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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 805 Computation of Nonlinear Turbulent Free Surface Flows Using the Parallel Uncle Code M.Beddhu, R.Pankajakshan, M.-Y.Jiang, M.Remotigue, C.Sheng, L.Taylor, W.Briley, D.Whitfield (Mississippi State University, USA) ABSTRACT A numerical approach is presented in this work suitable for the computation of nonlinear free surface flows over complex geometries such as ship hulls in a fast, reliable and robust manner. The governing equations solved are the incompressible Reynolds Averaged Navier-Stokes (RANS) equations coupled with the free surface kinematic condition and a two equation turbulence model. Simple no normal-gradient dynamic boundary conditions are used at the free surface. The governing equations are cast with respect to an unsteady (non-inertial) general curvilinear coordinate system. The numerical approach uses the modified artificial compressibility formulation. The governing equations are discretized using a finite volume approach where the numerical fluxes at cell interfaces are obtained using Roe's inviscid flux averages coupled with van Leer's MUSCL formulation for higher order flux extrapolation. Viscous fluxes are averaged using central differencing. Time is discretized implicitly using the first order Euler backward differencing. The resulting non-linear algebraic equations are solved using the discretized Newton-relaxation (DNR) approach with symmetrical Gauss-Seidel sweeps. To speed up the solution process a parallel implementation of the numerical algorithm that uses MPI for message passing is used. In order to accelerate the solution convergence process a multilevel approach coupled with the traditional multigrid approach is taken. The resulting algorithm has been applied to various ship geometries and comparisons with the sequential code solutions and experimental results are presented. The results show that the parallel version of the free surface UNCLE code accurately reproduces these earlier results. INTRODUCTION Nonlinear, turbulent free surface flows represent an important class of problems with immediate naval applications especially when such flows occur in the vicinity of a body. These problems are very challenging from a Computational Fluid Dynamical point of view in that they demand a very robust numerical algorithm, very large computational resources and very large amounts of computing time. More importantly, in addition to robustness, the CFD algorithm must model the correct physics. The original UNCLE (for UNsteady Computation of fieLd Equations) code (Taylor (1991), Whitfield and Taylor (1991)) was developed based on first principles to solve the unsteady Reynolds averaged Navier-Stokes (unRANS) equations without any further simplifying assumptions. This sequential version was further extended in (Beddhu et al (1994), (1999)) to include the effects of a free surface. In Beddhu et al (1999) the free surface governing equation was formulated in terms of surface curvilinear coordinates introduced on the actual evolving free surface for the first time. Previous efforts have introduced the surface curvilinear coordinates on a flat surface. The formulation introduced in Beddhu et al (1999) allows for the computation of steep and breaking waves. The sequential version of the free surface UNCLE code has been applied to various geometries with a good degree of success (see Beddhu et al (1999) and the references therein). However, these computations took enormous computing time. With the motivation of reducing the run-time of the sequential UNCLE code when applied for complex configurations, a parallel version of the UNCLE code without the free surface capability was developed by Pankajakshan and Briley (1996). It is designed to operate at coarse/medium grain parallelism levels for optimum performance and is based on Single Program Multiple Data (SPMD) model. It uses MPI for message passing. This version represents one of the first parallel CFD codes that has been used for solving practical problems in a routine manner. Numerous test cases were used to test the robustness and accuracy of the code (Pankajakshan (1997)). The effort presented in the present paper incorporates the free surface capability into the parallel version of the UNCLE code. This involved a complete re-write of the free surface code using FORTRAN-90 which is now included as a separate module in the parallel code. The approach adopted to compute free surface flows is to cast the governing equations with respect to a non-inertial frame so that the free surface can always be made to coincide with a coordinate surface. Thus, Navier-Stokes equations are cast with respect to a set of general unsteady curvilinear coordinates and are solved the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 806 along with the non-linear free surface kinematic boundary condition. The modified artificial compressibility method Beddhu et al (1994) is used to march the solution in time. Approximate inviscid dynamic boundary conditions are used at the free surface since the wind stresses and surface tension are neglected. A two equation turbulence model is used for closing the momentum equations. Characteristic Variable Boundary Conditions (CVBC) are used on the inflow and outflow boundaries. The numerical scheme in Taylor (1991) and Whitfield and Taylor (1991) uses a finite volume formulation and uses the Roe scheme for obtaining the first order fluxes and the MUSCL scheme of van Leer to obtain higher order fluxes at the cell faces. The flux Jacobians which are required in an implicit scheme are obtained numerically. The viscous flux Jacobians are obtained using the thin layer approximation. The resulting system of algebraic equations are solved using the Discretized-Newton Relaxation (DNR) scheme which is comprised of Newton iterations with symmetric Gauss-Seidel sub-iterations in each Newton iteration. A Full Approximation Scheme (FAS) multigrid algorithm is used to accelerate the convergence. The free surface kinematic boundary condition is solved using the same approach. Since the present approach falls under the category of front tracking methods, it is necessary to regenerate the free surface and the underlying grid at each time step. The free surface is assumed to be of the form y=Y(Î¾, Î¶, Ï) where Î¾ and Î¶ are surface curvilinear coordinates and Ï denotes time. Thus, in solving the kinematic boundary condition one obtains an increment ây at every point on the free surface for an increment âÏ in time. Using the ây's the new free surface position is obtained and the flow grid is updated using a background grid. The parallel code was verified by running the same cases as the sequential code and the results were compared with experiments. The majority of the results shown are for Series-60 CB=0.6 and Model 5415. The flow parameters for Series-60 CB=0.6 were specified by the experiments to be as follows: a Froude number of 0.316 and a Reynolds number of 4.2Ã106. For Model 5415, the Froude number was given as 0.2756 and the Reynolds number as 12.02Ã106. Some of the precursory works in free surface flow computations about surface ships were performed by Miyata and Nishimura (1985), Kodama (1989), Hino (1989) and Farmer et al (1993). Recent works are that of Tahara and Stern (1996) and Beddhu et al (1994), (1998), (1999). Tahara and Stern (1996) use a Poisson equation for pressure and use the pressure-implicit split-operator (PISO) algorithm to solve the mean flow equations and the Beam and Warming approach with artificial dissipation for solving the free surface equation. They use the so-called finite analytic method to discretize the governing equations. In addition, they use separate grids for solving the mean flow and the free surface and use interpolation for transferring data between the grids. Beddhu et al (1994) introduced the modified artificial compressibility method for solving the mean flow equations and use an explicit method for calculating the free surface motion. For the computation of mean flow they use a finite volume, implicit scheme patterning their numerical scheme after compressible flow solvers. Thus, Roe scheme is used for obtaining first order numerical fluxes and van Leer's MUSCL approach is used to obtain third order corrections. The viscous terms are approximated using second order central differences and the time derivative is approximated using either first order or second order backward differences. In contrast to Beddhu et al (1994), Beddhu et al (1999) use an implicit method for calculating the free surface. The method was chosen to be the same as the one used for the mean flow. In addition, a novel way of tracking the free surface in time was introduced which preserves the shape of the free surface at each time step during various grid operations. Since the geometries of actual ships are very complex such an approach is necessary for advancing the free surface along curvilinear coordinates. In essence, a background grid was used which is fixed in time. Grid points on the free surface are allowed to move along a particular family of coordinate lines, chosen a priori, of the background grid. The portion of the coordinate lines below the free surface is then used to rebuild the actual grid at the next time level using the arclength distribution of the actual grid line at the current time level. GOVERNING EQUATIONS The governing equations in this work are cast with respect to a non-inertial frame. In tensor invariant form the continuity equation in the modified artificial compressibility method (Beddhu et al (1994), (1999)) is given by (1) where Î² is the artificial compressibility parameter. The momentum equation for viscous, incompressible flows in a non-inertial frame of reference, in a gravitational field, in non-dimensional, vector invariant form is given by (Beddhu et al (1994), (1999)) (2) the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 807 In Eqs. (1) and (2), u=u*/U0, is the non-dimensional absolute velocity vector, v=u+w is the non-dimensional relative velocity vector, w is the non-dimensional grid velocity vector, Ï=tU0/L, is the non-dimensional time, is the Stokes tensor and pâ²=p+Ï/Fr2 where is the non-dimensional pressure and Ï is the body force potential due to gravity. Re0 is the Reynolds number, Re0=Ï0U0L/Âµ0, where, Ï0 is a reference density, U0 is a reference velocity, L is a reference length, and, Âµ0 is a reference coefficient of viscosity. Fr is the Froude number given by where, a is the acceleration due to gravity. Note that when the positive y-direction is aligned in the direction opposite to the gravity vector, one obtains Ï=y. A tilde over a quantity denotes that it is a tensor and an underscore denotes that it is a vector. The Stokes tensor is given by Fig. 1. Schematic illustrating the Cartesian and curvilinear coordinates chosen. (3) where, Âµ=Âµ*/Âµ0, is the non-dimensional coefficient of viscosity. The superscript âT' in Eq. (3) denotes the transpose operation. Casting the governing equations (1) and (2) with respect to a curvilinear coordinate system (Î¾, Î·, Ï, Ï) and using the so-called partial transformation in which the vectors and tensors that appear within the divergence operator are expressed with respect to the underlying Cartesian coordinates whereas the divergence operator itself is expressed in terms of the curvilinear coordinates, one can obtain the so-called numerical vector form which is as follows: (4) where, the authoritative version for attribution. u, v, and w, are the components of the absolute velocity vector with respect to a Cartesian coordinate system, Ïxx, etc., are the Cartesian components of the Stokes tensor, Î¾x, Î¾y and Î¾z, are the Cartesian components of the contravariant base vector grad Î¾. Expressions for G and H are similar to F and can be obtained from F by replacing Î¾ by Î· and Î¶ respectively. Similarly GV and HV can be obtained from FV. Figure 1 indicates the coordinate systems chosen. Equation (4) is the system of non-linear equations to be solved numerically, using a set of physical and numerical boundary conditions. The physical boundary conditions include the noslip condition

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 808 (v=0 â u=âw) on the body and the kinematic and dynamic free surface boundary conditions. The kinematic condition used in this work can be derived as follows. Assume that the free surface is represented by the function Then the kinematic condition implies (Warsi (1993)) (5) In terms of a set of inertial curvilinear coordinates Eq. (5) can be written as (6) u1 u2 u3 where the contravariant velocity component is defined below Eq. (4) and and can be similarly defined. On the other hand, in terms of a set of non-inertial curvilinear coordinates Eq. (5) can be written as (7) where w1, w2 and w3 are the contravariant components of the grid velocity vector. Choosing Î·=y, Eq. (7) becomes (8) Up to this point the notion of surface curvilinear coordinates is not needed. Note that there is no restriction placed on the coordinates Î¾ and Ï. In general, Î¾=Î¾(x, y, z, t) and Ï=Ï(x, y, z, t). Now, let the surface be represented by (9) Substituting, Eq. (9) in Eq. (8), one obtains (10) In Eqs. (9) and (10) the curvilinear coordinates Î¾ and Ï need to be interpreted as surface curvilinear coordinates. However, Î¾ and Ï are still of the form Î¾=Î¾(x, y, z, t) and Ï=Ï(x, y, z, t). The velocity components u1 and u3, and, the grid velocity components w1 and w3 are now surface contravariant components. Note that Eq. (10) is identical inform to the one obtained by Farmer et al (1993). However, they have explicitly assumed that Î¾=Î¾(x, z) and Ï=Ï(x, z). The formulation introduced here is completely general and is valid as long as Y remains a single valued function of Î¾ and Ï. Thus, the formulation introduced here is valid for tracking breaking waves up to the point of reentry (see Fig. 2). The conceptual difference between the present formulation and that of Farmer et al (1993) is that Farmer et al (1993) introduce the curvilinear coordinates on a flat surface (i.e. xz-plane) whereas the present formulation introduces them on the actual free surface. Fig. 2. Breaking wave representation. Y(x) is multiâvalued for x0â¤xâ¤x1. Y(Î¾) is however singleâvalued until reentry. Any book on differential geometry, Warsi (1998) for example, can be consulted for obtaining the metrics of the surface curvilinear coordinates. From these metrics the surface contravariant components of the flow velocity and the grid velocity can easily be obtained as outlined in Beddhu et al (1999). Since, Eq. (10) is cast in terms of curvilinear coordinates, the numerical scheme for solving it can be patterned after that of Eq. (4). Two two-equation turbulence models are available in the UNCLE code. They are the modified Shih and Lumley model (Shih and Lumley (1993), Yang et al (1995), Liou and Shih (1996)) and the q-Ï model (Coakley, 1983). The governing equations of the model are: (11) the authoritative version for attribution. (12)

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 809 where C1=1.44, C2=1.92, In this formulation the eddy viscosity is defined as usual as (13) However, fÂµ and CÂµ are defined as The governing equations of the q-Ï model are: (14) (15) In this formulation, and the eddy viscosity is defined as before as (16) where CÂµ=0.9 and fÂµ=1âexp(â0.0065qy/v). The other constants and functions are defined as C2= 0.92, C1=0.045 +0.405fÂµ, and ÏÏ=1.3. In the absence of surface tension, continuity of the stress vector across the interface is the exact dynamic free surface boundary condition that one needs to impose. This condition was originally obtained by Hirt and Shannon (1968). An efficient way of implementing these exact conditions is outlined in Beddhu et al (1997). However, in the present study exact dynamic boundary conditions are not used. Since the external tangential stresses are neglected and only the atmospheric pressure is considered in the normal stress component and also since the grid near the free surface is not fine enough to resolve the weak surface layer, the dynamic boundary condition has been approximately implemented by many authors for computing ship related flows. For example, commonly used approximate conditions are where, z is the direction opposite to the direction of the gravity vector. In the present work, a characteristic variable based approach is used which can be outlined as follows. First, every term in Eq. (4) is neglected except the unsteady term and the inviscid term in the Î·-direction. The resultant equation is linearized and cast in a diagonal form as outlined in Taylor (1991). Thus, one obtains (17) is the left eigenvector of the flux Jacobian âG/âQ and is where, is the characteristic variable vector, the diagonal matrix containing the eigenvalues of the flux Jacobian âG/âQ. The subscript â0' in denotes that it is treated as a constant matrix. The eigenvalues are given by (18) where, On a Î·=Î·max boundary which is typically chosen as the free surface, Î»1 and Î»2 are zero, Î»3 is positive and Î»4 is the authoritative version for attribution. negative. Thus the characteristic variable W4 needs to be prescribed and W3 needs to be extrapolated from within the computational domain. However, since Î»1 and Î»2 are zero one can either extrapolate or specify W1 and W2. In this work, W2 and W3 are extrapolated from within the computational domain, and instead of prescribing W1 and W4 one uses the conditions Î·t+uÎ·x+vÎ·y+wÎ·z=0 and pâ²=y/Fr2. A 3x3 matrix is solved for the velocity components at each grid point using W2, W3 and the kinematic condition. Even though, this is an inviscid approximation to the exact viscous dynamic boundary condition, it works quite well for the cases considered and very good agreement has been obtained with measured wave profiles. The numerical boundary conditions are imposed on artificial (outer) boundaries which are

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 810 introduced to truncate the computational domain to a finite size so that the resulting problem can be solved using a computer. The assumption is that the artificial boundaries are far removed from the physical body that they won't affect the accuracy of the solution in the vicinity of the body. In this work the fluid flow is assumed to be along the positive x direction with the body fixed at the origin. Thus, far upstream of the body one uses a characteristic variable based inflow boundary condition and a characteristic variable based outflow boundary condition far downstream of the body. The up stream and downstream boundaries are located at least at 5 body lengths each from the origin. At the far away side boundary and the bottom boundary either characteristic variable based inflow or outflow boundary condition is used depending upon the local velocity vector. The side boundary and the bottom boundaries are located at 5 body lengths from the origin. At the z=0 boundary, flow symmetry is imposed. In addition, the computational domain is subdivided into arbitrary sub-domains and at the boundaries of these domains a two point symmetry condition is used. The following procedure is used to advance the solution from time step n to time step n+1: 1) Solve for the interior of the flow field using the Discretized Newton Relaxation method. 2) Update the boundary conditions on the flow variables on all surfaces. 3) Use the kinematic condition to find the new position of the free surface. 4) Find the intersection of the background Î· lines with the updated free surface using a nearest point approximation. 5) Recreate the volume grid from the free surface obtained in step 4. 6) Obtain the new metric coefficients and grid speeds. 7) Update the free surface dynamic boundary condition once again. This is done since the shape of the free surface has changed (in step 2 the free surface at the previous time level n was used) and is found to enhance the stability of the scheme. 8) go to step 1. An integral quantity of immense practical interest is the total resistance of a ship. The total resistance is calculated using the following expression: where In the actual computation, the integral is replaced by summation over all the surface grid cells. nx is the component of the unit normal in the x direction at any point on the body. The quantity |âÏ|dÎ¾dÎ· is the surface elemental area where Note that the body is represented by a Ï=constant surface everywhere except the transom surface which is a Î¾=constant surface. This fact is properly accounted for in the results presented. NUMERICAL PROCEDURE The numerical scheme used in this study is similar to that proposed by Pan and Chakravarthy (1989) and is discussed in detail by Taylor (1991), and, Whitfield and Taylor (1991). An extensive discussion of the methodology has been presented by Whitfield and Taylor (1994) applicable to two dimensional flows. The approach taken in this work is to solve Eq (4) implicitly using the Discretized Newton-Relaxation (DNR) scheme (Ortega and Rheinboldt (1970), Whitfield and Taylor (1991)), where the fluxes at the cell faces are obtained using the Roe scheme (1981) with higher order accuracy achieved using the MUSCL approach (van Leer (1979); Whitfield and Taylor (1991)). Writing Eq. (4) in discrete form, (19) where and so on. Note that for a higher order flux representation depends on the authoritative version for attribution. as well. If Eq. (19) is expanded for each grid cell, a system of algebraic equations are obtained in terms of qn and +1 at each grid cell where Strictly speaking Fn+1 is a function of both qn+1 and the metrics at n+1. Since the metrics at n+1 are known, no linearization needs to be done with respect to the metrics. Hence

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 811 Eq. (19) is regarded as a function of qn+1 alone. In functional form, Eq. (19) can be represented as (20) Solving Eq. (20) involves finding the roots of a system of non-linear algebraic equations. Using Newton's method, the solutions of Eq. (20) are obtained from the following linear equations: Fig. 3 Schematic of the Parallel Iteration Hierarchy (21) In order to limit the band width of the matrix, the operator (âX/âq) is obtained using higher order fluxes in a special manner and is rearranged along the lines of Whitfield and Taylor (1994), into a strong diagonal form. The viscous flux Jacobians are obtained using the thin layer approximation whereas the residue X(qn+1, m) contains all the viscous terms. Within each Newton iteration symmetric Gauss-Seidel passes are used. The resulting algorithm is termed the Discretized Newton-Relaxation (DNR) procedure. When the iteration in m converges, qn+1 is obtained and the calculation procedure is extended to the next time level. As the iteration in m converges, the LHS of Eq (21) goes to zero. Hence time accuracy is introduced into the scheme by multiplying the local time derivative term in X(qn+1, m) by a conditioning matrix Ia where Ia= diag (0, 1, 1, 1). The inviscid fluxes on the RHS of Eq (21) are obtained using a third order MUSCL-type flux and the viscous fluxes using a second order central differencing. PARALLEL ITERATION HIERARCHY The parallel implicit algorithm used here was developed by Pankajakshan and Briley (1996) and Pankajakshan the authoritative version for attribution. (1997) in terms of an iteration hierarchy in which the Newton, multigrid and relaxation schemes are re-examined in the context of a domain decomposition that exploits coarse/medium grain parallelism. The basic iteration hierarchy was extended with the addition of multilevel acceleration (Whitfield and Taylor (1998)) which is used with an embedded multigrid scheme at the finer grid levels. A schematic of the parallel iteration hierarchy is shown in Figure 3. The linearized approximations at each time step are solved using a block Jacobi symmetric Gauss-Seidel (BJ-SGS) relaxation procedure that involves communication at each sub-iteration level. The sequencing of forward and backward sweeps interspersed with data exchange is shown in Figure 4. The code uses MPI for all communication primarily through point-to-point messages in asynchronous non-blocking mode. There are separate message-passing routines for the solver, two-equation model and the free surface module. Parallel runs typically have communications overheads of 10â15%. Scalability studies using heuristic performance estimates that account for algorithm characteristics and system resources indicate that parallel efficiencies of 80% can be

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 812 achieved on up to 500 processors using appropriately sized grids of up to 50 million points. MULTIGRID STRATEGY The multigrid approach (Sheng et al (1995)) used to accelerate the iterative convergence at each multilevel level is described schematically in Figure 5. It is seen that a time-dependent solution is obtained through the use of Newton sub- iterations, which are the most costly part of the overall work. However, this effort can be reduced with the aid of an efficient multigrid method (FAS). The unsteady multigrid method is basically the same as the steady, except that both the fine grid and coarse grid equations must be solved at the same time level to ensure temporal consistency, while in the steady multigrid approach, time is advanced in the fine grid as well as the coarse grid to achieve full efficiency. The two- level multigrid method for a general unsteady equation can be briefly described as follows: 1. Iterate Nh(Qh)=0 times on the fine grid h by Newton's method. 2. Restrict the residual and solution to the coarser grid 2h, and iterate N2h(Q2h)=Ï2h times, where Ï2h=N2h(I2hhQh) 2h (NhQh) is the relative truncation error between the grids h and 2h. âR h 3. Interpolate the correction from the coarser grid to the fine grid and update the solution Fig. 5. Schematic of the Multigrid Cycle Fig. 4. Sequence of Operations for One BJ-SGS/1 Sweep in Asynchronous Mode times at the same time level, using Qh as the new approximation to Qn+1. 4. Repeat steps 1~3 for In the above procedure, is the number of Newton sub-iterations for the fine grid and coarser grids, and is the number of multigrid cycles implemented at each time step. Choosing different values of and may form different multigrid strategies and result in different effects. However, an important fact is that the cost of CPU time is proportional to the multigrid cycles ( ) at each time step. GRID GENERATION The grids presented in this work are generated using Graphical Unstructured Multi-Block (GUMB) structured grid system (Jiang (2000), Remotigue (1999), Jiang and Remotigue (1998)) which is being developed in house. GUMB retained and further enhanced both the initial General Topology Model (GTM) data structure and geometry engine based on Non-Uniform Rational B-Splines (NURBS) which were originally developed for NGP (Thompson (1992), Remotigue (1994)). They are both coupled with a structured grid the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 813 generation library. A schematic of the organization of the libraries of GUMB is shown in Figure 6. Fig 6. GUM-B Schematic of Library Layout GUMBO, Graphical Unstructured Multi-Block Omnitool, is a graphical user interface, developed at Mississippi State University, that has flexible and general repartition and manipulation algorithms that automatically account for the boundary conditions and connectivity. Furthermore, the application of boundary conditions and connectivity is simple and straight forward. In addition, grid assurance and quality measures are included to validate the grid, boundary conditions, and connectivity. The schematic of GUMBO can be seen in Figure 7. Fig. 7. GUMBO Schematic of Library Layout For all the cases presented, load balancing through domain decomposition, specification of boundary conditions and automatic detection of inter-block boundaries were done using GUMBO. For nonlinear free surface calculations it becomes necessary in the present method that the flow grid be adapted to the free surface such that the top surface of the flow grid coincides with the free surface. Beddhu et al (1998b) took an intersection approach where the every Î·-line of the background grid is intersected with the actual free surface to redistribute points on the free surface in order to generate the flow grid for the next time step. Thus, this approach preserves the shape of the free surface during the grid regeneration process and is suitable for unsteady free surface flows. Since the present work is focussed on steady free surface flows, the following approach is taken to redistribute points on the free surface: For each point (i, k) on the free surface find the closest point on the corresponding Î·-line of the background grid. Note that this approach distorts the free surface. Since only steady state is of interest, one expects the free surface to move little as the iteration count increases and thus one expects the free surface to reach the same shape as it would have reached via the intersection approach. However, in practice, though very little free surface movement is noticed at large iteration counts, the final shape is slightly inferior to the one obtained through the intersection approach. Further testing and an intersection approach based on NURBS are underway. Fig. 8 Hull Profile Comparisons for Wigley Hull the authoritative version for attribution. Fr=0.289; Re=3300000 â Expt âââ Sequential â - â Parallel RESULTS The free surface version of the parallel UNCLE code described in the earlier sections was used to compute the free surface flow fields around three popular hulls: (1) Wigley, (2) Series 60 CB=0.6 and (3) Model 5415. These results are presented below. Wigley hull is a fairly simple geometry and has an analytical description to it. Referring to the coordinate system in Fig. 1, the equation for the hull is given

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 814 by: where B is the breadth, L is the length and D is the draft of the ship. The following ratios were taken L/B=10 and L/D=16. The flow conditions are Fr=0.289 and Re=3.3Ã106. Computed results using the sequential UNCLE code for the Wigley hull were reported in Beddhu et al (1998a). Hull profile for Wigley hull is shown in Fig. 8. Note that the parallel code produces a profile that is in very good agreement with the sequential code and the experiment. Fig. 9 Comparison of Hull Wave Profiles for Series 60 CB=0.6 Fr=0.316; Re=4.02Ã106 â Expt âââ Sequential â - â Parallel Fig. 10 Transverse Wavecut Comparison in the Bow Region for Series 60 CB=0.6 Fr=0.316; Re=4.02Ã106 â Expt âââ Sequential â - â Parallel Fig. 11 Transverse Wavecut Comparison in the Stern Region for Series 60 CB=0.6 Fr=0.316; Re=4.02Ã106 â Expt âââ Sequential â - â Parallel Fig. 12 Comparison of Experimental and Computed (Parallel UNCLE) Wave Contours for Series 60CB=0.6 the authoritative version for attribution. Fr=0.316; Re=4.02Ã106 Series 60 CB=0.6 is a geometry for which a lot of experimental data is readily available (Longo et al (1993)). The flow conditions chosen for running the parallel UNCLE code were as follows: Fr=0.316 and Re=4.02Ã106. Results using the sequential version

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 815 of the UNCLE code were reported in Nichols (1998) and Beddhu et al (1998c). Comparison of the hull profile between the sequential and parallel versions with the experiment is shown in Fig 9. It can be seen that the prediction of the parallel version is similar to the sequential version. This similarity can be further seen in the transverse wavecuts presented in Figs 10 and 11. In Fig. 12, comparison of the overall wave contours between experiment and computation using the parallel UNCLE code is shown. This figure shows that there is reasonable overall agreement. In Fig. 13 the u-velocity contours on the propeller plane of the parallel code is compared with the experiment. Here, it can be seen that the agreement is quite good. A similar agreement for the sequential code is shown in Beddhu et al (1998c). Fig. 13 Comparison of Experimental and Computed (Parallel UNCLE) u-velocity contours on the Propeller Plane for Series 60 CB=0.6 Fr=0.316; Re=4.02Ã106 Fig. 14 Comparison of Hull Wave Profiles for Model 5415 Fr=0.2756; Re=12.02Ã106 Fig. 15 Comparison of Longitudinal Wavecuts at z=0.0965 for Model 5415 Fr=0.2756; Re=12.02Ã106 â Expt âââ Sequential â - â Parallel the authoritative version for attribution. Fig. 16 Comparison of Wave Contours for Model 5415 Fr=0.2756; Re=12.02Ã106 The next test case to be presented is the Model 5415. Extensive experimental results for this model is reported in Ratcliffe and Lindenmuth (1990) and Oliv

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 816 eri et al. The flow conditions are Fr=0.2756 and Re=12.02Ã106. The computed wave profile has been plotted against the available experimental wave profiles in Fig. 14. Note that the experimental wave profiles themselves do not fall in each other's uncertainty limits. Thus, the implication of such disagreements between experimental results to the efforts for verification and validation of numerical results is not clear. However, the computed profiles agrees reasonably well with either measurement. In Fig. 15, the comparison of the longitudinal wavecut at z=0.0965 between experiment and the two versions of the UNCLE code is shown. It can be seen that the agreement is quite reasonable. The overall comparison of the computed and experimental wave contours shown in Fig. 16 also shows good agreement. Fig 17 compares the contours of u-component of velocity between the parallel version and the experiment. It can be seen that the agreement is excellent. Computed and measured total resistance comparisons are shown in Fig. 18. It can be seen that though the sequential and parallel versions follow different paths of convergence they both converge to the experimental value in about 4000 cycles. The difference in the paths of convergence is due to algorithmic differences between the two versions. For example the sequential version has no multilevel capability. Finally, the computed and measured stern wave patterns are compared in Fig. 19. Since the transom is wet at this Froude number, computing and matching the stern wave system is supposed to be a tough challenge. It can be seen that the parallel version of the UNCLE code does a very good job of predicting this tough flow feature. A similar comparison with the sequential code is shown in Beddhu et al (1998b, 1999). Fig. 17 Comparison of Contours of U-Velocity Component at the Propeller Plane for Model 5415 Fr=0.2756; Re=12.02Ã106 Fig. 18 Resistance Comparisons for Model 5415 Fr=0.2756; Re=12.02Ã106 Fig. 19 Comparison of Stern Wave Contours for Model 5415 Fr=0.2756; Re=12.02Ã106 â¢ Expt âââ Sequential â - â Parallel CONCLUSION The free surface version of the parallel UNCLE code was designed to be a production mode code that is fast, robust the authoritative version for attribution. and reliable. It has been tested against various geometries and has produced similar agreements with experiments as the free surface version of the sequential UNCLE code. A NURBS based intersection algorithm is being developed that will improve the accuracy of the computation of free surfaces. This code has been successfully transitioned to DTMB and has undergone extensive user trials. Currently, efforts are being focussed in solving unsteady free surface flows. ACKNOWLEDGEMENTS This work was supported by grant N00014â97â1â0959 from the Office of Naval Research. The grant monitor is Dr. Edwin Rood. This support is greatly appreciated. REFERENCES Beddhu, M., Taylor, L.K. and Whitfield, D.L., âATime Accurate Calculation Procedure for Flows with a Free Surface Using a Modified Artificial Compressibility Formulation,â Applied Mathematics and Computation, Vol. 65, 1994, pp. 33â48.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 817 Beddhu, M., Jiang, M.Y., Whitfield, D.L, Taylor, L.K., and Arabshahi, A. âComputational Physical OceanographyâA Comprehensive Approach based on Generalized CFD/Grid Techniques for Planetary Scale Simulations of Oceanic Flows,â MSSU-EIRS-ERC-97â5, Mississippi State, MS 39762, Feb., 1997. 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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 818 na dislocante veloce. Parte II: rilievo della formazione ondosa a Fr=0.28 e Fr=0.41,â Rapporto Tecnico INSEAN (in Italian). Remotigue, M. and the NGP Team, âThe National Grid Project: Making Dreams into Reality,â Proceedings of the 4th International Conference on Numerical Grid Generation in Computational Fluid Dynamics and Related Fields, Swansea, Wales, 1994, pp. 429â440. Ratcliffe, Toby J., and Lindenmuth, William T., âKelvin Wake Measurements Obtained on Five Surface Ship Models,â DTRC-89/038, January 1990. Remotigue, M.G., âStructured Grid Technology to Enable Flow Simulation in and Integrated System Environment,â Ph.D. Dissertation, Mississippi State University, December 1999. Roe, P.L., âApproximate Riemann Solvers, Parameter Vectors, and Difference Schemes,â Journal of Computational Physics, Vol. 43, 1981, pp. 357â372. Sheng, C., Taylor, L.K. and Whitfield, D.L., âMultigrid Algorithm for Three-Dimensional Incompressible High Reynolds Number Turbulent Flows,â AIAA Journal Vol. 33, No. 11, November, 1995, pp. 2073â2079. Shih, T.H. and Lumley, J.L., âKolmogorov Behavior of Near-Wall Turbulence and its Application in Turbulence Modelling,â Comp. Fluid Dyn.. Vol 1, 1993, pp. 43â56. Tahara, Y., and Stern, F. âA Large-Domain Approach for Calculating Ship Boundary Layers and Wakes and Wave Fields for Nonzero Froude Number,â Journal of Computational Physics, Vol. 127, 1996, pp. 398â411. Taylor, L.K. âUnsteady ThreeâDimensional Incompressible Algorithm Based on Artificial Compressibility,â Ph. D. Dissertation, Department of Aerospace Engineering, Mississippi State University, Mississippi State, May 1991. Thompson, J.F., âThe National Grid Project,â Computing Systems in Engineering, Vol. 3, Nos 1â4, pp. 393â399, 1992. van Leer, B., âTowards the Ultimate Conservation Difference Scheme V, A Second Order Sequel to Gudunov's Method,â Journal of Computational Physics, Vol. 32, 1979, pp. 101â136. Whitfield, D.L. and Taylor, L.K., âDiscretized Newton-Relaxation Solution of High Resolution Flux-Difference Split Schemes,â AIAA-91â1539, June 1991. Whitfield, D.L. and Taylor, L.K., âNumerical Solution of the Two-Dimensional Time-Dependent Incompressible Euler Equations,â MSSU-EIRS- ERC-93â14, Mississippi State University, Mississippi MS 39762 April, 1994. Whitfield, D.L. and Taylor, L.K., âVariants of a Two-Level Method for the Approximate Numerical Solution of Field Simulation Equations,â MSSU- EIRS-ERC-98â09, Mississippi State University, Mississippi MS 39762 July, 1998. Warsi, Z.U.A., âFluid Dynamics: Theoretical and Computational Approaches,â 1st. ed., CRC Press, Boca Raton, p. 10, 1993. Warsi, Z.U.A., âRuminations on Applied Tensor Analysis,â MSSU-EIRS-ASE-98â1, Mississippi State, MS 39762, Oct. 1998, p. 78. Yang, Z., Georgiadis, N., Zhu, J. and Shih, T.H., âCalculations of Inlet/Nozzle Flows Using a New k- model,â AIAA Paper 95â2761, 1995. the authoritative version for attribution.

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. S.Cordier None Received. Bassin d'Essais des Carenes, France DISCUSSION AUTHOR'S REPLY for example). Can you tell us what your plans are to address this limitation? COMPUTATION OF NONLINEAR TURBULENT FREE SURFACE FLOWS USING THE PARALLEL UNCLE CODE 819 The use of grids fitted to the free surface has limits when simulating non-linear free surface flows (braking and jets