Eddington permutation symbol. Since the elliptic grid generation system used in the present study works for smoothing the grid distribution rather than for clustering grid lines in the regions of interest, a great number of grid points are required when a single grid system is employed for a high Reynolds number flow. In the present study, a grid system, which is generated in the elliptic method with more than 250,000 (170 x 30 x 50) grid points, provides satisfactory resolution for the viscous flow only at the Reynolds number 105. In order to alleviate the grid-refinement problems in the vicinity of the ship hull, a zonal method is adopted in this study so that sufficient grid resolution is achieved in the turbulent boundary layer of ship at the Reynolds number lO8. The zonal method is applied only in the ~ 2 direction. Therefore, the inner zone with finer spacing is located in the vicinity of the hull surface and centerplane. In this study, the original grid system generated by the elliptic grid generation procedure is called "coarse grid", while a finer grid system in the vicinity of the hull surface is called "fine grid". In the fine grid system about 12 grid points of the coarse grid system are subdevided into 40 grid points overlapping a region with the thickness about 2y/B=0.4 from the ship hull. The locations of the fine grid points is set so that they may accord with the grid points of the coarse grid system. Therefore, at the boundary of two zonal regions called "zonal- boundary" only metric discontinuity exists. Figure 2.2 illustrates the methodology of the grid- refinement along the ~ 2-grid lines and shows that the coarse grid points are at the same locations with the correspondent fine grid points. Figure 2.3 shows a pair of the coarse grid system and the fine grid system of a cross section of the ship. The flux conservation across the zonal-boundary is on the satisfactory degree. The details of the boundary condition at the zonal-boundary will be described in the subsequent section. 3. Computational Procedure and algorithm Time-dependent Navier-Stokes equations in rotational form and the continuity equation are the governing equations. [6] [10] 8~' = grad(P + 2 u u) + u X ~veal (lo) + R div(u) = 0 where u is the velocity vector, t is the time, P is the pressure divided by the density, v is the kinematic viscosity, and R is the net contribution of the turbulent fluxes described in the following section. All of the physical values are defined in the regular grid system of the general curvilinear coordinates. Since the vectors are expressed with contravariant components, the governing equations are written by using the notation of metric tensors as follows. at g b: ~ ( ~ ~gk/~ I' ~ + g iCiklUkw! VE j 0< j (gk`~) ) + R g _ i/2 ~ (gl/2~`i) = 0 where ~ i is the contravariant component of vortici ty vector, wi _ Eii' d: j ( g'.,u ) and aid k 120 is a permutation third-order tensor ink -- I/2cijk

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The well-known MAC method is employed as the computational procedure. The time derivative term of Navier-Stokes equations is approximated explicitly by the forward difference. From the mass conservation condition, a Poisson equation is derived for the Bernoulli-like scalar field [Gig After solving the Poisson equation iteratively by the Richardson relaxation method, a correspondent Bernoulli-like field is given. The details of the computational procedure is described in E61 Since the tonal method is used in the present study, the time increment for the fine grid system is set at one fifth of that of the coarse grid system for the safety of the computational stability. Therefore after one step of time-marching is conducted in the coarse grid system, five steps of time- marching are conducted in the fine grid system. The zonal-boundary conditions seem to be of crucial importance in this algorithm and w111 be discussed in the subsequent section. Tile calculation is started with the f low field of uniform velocity and constant pressure. 4.Differencing scheme The accuracy of the differencing scheme is very important in the finite-difference method, especially in the calculation of a turbulent flow at a high Reynolds number. Since the dominant equations are written in the form with conjugate components of the transformed coordinates, it is possible to adopt various high- order differencing schemes so far known for the Cartesian coordinates. The authors have employed the third-order upwind difference scheme for variable mesh system and suggested that one may change the factor of the fourth-order velocity differential derivative depending on the mesh size and the Reynolds number to compromise the accuracy with the stability since the third-order upwind scheme is composed of the fourth-order centered scheme and the artificial dissipation of the fourth- order derivative of velocity [21] E22]. In this study, the above scheme is used in the transformed coordinates. Although the convection term is in the rotational form, the dissipation term is derived from the convection term in the gradient form. In the curvilinear coordinates, the covariant derivative of the contravariant velocity component is written by using Christoffel symbol as of =/ + pf~t 'a 8~ ~ and the convection term in the gradient form becomes "J"~ = ~j~ + at then using the differencing scheme recommended by Baba and Miyata [23], the dissipation term is obtained as follows ~ age' Eli; lag where ~ is the factor for the artificial dissipation term Further details of the derivation are referred to Miyata, Zhu et al. [21] [22] and Baba and Miyata [231 The derivatives of the convection term at and near the boundaries where sufficient grid points are not available are approximated by one-sided upwind diffarencing scheme and second-order centered differencing scheme with the artificial diffusion term. The other . derivatives are approximated by the second-order centered differencing scheme. The time derivative term is approximated by the forward differencing scheme. S.Turbulence models In the numerical simulation it is considered that because of the machine ability of the temporary computer it is impossible to calculate the turbulent flow of high Reynolds number without turbulence model except few cases of direct simulation of a flow with very simple geometry. The choice of the turbulence model as well as the computational procedure depends on the purpose of the simulation. Although the turbulent flow is substantially unsteady , only averaged steady flow field is required in some of the engineering problems. However, for scientific purposes and in some engineering problems the detailed unsteady flow should be simulated. Many simulations of turbulent flow around ship hull conducted so far use Reynolds-averaged Navier-Stokes equations and turbulence models such as algebraic turbulence model, ~ equation, K- E model or their combination [8] E14] E151 It is a general approach in this area that when the solution of simulation converges, the results are compared with the experimental ones which are the averaged data of the real physical values. Some simulations have shown excellent agreement with the averaged experimental results. But nobody so far answered several fundamental questions how the flow of the boundary layer is deformed to develops into wake and how is the transition from laminar to turbulent flow on the surface of the forepart of a hulL In order to investigate into the fundamental physical features of the turbulent flow around a ship, the authors employ a zonal method near the ship hull surface, and on the other hand adopt a computational procedure with SGS turbulence model, the latter of which is similar to large eddy simulation and is called LES-like procedure by the authors. Contrary to the methods for an averaged flow this method is 121

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supposed to resolve unsteady turbulent fluid motions of smaller scale and it will provide useful information for the understanding of the fundamental features of the turbulence structure of the ship boundary layer. The present study from above-mentioned standpoint will hopefully to throw a light to the research of this area. In this study two turbulence models are used, one is the algebraic turbulence model and the other is subgrid-scale turbulence model. In order to exmamine the possibility of applying the SGS turbulence model to the turbulent flow around ship by comparing the computational results of the two turbulence models with the experimental data. Then the details of the physical features from the computation are discussed. The formulation of the subgrid-seale turbulence model is essentially same with the previous study by Deadroff-type ill] and is based on the eddy viscosity concept, that is to say, the subgrid-seale stresses used in this study are isotropic ones. The SGS eddy viscosity is defined as Vs = (Co ~)2 (2`,jj em and the SGS stresses R;i are written as Rii- 2`i'ui' = ~gij "us" g,,,, - 2v' eiJ However, using this formulation of SGS turbulence model without any special treatment near the ship hull surface, the turbulent production may be insufficient and be diffused out in the vicinity region inside the laminar viscous sublayer [12] tl3]. This is because the essential turbulence generation near the hull surface is due to the inhomogeneous wall turbulence, which is characterized by a mixing length of the scale of sublayer thickness and of boundary layer thickness. The effects of the curvature of a wall as well as the pressure gradient should also be considered in the turbulence models. However, they are postponed to the future study. In order to take into account the inhomogeneous effect of the wall turbulence in the subgrid-scale turbulence model, the Prandtl-van Driest formulation is introduced for the reduction of the turbulence scale near the hull surface by multiplying the subgrid-seale in the Smagorinsky eddy viscosity by the exponential damping function, that is, a = a [ I - e-Y'/~ ] 122 where A is set constant at 26.0. The details of the formulation of subgrid-scale turbulence model are described in t6]. The algebraic turbulence model used in this study is a modified Cebeei-Smith type tl7] [18]. For the inner region of the boundary layer the Prandtl-van Driest formulation is used as ~ _ 12ldUI I=ky[le~Y+~ ] and for the outer region the Clauser's formulation together with Klebanoff's intermitteney function is applied as follows. ~ l + 5.5( ye/) 63 The correspondent boundary layer displacement thickness ~ and boundary layer thickness ~ in the ease of zero-pressure gradient are determined by ye 8x of the maximum point of the root of the shear stress where the velocity is defined from the law of the wall of the Coles formulation [19] ~=Yleyl[l-e-y+/A ] and the boundary layer thickness ~ and the dispalcement thickness ~ are obtained as ~ 1 .93 6ymnX Also both the accelerated and the decelerated flows including separated flows are considered in this formulation of the modified Cebeci- Smith model, see Stock and Hasse [17]. 6.Boundary Condition No-slip velocity condition is implemented at the ship hull surface. In this study the first grid point in the fluid region is set inside the viscous sublayer, and the velocity at this point is interpolated by the velocity profile of van Driest [18] given as

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+2 it+ I+=KY+11e Y / ] where us = u/u~ and y. = y u~/v. As shown in Figure 6.1, point A is the nearest point and point B is the second point. At each time- step the friction velocity on the ship hull surface is calculated so that the velocity at point B satisfies the above equation. And then the velocity at point A is interpolated by the following equation. t`A t`A qA = = UB tiB qB where u ~ is the component of velocity along ~ ~ grid line and U3 iS along ~ 3 grid line, both of the lines are parallel to the ship hull surface. And q is the velocity magnitude at the grid point, which is calculate] by the ordinary procedure at point B and by the equation of van Driest's velocity profile at point A. At the other boundaries, the uniform stream velocity is set at the inflow boundary and zero normal-gradient condition is set at the side and outflow boundaries. The pressure is fixed at the bottom boundary and zero normal-gradient condition of pressure are set at the other boundaries. The zonal-boundary that connects two zones described in the previous section is placed along the ~ 2 grid line. As shown in figure 6.2, the velocities and momentum terms of the Navier-Stokes equation in the overlapping region of the coarse grid system are set at the same values with those of the fine grid system for the mass and momentum conservation. The algorithm of the calculation with the zonal-boundary is as follows. 1. Calculation in the coarse grid (zone 2): a. Calculate the momentum terms of the Navier-Stokes equation while they are interpolated from zone 1 in the overlapping region by using the updated velocity in zone 2. b. Calculate the source term of the Poisson equation in zone 2 and iterate the pressure solution loop under the zonal-boundary condition, which the pressure at the point A' (figure 6.2) is set at the same values with the pressure at point A of the fine grid (zone 2). c. Update the velocity in the corse grid (zone 2). 123 2. Calculation in the fine grid (zone 1): a. Calculate the momentum terms of the Navier-Stokes equation by using the velocity of the f ine grid (zone 1). b. Compose the Poisson equation for the fine grid ozone 1) and solve it under the zonal- boundary condition that the pressure at point B (f igure.6.2) is set at the same value with that at point B' in the coarse grid (zone 1). c. Update the velocity at the fine grid (zone 1) and update the velocities in the overlapping region of the coarse grid system. 7. Computed results Computations are performed for a flow about a Wigley hull at the Reynolds number (Re) 108 with the algebraic turbulence model (modified Cebeci-Smith model) (Case 1) or the subgrid- scale model (Case 2). It is noted that in the computed results the viscous flow about a hull is not wholly developed but it is on the transition stage, since the computations are continued only for T=1.2 Dimensionless time, T=tU0 /L, Us is uniform flow velocity and L is ship length) in the Case 1 and for T=0.8 in the Case 2. The grid system shown in Fig.2.3 is used and the number of grid points is 255,000 for the coarse grid system and 340,000 for the fine grid system. The smallest grid spacing in the ~ 2 direction about 0.005% of the ship length. The time increment At is 0.00005 for the coarse grid system and 0.00001 for the fine grid system, respectively. The factor of the artificial dissipation term a is set at 6.0. The computations are conducted on HITAC S820/80 supercomputer with almost 20 hours of CPU time. The vectorization ratio of CPU time is 98% for both cases. The pressure distribution on the ship hull surface (x3=o.O) computed with the algebraic turbulence model is compared in Fig.7.1. The agreement with the measured results by Sarda [20] is not very satisfactory since the flow is not f ully developed and f urthermore in the algebraic turbulence model used in this study the displacement thickness of the boundary layer is determined by the well-known Coles velocity profiles for the zero-pressure gradient [17] while the decelerated flow near the after end of ship is involves large pressure gradient. However it is noted the overall flow field is approximately realistic. In order to examine the detailed flow field comparison is made of the f low variables at two longitudinal location x ~ /L=0.8012 (E ~ =110) and x ~ /L=0.9218 (E ~ =120>, where the viscous flow along the hull surface may gradually develops and three-dimensional motions may become important. The data are illustrated in Fig.7.3 and 7.4 for Case 1, and in Fig.7.5 and 7.6 for Case 2. All variables are made dimensionless following the equations described in the previous sections. The distribution of velocity components,

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vorticity components, eddy viscosity coefficient and Reynolds shear stresses along the lateral ~ 2 grid line are presented at two vertical location x3=-0.0463 (~=30) and X3=- 0.0341 (E 3 =35) while the water plane is at X 3 =0.0 and the keel is at x 3 =-0.0625. The contour maps of u ~ and c~ ~ indicate that the boundary layer is still developing and the streamwise vortex is going to be formed in it. The thickness of boundary layer in Case 1 is much thinner than Case 2. This is mostly due to the small magnitude of the eddy viscosity, which may deteriorate the diffusive effect of turbulent flow. It is approximately one seventh of the case of the subgrid-scale model. At the bottom of Fig.7.3 to 6 the computed Reynolds shear stresses are shown. Since their magnitude reaches the maximum value of 10 3 according to the measured results t20], the turbulent flow is not wholly developed in the computations. However it is shown that two components of the shear stresses which are not considered in Case 1 are important in this flow field and hence the use of the algebraic zero-equation model is questionable. The relation between the location of the maximum shear and the boundary layer thickness is shown in Fig.7.2. According to Stock and Haase [17], the relation should be given by the linear equation 6=1.936Y~ .x . This indicates that the boundary layer is excessively enlarged by the subgrid-scale model and on the contrary it is supressed by the zero-equation model. This tendency is amplified at ~ ~ =120 more than at ~ ~ =110. It is supposed that the subgrid-scale turbulence model overestimates the turbulence stresses and the zero equation model underestimates in this region. The figures for vorticlty components indicates that all three components play some important roles and their interactions may be significant. Notwithstanding the difference in the thickness of the boundary layer it is common to Case 1 and 2 that the inflection of vorticity profile appears at ~ ~ =120. 8. Concluding remarks The improved version of the WISDAM-II method is still under development and comprehensive comparison with the measured results or interesting elucidation of the complicated turbulent motions are beyond the scope of the paper. Since the present method avoid approximation as far as possible, the accuracy seems to be on a high level but it is of ten accompanied by the extremely long CPU time even by the supercomputer. The purpose of the present method is to solve both wave and viscous motions simultaneously. It is already demonstrated that this is achieved by use of the moving grid system. t6] However the difference of wave length between free-surface waves and viscous turbulent motions is tremendous. The use of zonal method described here will be one 124 of the promising approach. For the numerical simulation of the detailed viscous flows on the hull surface we must be very careful as suggested by the present test computations The zero-equation model ignores some of the Reynolds stresses which may not be sufficiently small in the real flow. The two- equation model is said to be insufficient for the separated flow. The subgrid-scale model may give excessive turbulence stresses when it is used in the grid system of which spacing is not sufficiently small. The subgrid-scale model will be useful not only for the large eddy simulation but also for the flow simulation of engineering purposes. However, the coefficients and scales for the model must be carefully chosen. This reseach is supported partly by the Grant-in-Aid for Cooperative Reseach of the Ministry of Education, Science, and Culture and partly by the LINEC group of shipbuilders in Japan. References - l.Miyata, H., Nishimura, S. and Masuko, A., "Finite difference simulation of nonlinear waves generated by ships of arbitrary three- dimensional configuration", J. Computational Phys., Vol.60, No.3 pp.391-436 (1985a) 2.Miyata, H. and Nishimura, S., "Finite- difference simulation of nonlinear waves", J. Fluid Mech., Vol.157, pp.327-357 (1985b). 3.Miyata, H., Nishimura S. and KaJitani, H., "Finite-Difference simulation of non-breaking 3-D bow waves and breaking 2-D bow waves", Proceedings of Fourth International Conference on Numerical Ship Hydrodynamics, Washington D.C., pp.259-291 (1985c). 4.Maekawa, Y., Kawasumi, K. and Miyata, H., "A method of optimizing hull-forms by use of the finite-difference technique TUMMAC-IV", Proceedings of International Symposium on Ship Resistance and Powering Performance, Shanghai, PP.70-77 (1989). 5.Kim, J. and Larsson, L., "Comparison between first and higher order methods for computing the boundary layer and viscous resistance of arbitrary ship hulls", Proceedings of International Symposium on Ship Resistance and Powering Performance, Shanghai, pp.17-24 (1989). 6.Miyata, H., Sato, T. and Baba, N.," Difference solution of a viscous flow with free-surface wave about an advancing ship", J. Computational Phys., Vol.72, No.2, pp.393-421 (1987). 7.Chen, H. and Patel, V. C.,"Calculation of trailing-edge, stern and wake flows by a time- marching solution of the partially-parabolic equations", IIHR Report No.285 (1985).

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8.Chen, H. and Patel, V. C., "Practical near- wall turbulence models for complex flows including separation", AIAA paper 87-1300 (1987). 9.Thompson, J. F. et al.,"Boundary-fitted coordinated systems for numerical solution of partial differential equation - id review", J. Computational Phys., Vol.47, pp.1-108 (1982). lO.Warsi, Z. U. A., "Conservation form of the Navier-Stokes equations. in general nonsteady coordinates", AIAA J., Vol.19, No.2, pp.240- 242 (1980). ll.Deardorff, J. W., "A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers", J. Fluid Mech., Vol.41, pp.453-480 (1970). 12.Moin, P. and Kim, J., "Numerical investigation of turbulent channel flow", J. Fluid Mech., Vol.118, pp.341-377 (1982). 13.Shumann, U., "Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annul)", J. Computational Phys., Vol.18, pp.376-404 (1975). 14.Kodama, Y., "Computation of high Reynolds number flows past a ship hull using the IAF scheme", J. the Society of Naval Architects of Japan, Vol.161, pp.22-33 (1987). 15.Masuko, A. et al.,"Numerical simulation of viscous flow around a series of mathematical ship models"( in Japanese ), J. the Society of Naval Architects of Japan, Vol.162, pp.1-10 C1987). 16.Rai, M. M., " A conservative treatment of zonal boundaries for Euler equation calculations", J. Compuational Phys., Vol.62, pp.472-503 (1986). 17.Stock, H. W. and Haase, W., "Determination of length scales in algebraic turbulence models for Navier-Stokes methods", AIAA J., Vol.27, pp.5-14(1989). 18.Cebeci, T. and Smith, A. M. 0., Ana I ys ~ s o f Turbu ~ e n ~ Boundary Laye r, Academic Press, New York (1974). l9.Baldwin, B. S. and Lomax, H., "Thin layer approximation and algebraic models for separated turbulent flows", AIAA -78-257 (1978). 20.Sarda, 0. P., "Turbulen hulls - an experimental study", Ph.D thesis of Univ. t f low past ship and computational of Iowa (19863. 21.Miyata, H., KaJitani, H., Zhu, M. and Kawano, T., "Nonlinear forces caused by breaking waves", Proceedings of 16th Symposium on Naval Hydrodynamics, Berkeley, pp.514-536 (1986). 22.Miyata, H., Kaiitani, H., Zhu, M., Kawano, T. and Takai, M, "Numerical study of some wave-breaking problems by a finite-difference method", J. Kansai Soc. N.A., Japan, No.207, pp.11-23 (1987~. 23.Baba, N. and Miyata, H., "Effect of the form of Navier-Stokes equation on a separated flow simulation", J. Computational Phys. C1989) Csubmitted). 125

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aG' aGs Ir.-:.~ .....,~~.. . l Y~ :.-:.-- :-:-:-. :.-:.-2-:.-.~1: ~ ~ l ~ .:: :.~ :.:-:.- :::.: ..~ a`7, --lo I l ... .. ~ ^ , ~ ~ oQ ~W _ . aG6 ,~ ~ / ,, / ; /Physical Region aG7 ~ aG2 Transformation ~ al, /Hs 1 ~ ~ ~ ::. / . . , I .: :.:::: . :. : ::.: :. ail, , ~ , ........... l A ~ahoy ~ ::ir.::: 1 ~ j3 ~ eH6 ~ __ i / ~ /Transformed Region ~H' aH2 Fig.2.1 Coordinate transformation. OVERLAPPING REGION O ll Fig.2.2 Definition sketch for the interfacing region Fig.2.3 Grid system for the zonal method (transverse section), coarse (left) and fine (right) grids. 126

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~2 \ _ ;~ \ ~ w e\\ / ~\\N'''- 0.4 _ 0 2 V o _ -o.< ,,,, 1 0 0.2 Fig.6.1 Definition sketch for the treatments on the wall. I 1 1 ~ 1 1 ~ I ~ 1 ~ ' 1 ~ 1 ' ' ' ' 1 ' ' ' ' - ~. ~ __- i ,~ it.,. - ! - s t ,,, 1,,,, 1 0.4 2x/L o o ,, ,_' 1_, .,, 0.6 0.8 1 Fig.7.1 Longitudinal distribution of pressure on the hull surface, ; potential theory, O; experiment by Sarda {20] at Re=4.5x1 O8 , - - - - ; present computation with zero-equation model at T=0.6, x ; do at T=1.2. 0.~4 2 SIB 0 t2 0.10 0.08 0.~S 0.04 0.02 120, 35) __ /=1~ 938Ymax / MU,/ / /, / / 'l'0 ~~ 0~ / (120,35) (120,30) / ~ /,,,110,35tt(120,30) / / ,, / ~(1~0,30) 0.02 0.03 0.04 B.05 0.06 0.07 2Ymax /B Fig.7.2 Relation between the boundary layer thickness (S ) and the location of maximum Reynolds stress (Y max), blank and black marks are for the SGS and the zero-equation model,respectively, and numbers in parentheses indicate location. 127

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CElNTClUR I NTERVf3L" 0.1000 1 (a) CONT1IUR I NlERVRL~ 20.0000 1 (b) 0.4~ 0.3 _ m \ C\2 0.2 0.1 _ O l O- O .. ., ~ O- O 0l 0 O. 0 O 0, 0 0 ~ 0 0 <, O O 0 0 O 0 0 4 0 o o O O o o O ~ O O O o o o ~ o o ~ o O o 0.25 0.5 0.75 1 Velocity 0.3 - 0.2: 0.1~ O ~ (d) t ~ ~ ol~- O I f ~~A | ~ I l l l l -500 0 Vorticity (e) (f) 0.4 _ 0 0.25 0.5 0.75 1 Velocity 500 o~ l l 1 ~l l l l 1 -500 0 500 Vorticity o ~=~! ~ ~ I ~ ~ ~ ~ I ~ ~ ~ ~ I ~ ~ ~ ~ ~ I , , , , I , , , , I , , , ,~ O 0 05 0.1 0.15 0.2 0.25 0.3 n Eddy Viscosity Vt#105 n 1 L ~ ~ o ,,,, 1,,,, -400 -200 0 200 400 Reynolds Stresses l~,, 1,,,, 1 o l 0 0.05 O. 1 O. 15 0.2 0.25 0.3 Eddy viscosity Vt#105 O - , , , , t , , , , :, , I -400 -200 o 200 Reynolds Stresses Fig.7.3 Computed results with the zero-equation model at ~ 1=110, T=1.2. (a) contour map of ul, (b) contour map of ~1, (c) velocity profiles, 5;U3 (left;: 3=30, right;? 3=35), (d) vorticity profiles, O;@l, 0;~)2, (e) turbulent eddy viscosity, (f) Reynolds shear stress - R 1 2/ U r2 128 . . . . 400 O;u:, O;U2, 2;03 (d_),

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(a) C9NTOUR INlERV8L. O. 1000 r / 03~ . m \ :^ C\2 0.2 _ 0.1 _ 0 , (d) n , , I 0.1 ~ - 500 (e) ~ ~~,,, I ,,,, I,,, , ~ 0 0.05 0.1 0.1 5 0.2 0.25 0.3 Eddy Viscosity Vt~105 01 _ 0 _ -400 - 200 0 200 400 Reynolds Stresses (f) OONTOUR ] NlERVRL- 20.0000 (b) 0.4 . O. O . . O ., O . O ! o t o o . o 1 0 ' o o o, o o: o o. o o o o: 9 1 ~ I ~ o t~~ ] 0 0.25 O 1 O O O o O O o O o O o O _ o oo O O , 1,,, 0.75 1 o o ,,_1,,,,1,,, 0.5 Velocity o.b 0.4 0.3 \ C\2 0.2 O.] O _ . . . o . o , o . o ~ ~ o ! o . o o, o . ', o , o . o . o o . g - I OIO,,, ,_1_,,,, I 011- ~ ~ It ~ Vorticity ~500 Vorticity 500 -snn 0.1 ~ - 0.25 0.5 0.75 1 Velocity 1,, 1 500 o ~ , 1 , , , , 1 , , , ,'1 , , , , 1 , , , , 1 , , , ,~ 0 0.05 o. 1 o. 15 0.2 0.25 0.3 Eddy Viscosity Vt~105 0.1 _ o ,,,, 1, . . . -400 - ~nn Fig. 7.4 same as Fig. 7.3 at ~ 1 =120, T=1 .2. 129 ] ___ 0 200 400 Reynolds Stresses

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(a) CONTOUR INlERV~L" 0.1000 1 - 1 0.3 m \ o.: o 0.5 ~ ! 1 - o ., . I O -;V OIO , I I I 0 0.25 0.5 0.75 1 Velocity o o o o (d) n,: f ~ 1i ~\ l 11 ~ ~ ~ _ 0 500 Vorticity (e) (f) 0.1 _ ~ oL , , ~ I ~ I ~ 1 1 , , , , 1 , , , ,~ Eddy Viscosity VL*104 2 0.~ o 1,,,, 1,... -200 - 100 0 Reynolds Stresses O b o t ~ o g ~ P 1 D D r t; jr~ 'Oo ~ 0 _ o , 1 , , , ~ 1 00 200 n ~ 0 ~ n o i 0.1: (b) CONTOUR ~ NlERV9L. 20.0000 - o o o o 0.25 0.5 0.75 1 Velocity _~0 Or I I I I I ~ (! I I I I I I ~ -500 0 500 Vorticity 0.1~- _ oW~ 1 , , , , 1 , , , ,~ Eddy Viscosity Vt*103 2 0.1 ~ n I v ~ ~ ~ J 1 ~ ~ ~ -200 - 100 0 100 200 Reynolds Stresses F:g.7.5 Computed results with the SGS model at ~ ~=110, T=0.8. (a) contour map of u~, (b) contour map of ~, (c) velocity profiles, O;u:, O;u2, 3;U3 (left;? 3=30, right;? 3=35), (d) vorticity profiles, 0;~, 0;02, 0;03 (do), (e) turbulent eddy viscosity, (f) Reynolds shear stresses, 0;- R 23/~ ~2 , 0;- R 3~/ ~ r2 , 0;- R t2/ ~ r2 130

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(a) (c) (e) CONTOUR IN7ERVRL- O. 1000 n no n n n CONTOUR I N7ERVRLe 20.0000 n A no no -500 o Vorticity I I i I I 1 of 500 Joc n ~ ~ I,' O p ~ jO 1,, -500 o Vorticity 01 ~ _ ~ n.C - o ~,, 1,,,, l , 1 1~ Eddy Viscosity Vt~105 2 ~ WoO 1 lt- (f) ~ ~ - 200 - 1 00 0 1 00 Reynolds Stresses 500 1, 01 _ ~ o A, 1,, 1 1 1 1 1 1 1 1,, , o 05 1 1 5 Eddy Viscosity Vat l 0~ o o o o o 200 Fig. 7.6 same as Fig. 7.5 at ~ 1 =l 20/ T=0 .8. 131 1 2

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