The gas-kinetic methods have become popular for the simulation of compressible fluid flows in the last

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1 Parallel Implementation of Gas-Kinetic BGK Scheme on Unstructured Hybrid Grids Murat Ilgaz Defense Industries Research and Development Institute, Ankara, 626, Turkey and Ismail H. Tuncer Middle East Technical University, Ankara, 653, Turkey The gas-kinetic BGK scheme on unstructured hybrid grids is presented. In order to accurately resolve the boundary layers in wall bounded viscous flow solutions, quadrilateral grid cells are employed in the boundary layer regions normal to solid surfaces while the rest of the domain is discretized by triangular cells. The computation time, which is a significant deficiency of the gas-kinetic schemes, is improved by performing computations in parallel. The parallel algorithm for hybrid grids is based on the domain decomposition using METIS, a graph partitioning software. Several numerical test cases are presented to show the accuracy and robustness of the proposed approach. I. Introduction The gas-kinetic methods have become popular for the simulation of compressible fluid flows in the last decade. The most promising ones are the Equilibrium Flux Method, the Kinetic Flux Vector Splitting 2, 3 and the Gas-Kinetic BGK method. 4 Due to the inclusion of intermolecular collisions with BGK simplification, the gas-kinetic BGK scheme gives a more complete and realistic description of the flow and has been well studied. 5 2 Although the first studies about the gas-kinetic BGK method were based on finite volume formulation on structured grids, the obvious advantages of unstructured grids over structured counterparts for complex configurations have led researcher to adopt the scheme on unstructured meshes. 3 5 Recently, in the previous work of the present authors, the gas-kinetic flux vector splitting and the gas-kinetic BGK schemes on unstructured grids were given. 6 The solutions were obtained in parallel in order to reduce the computational inefficiency common in gas-kinetic schemes. Several numerical test cases were presented and it was shown that the gas-kinetic schemes are efficient in parallel computations and do not experience numerical instabilities as their classical counterparts. However, in general, unstructured grids are not so suitable for the simulation of viscous flows especially for the well resolution of the boundary layers. In the present work, a high-order gas-kinetic scheme for the Navier-Stokes equations on unstructured hybrid grids is given. The finite volume formulations are combined with explicit Runge-Kutta time-stepping scheme. The parallel algorithm and the domain decomposition for unstructured hybrid meshes are explained and several numerical test cases are presented to show the accuracy and robustness of the present approach. II. Gas-Kinetic Theory In gas-kinetic theory, gases are comprised of small particles and each particle has a mass and velocity. Due to the existence of large number of particles in a small volume at standard conditions (e.g. at STP, 9 air molecules in cm 3 volume), it is very difficult to trace individual particle motion. Instead, a particle distribution function is defined to describe the probability of particles to be located in a certain velocity interval Senior Research Engineer, Aerodynamics Division, AIAA Student Member. Professor, Department of Aerospace Engineering, AIAA Member. of 3

2 f(x i, t, u i ) () Here x i = (x, y, z) is the position, t is the time and u i = (u, v, w) are the particle velocities. The macroscopic properties of the gas can be obtained as the moments of the distribution function. For example, the gas density can be written as ρ = m n i (2) i where m is the particle mass, n i is the number density. Since, by definition, distribution function is the particle density in phase space, it is concluded that m n i = f(x i, t, u i ), (3) ρ = f dudvdw. (4) The time evolution of the distribution function is governed by the Boltzmann equation 7 f t + u i f xi + a i f ui = Q(f, f). (5) Here a i shows the external force on the particle in the ith direction and Q(f, f) is the collision operator. When the collision operator is equal to zero, collisionless Boltzmann equation is obtained and the solution of this equation gives the Maxwellian (equilibrium) distribution function g = ρ ( λ π ) N+3 2 exp { λ [(u i U i ) 2 + ξ i 2 ]} (6) where ξ i = (ξ, ξ 2,..., ξ N ) are the particle internal velocities, N is the internal degrees of freedom, U i = (U, V, W ) are the macroscopic velocities of the gas and λ is a function of temperature given by R being the gas constant. λ = 2 R T (7) III. Numerical Implementation The proposed numerical method is based on the cell-centered finite volume formulation. The numerical fluxes are calculated at the edges of the cells and the conservative variables at the cell centers are updated using a third-order explicit Runge-Kutta time-stepping scheme. In the following, the numerical implementation of the proposed approach is presented in terms of the initial reconstruction of the conservative flow variables and gas evolution stages. A. Initial Reconstruction The initial reconstruction of conservative variables is based on the method proposed by Frink. 8 In this method, a universal expression for the Taylor series expansion within a triangular cell is derived, which requires the knowledge of the state at the nodes. The Taylor series expansion of the conservative variables in the neighborhood of each cell center (x cc, y cc ) is Q(x, y) = Q(x cc, y cc ) + Q cc r + O( r 2 ) (8) where Q = ρ ρu ρv ρe. (9) 2 of 3

3 Figure. Arbitrary triangular cell. This formulation requires the gradients of the flow variables at the cell centers. The evaluation of the gradient vectors is based on the Green s theorem: Q da = Q ˆn dl () Ω The average gradient vector at the cell center, Q cc, is then given by the closed boundary integral over the cell Q cc = Q ˆn dl () A Ω where the unit normal ˆn is assumed to point outward from the domain. Finally, the flow variables at the middle of the cell edges, Q ci, (Fig. ) is given by Frink 8 Ω Ω Q ci = Q cc + 3 [ 2 (Q 2 + Q 3 ) Q ] (2) where the nodal values are calculated from the surrounding cell center values. Mitchell 9 stated that this expression is an average of one central difference term and two upwind terms and recovers a family of schemes common with structured grid schemes. Rewriting Eq. (2) Q ci = 3 [ 2 (Q 2 + Q 3 ) + 2( 3 2 Q cc 2 Q )], (3) Q ci = w[c.d.] + ( w)[upwind] (4) where w = is fully central differencing, w = is fully upwinding. Note that w = /3 recovers Eq. (2) which is incidently suggested by Mitchell as the optimum value for a variety of test functions and mesh distributions. 9 The reconstruction algorithm given for triangular cells may similarly be applied to quadrilateral cells. For an arbitrary quadrilateral cell shown in Fig. 2, the triangular cell in red is used for the reconstruction of the flow variables on cell edge 2-3: Q ci = Q cc + 3 [ 2 (Q 2 + Q 3 ) Q cc ] (5) For the remaining edges, the corresponding triangular cells within the quadrilateral cell are used for reconstruction. It should be noted that since the reconstruction procedure involves the simple averaging of the flow variables, the formal accuracy of the method may be less than second-order. B. Gas Evolution The gas evolution is based on the gas-kinetic BGK scheme. The gas-kinetic BGK scheme is based on the Boltzmann BGK equation where the collision operator (see Eq. (5)) is replaced by the Bhatnagar-Gross- Krook model. 2 The Boltzmann BGK equation in two-dimensions can be written as f t + u f x + v f y = g f τ (6) 3 of 3

4 Figure 2. Arbitrary quadrilateral cell. where f is the particle distribution function, g is the equilibrium state approached by f over particle collision time τ, u and v are the particle velocities in x- and y-directions, respectively. The equilibrium state is usually assumed to be a Maxwellian g = ρ ( λ π ) K+2 2 exp { λ [(u U) 2 + (v V ) 2 + ξ 2 ]} (7) where ρ is the density, U and V are the macroscopic velocities in x- and y-directions and K is the dimension of the internal velocities. The general solution of the particle distribution function f at the cell edge ci and time t is f(s ci, t, u, v, ξ) = τ t g(s, t, u, v, ξ) exp [ (t t )/τ] dt + exp ( t/τ) f (s ci ut vt). (8) Here s = s ci u (t t ) v (t t ) is the particle trajectory of a particle and f is the initial gas distribution function at the beginning of each time step. The relation between mass ρ, momentum ρu, ρv and total energy ρe densities with the distribution function is defined based on Eq. (4) ρ ρu ρv ρe where ψ represents the vector of moments for the distribution function u ψ = v 2 (u2 + v 2 + ξ 2 ) = f ψ dξ (9) and dξ is the volume element in phase space. Since mass, momentum and energy are conserved during particle collisions, f and g must satisfy the conservation constraint of (f g) ψ dξ =, (2) at all (x, y) and t. The high-order gas-kinetic BGK scheme on unstructured hybrid meshes is implemented as follows: Consider the control volumes (triangular or quadrilateral), their neighbors and local coordinates given in Fig. 3 where dots represent cell centers, the triangles or quadrilaterals in red show the control volumes, L and R the left and right states, x n and x t the local coordinate system normal and tangent to the cell edge ci, respectively, x and y the global coordinate system. In cell-centered, finite volume gas-kinetic BGK scheme on unstructured hybrid meshes, the velocity components U n and U t in local (normal and tangential) coordinate system are found from global coordinate system counterparts U and V for both mesh types. (2) 4 of 3

5 Figure 3. Control volumes, their neighbors and local coordinate system. In the present work, the initial gas distribution function f and the equilibrium state g are assumed to be { g f = L [ + a L (x n x nci )], if x n x nci (22) g R [ + a R (x n x nci )], if x n x nci and g = { g [ + a L (x n x nci ) + A t], if x n x nci (23) g [ + a R (x n x nci ) + A t], if x n x nci where g L, g R and g are the equilibrium distribution functions to the left, right and in the middle of the cell edge ci, respectively, of the form g = ρ ( λ π ) K+2 2 exp { λ [(u n U n) 2 + (u t U t ) 2 + ξ 2 ]}, (24) and a L, a R, a L, and a R are the spatial slopes and A is the time slope, given by a L = a L + a L 2 u n + a L 3 u t + 2 al 4 (u 2 n + u 2 t + ξ 2 ) a R = a R + a R 2 u n + a R 3 u t + 2 ar 4 (u 2 n + u 2 t + ξ 2 ) a L = a L + a L 2 u n + a L 3 u t + 2 al 4 (u 2 n + u 2 t + ξ 2 ) a R = a R + a R 2 u n + a R 3 u t + 2 ar 4 (u 2 n + u 2 t + ξ 2 ) A = A + A 2 u n + A 3 u t + 2 A 4 (u 2 n + u 2 t + ξ 2 ) (25) Here u n and u t are the particle velocities normal and tangent to the cell edge, respectively. With the reconstructed values obtained in III.A, the equilibrium distribution functions to the left and right of the cell edge as well as their slopes can be determined g L ψ ci dξ = ρ L ci ρu L n ci ρu L t ci ρe L ci, g R ψ ci dξ = ρ R ci ρu R n ci ρu R t ci ρe R ci (26) 5 of 3

6 g L a L ψ ci dξ = (ρ L ci ρl cc)/ s (ρu L n ci ρu L n cc )/ s (ρu L t ci ρu L t cc )/ s (ρe L ci ρel cc)/ s, g R a R ψ ci dξ = (ρ R cc ρ R ci )/ s (ρu R n cc ρu R n ci )/ s (ρu R t cc ρu R t ci )/ s (ρe R cc ρe R ci )/ s (27) where the subscript ci corresponds to the reconstructed values at the cell edge, the subscript cc refers to the cell center values, s is the distance from the cell edge to the cell center and ψ ci stands for the vector of moments at the cell edge u ψ ci = n (28) u t 2 (u2 n + u 2 t + ξ 2 ) Considering Eq. (24), all the parameters in g L and g R can be uniquely determined in Eq. (26). Once g L and g R are obtained, the slopes a L and a R can be computed from Eq. (27). After determining f, the equilibrium state g can be found using the compatibility condition g ψ ci dξ = g L ψ ci dξ + g R ψ ci dξ (29) u> from which the values of ρ, U n, U t and λ in g are determined. Then the slopes a L and a R can be obtained through the relations (ρ ρ L cc)/ s (ρ R cc ρ g a L (ρu ψ ci dξ = n ρun L )/ s cc )/ s (ρu t ρut L cc )/ s, g a R (ρu ψ ci dξ = n cc ρu n )/ s (ρu R t cc ρu t )/ s (3) (ρe ρecc)/ s L (ρecc R ρe )/ s Up to this point, the equilibrium states to the left, to the right and in the middle of the cell edge as well as the corresponding spatial slopes are determined. The only unknown term is the time slope term A. Since both f and g contain the time slope A, the conservation constraint (Eq. (2)) at the cell edge ci can be applied and integrated over the time step t t (f g) ψ ci dξ dt =, (3) from which A can be calculated. Substituting Eqs. (22) and (23) into Eq. (8), the final gas distribution function at the cell edge ci is expressed as u< f(x nci, t, u, v, ξ) = [ exp ( t/τ)] g +{τ [ + exp ( t/τ)] + t exp ( t/τ)} [a L H(u) + a R ( H(u))] u g +τ [t/τ + exp ( t/τ)] A g +exp ( t/τ) [( u t a L ) H(u) g L + ( u t a R ) ( H(u)) g L ] (32) where H(u) is the Heaviside function H(u) = {, if u <, if u >. (33) The local numerical fluxes for the mass, momentum and total energy across the cell edge ci can then be computed as F ρ F ρun F ρut F ρe = t u n f xnci ψ ci dξ dt (34) The fluxes at the cell faces are first computed and the total flux for both triangular and quadrilateral cells is then obtained by adding the flux contributions. 6 of 3

7 IV. Parallel Processing Parallel processing is based on domain decomposition. The unstructured hybrid grid is partitioned using METIS software package. METIS needs the graph file for the unstructured hybrid mesh, which is actually the neighbor connectivity of the cells. A sample unstructured hybrid mesh and the corresponding graph is given in Fig. 4. The first line in the graph represents the number of cells, number of nodes, weighting option and number of weights, respectively. The remaining lines show the number of faces and the indices of the neighbors of the corresponding cell. Partitioning of the graph is performed using kmetis program. During the partitioning, each cell is weighted by its number of edges so that each partition has about the same number of total edges to improve the load balancing in parallel computations. Figure 4. Sample computational mesh and the corresponding graph. Parallel Virtual Machine (PVM) message-passing library routines are employed in a master-worker algorithm. The master process performs all the input-output, starts up pvm, spawns worker processes and sends the initial data to the workers. The worker processes first receive the initial data, apply the interface and the flow boundary conditions, and solve the flow field within the partition. The flow variables at the interface boundaries are exchanged among the neighboring partitions at each time step for the implementation of inter-partition boundary conditions. V. Results and Discussions In order to show the accuracy and robustness of the present approach, the gas-kinetic BGK scheme is applied to a laminar flow over a flat plate and a transonic viscous flow over an airfoil. The results are compared to the analytical solutions and available experimental data. A. Case : Laminar Flow over a Flat Plate A flat plate at zero angle of attack is selected as a first test case for which the analytical Blasius solution is available. The freestream conditions and the computational mesh for this case are given in Table and Fig. 5, respectively. Table. Freestream conditions for Case. Mach Number Angle of Attack, deg Pressure, psia Temperature, R The above conditions correspond to a Reynolds number of 2. The flat plate is placed between and and the boundary layer region is meshed with quadrilateral cells while triangular cells are used outside the boundary layer. Fig. 6 shows the velocity vectors inside the boundary layer on the flat plate and the 7 of 3

8 .8.6 y x Figure 5. Computational mesh for Case (zoomed). y.6 Mach x Figure 6. Mach number contours and velocity vectors (zoomed). 8 of 3

9 8 7 6 Exact x =.25 x =.4 x =.55 5 η u/u Figure 7. u-velocity profile in the boundary layer. comparison of u-velocity profile at three different locations with the Blasius solution is given in Fig. 7. The results compare quite well with the exact Blasius solution. It should be noted that the gas-kinetic BGK method produces Navier-Stokes solution in smooth flow regions as expected. Moreover, the use of quadrilateral cells near the wall region leads to the better resolution of the boundary layer profile while the use of triangular cells outside reduce the overall number of cells. The laminar flow solution given above verifies the accuracy and robustness of the present approach. B. Case 2: Transonic Flow over an Airfoil RAE 2822 transonic airfoil, which has available experimental data, is selected as a second test case. The freestream conditions for this case are given in Table 2. The computational mesh and the partitions generated using the METIS software is presented in Fig. 8. Table 2. Freestream condition for Case 2. Mach Number Angle of Attack, deg Pressure, psia Temperature, R Fig. 9 shows the Mach number contours obtained from the gas-kinetic BGK scheme on the unstructured hybrid grid. As shown, the flow solution is smooth and the shock is captured well. The comparison of the surface pressure distribution with the experimental data is given in Fig.. As seen, the shock location and the strength compare quite well and the overall pressure distribution is in excellent agreement. In Fig., the present viscous solution is compared to the inviscid Euler solution. The viscous effects are clearly seen in the prediction of shock location and the pressure distribution downstream of the shock. The parallel computations are performed on an Itanium cluster running on Linux. Dual Itanium2 processors operate at.3ghz with 3MB L2 cache and 2GB of memory for each. The parallel efficiency of 9 of 3

10 Figure 8. Computational mesh and partitions for Case 2 (zoomed)..2 y Mach x Figure 9. Mach number contours. of 3

11 - Pressure Coefficient Experiment Gas-Kinetic BGK x Figure. Pressure distribution on the airfoil surface (every other three data point is displayed). - Pressure Coefficient Inviscid Solution Viscous Solution x Figure. Viscous versus inviscid solution (every other three data point is displayed). of 3

12 Table 3. Parallel efficiency of computations. Number of Nodes Computational Efficiency, sec/iter the computations is given in Table 3 and Fig. 2. It is observed that the high parallel efficiency of the gaskinetic BGK scheme is maintained as the number of processors is increased, which is attributed to the high computing to communication ratio. Although the gas-kinetic BGK schemes are computationally expensive and require longer computational time than classical schemes in serial computations, 6 it may not be an issue in parallel computations Ideal Gas-Kinetic BGK Speed Up Number of Processors Figure 2. Computational speed up. VI. Conclusion In this paper, the gas-kinetic BGK scheme on unstructured hybrid grids are presented. The high-order finite volume formulations are given. The solutions are obtained in parallel and the results are compared to the available analytical/experimental data. The unstructured hybrid grids used in the present study removes the difficulties faced in viscous flow computations with triangular grids. The viscous flow solutions with the gas-kinetic BGK scheme on unstructured hybrid grids agree well with analytical/experimental data. In addition, the parallel computations improve the computational efficiency of the gas-kinetic BGK schemes significantly. 2 of 3

13 Acknowledgments The authors acknowledge the support of the Defense Industries Research and Development Institute (TUBITAK-SAGE) under project SAM. References Pullin, D. I., Direct Simulation Methods for Compressible Inviscid Ideal Gas Flow, J. Comp. Phys., Vol. 34, 98, pp Mandal, J. C., and Deshpande, S. M., Kinetic Flux Vector Splitting for Euler Equations, Comp. Fluids, No. 23-2, 994, p Chou, S. Y., and Baganoff, D., Kinetic Flux Vector Splitting for the Navier-Stokes Equations, J. Comp. Phys., Vol. 3, 997, pp Prendergast, K. H., and Xu, K., Numerical Hydrodynamics from Gas-Kinetic Theory, J. Comp. Phys., Vol. 9, 993, pp Xu, K., Martinelli, L., and Jameson, A., Gas-Kinetic Finite Volume Methods, Flux Vector Splitting and Artificial Diffusion, J. Comp. Phys., Vol. 2, 995, pp Xu, K., and Jameson, A., Gas-Kinetic Relaxation (BGK-Type) Schemes for the Compressible Euler Equations, AIAA Paper , Xu, K., BGK-Based Scheme for Multicomponent Flow Calculations, J. Comp. Phys., Vol. 34, 997, pp Xu, K., A Gas-Kinetic Scheme for the Euler Equations with Heat Transfer, SIAM J. Sci. Comp., Vol. 2-4, 997, pp Xu, K., and Hu, J., Projection Dynamics in Godunov-Type Schemes, J. Comp. Phys., Vol. 42, 998, pp Lian, Y. S., Xu, K., A Gas-Kinetic Scheme for Multimaterial Flows and Its Application in Chemical Reactions, J. Comp. Phys., Vol. 63, 2, pp Chae D., Kim C., and Rho O., Development of an Improved Gas-Kinetic BGK Scheme for Inviscid and Viscous Flows, J. Comp. Phys., Vol. 58, 2, pp Xu, K., A Gas-Kinetic BGK Scheme for the Navier-Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method, J. Comp. Phys., Vol. 7, 2, pp Kim, C., and Jameson, A., A Robust and Accurate LED-BGK Solver on Unstructured Adaptive Meshes, J. Comp. Phys., Vol. 43, 998, pp May, G., and Jameson, A., Improved Gaskinetic Multigrid Method for Three-Dimensional Computation of Viscous Flows, AIAA Paper 25-56, May, G., Srinivasan, B., and Jameson, A., Three Dimensional Flow Calculations on Arbitrary Meshes Using a Gas- Kinetic BGK Finite-Volume Method, AIAA Paper , Ilgaz, M., and Tuncer, I., H., Parallel Implementation of Gas-Kinetic Schemes for 2-D Flows on Unstructured Grids, 3rd Ankara International Aerospace Conference [CD-ROM], AIAC-25-8, Ankara, Turkey, Cercignani, C., The Boltzmann Equation and Its Applications, Springer-Verlag, Frink, N., T., Three-Dimensional Upwind Scheme for Solving the Euler Equations on Unstructured Tetrahedral Grids, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Mitchell, C. R., Improved Reconstruction Scheme for the Navier-Stokes Equations on Unstructured Meshes, AIAA Paper , Bhatnagar, P. L., Gross, E. P., and Krook, M., A Model for Collision Processes in Gases I: Small Amplitude Processes in Charged and Neutral One-Component Systems, Phys. Rev., Vol. 94, pp. 5, of 3