The use of gas-kinetic schemes for the simulation of compressible flows become widespread in the two last

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1 A Gas-Kinetic BGK Scheme for Parallel Solution of 3-D Viscous Flows 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 In this study, a gas-kinetic BGK scheme for the solution of 3-D viscous flows on unstructured hybrid grids is presented. The second order accurate finite volume formulations are given. In order to accurately resolve the boundary layers in wall bounded viscous flows, hexahedral grid cells are employed in the boundary layer regions normal to solid surfaces while the rest of the domain is discretized by tetrahedral cells. The computation time, which is an important deficiency of gas-kinetic BGK schemes, is improved by performing computations in parallel. The parallel algorithm for unstructured hybrid grids is based on the domain decomposition. Several numerical validation cases are presented to show the accuracy and robustness of the proposed approach. I. Introduction The use of gas-kinetic schemes for the simulation of compressible flows become widespread in the two last decades. Nowadays, the most promising ones are the Equilibrium Flux Method (EFM), the Kinetic Flux Vector Splitting (KFVS) 2, 3 and the Gas-Kinetic BGK method. 4, 5 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. 6 3 Although the gas-kinetic BGK scheme was first applied on structured grids, later, the scheme has been adapted on unstructured grids in order to simulate flows around complex bodies. 4 6 Recently, Ilgaz and Tuncer successfully applied the kinetic flux vector splitting and gas-kinetic BGK schemes for 2-D flows on unstructured grids. 7 In this work, the computation time, the basic deficiency of the gas-kinetic schemes, has been improved by performing computations in parallel. Various test problems were analyzed and it has been shown that the computational efficiency of the gas-kinetic schemes is superior and the gas-kinetic schemes do not experience numerical instabilities as their classical counterpart. 7 The authors further developed the gas-kinetic BGK scheme for 3-D inviscid flows on unstructured grids and applied to the missile flows. 8 Although inviscid flow solutions can successfully be obtained on unstructured grids, viscous flow solutions are more demanding. Wall-bounded viscous flow solutions require the resolution of boundary layers, which is quite difficult to achieve by employing only unstructured triangular/tetrahedral grids. It is well-known that high-resolution hybrid grids are needed for an efficient computation of viscous flows. The present authors have already developed a gas-kinetic BGK scheme for the parallel computation of 2-D viscous flows on unstructured hybrid grids, and showed the accuracy and robustness of the method. 9 In the present work, the method developed earlier is extended to the solution of 3-D viscous flows on unstructured hybrid grids. The second order accurate 3-D finite volume formulations are given and the solutions are obtained in parallel. Several viscous validation test cases on unstructured hybrid grids are presented to show the accuracy and robustness of the present approach. Chief Research Engineer, Ph.D., Aerodynamics Division, AIAA Member, milgaz@sage.tubitak.gov.tr Professor, Department of Aerospace Engineering, AIAA Member, tuncer@ae.metu.edu.tr of 3

2 II. Finite Volume Gas-Kinetic Method In gas-kinetic theory, gases are comprised of small particles and each particle has a mass and velocity. At standard conditions, the motion of large number of particles in a small volume is defined by the gas distribution function, which describes the probability of particles to be located in a certain velocity interval 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 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 is given in terms of 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. A. Gas-Kinetic BGK Scheme λ = 2 R T (7) 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 3-D can be written as (ignoring external forces) f t + u f x + v f y + w f z = g f τ where f is the gas distribution function, g is the equilibrium state approached by f over particle collision time τ; u, v and w are the particle velocities in x-, y- and z-directions, respectively. The equilibrium state is usually assumed to be a Maxwellian g = ρ ( λ π ) K+3 2 exp { λ [(u U) 2 + (v V ) 2 + (w W ) 2 + ξ 2 ]} (9) where ρ is the density, U, V and W are the macroscopic velocities in x-, y- and z-directions and K is the dimension of the internal velocities. The general solution of the gas distribution function f at the cell interface ci and time t is (8) f(s ci, t, u, v, w, ξ) = τ t g(s, t, u, v, w, ξ) exp [ (t t )/τ] dt +exp ( t/τ) f (s ci ut vt wt). () 2 of 3

3 Here s = s ci u (t t ) v (t t ) w (t t ) is the particle trajectory of a particle and f is the initial gas distribution function at the beginning of each time step. Since mass, momentum and energy are conserved during particle collisions, f and g must satisfy the conservation constraint of (f g) ψ dξ =, () at all (x, y, z) and t. III. Numerical Methodology The numerical methodology developed is based on the cell-centered finite volume formulation. The conservative flow variables at the cell boundaries are first reconstructed with a second-order accuracy. 2 The gas-kinetic BGK scheme is then used for the time evolution of the gas distribution function and evaluating the numerical fluxes through the cell faces. The macroscopic equations are integrated in time using a third-order accurate Runge-Kutta time-stepping scheme. A. Reconstruction of Flow Variables The second order reconstruction of conservative flow variables is achieved by expanding the cell-centered solution to each cell face with a Taylor series Q ci = Q cc + Q cc s + O( s 2 ) (2) where cc refers to the cell center and s is the vector from the cell center to the centroid of the cell face. The midpoint pointwise values of Q are first estimated by forming an arithmetic average of the two cells that share a common face and then the gradient at the cell center is calculated using the midpoint trapezoidal rule. 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. Flux Evaluation The numerical fluxes at the cell faces are calculated using a second order gas-kinetic BGK scheme. Figure shows the sample tetrahedral control volumes, cell interface and coordinate systems where dots in black represent cell centers, the dot in red shows the interface centroid, L and R the left and right states, x, y and z the local coordinate system normal and tangent to the cell interface ci, respectively, X, Y and Z the global coordinate system. U x, U y and U z are the macroscopic velocity components in the 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 x ci )], if x x ci (3) g R [ + a R (x x ci )], if x x ci and g = { g [ + a L (x x ci ) + A t], if x x ci (4) g [ + a R (x x ci ) + A t], if x x ci where g L, g R and g are the equilibrium distribution functions to the left, right and at the centroid of the cell interface ci, respectively, of the form g = ρ ( λ π ) K+3 2 exp { λ [(u x U x) 2 + (u y U y ) 2 + (u z U z ) 2 + ξ 2 ]} (5) 3 of 3

4 Figure. Sample tetrahedral control volumes, cell interface and coordinate systems. 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 x + a L 3 u y + a L 4 u z + 2 al 5 (u 2 x + u 2 y + u 2 z + ξ 2 ) a R = a R + a R 2 u x + a R 3 u y + a R 4 u z + 2 ar 5 (u 2 x + u 2 y + u 2 z + ξ 2 ) a L = a L + a L 2 u x + a L 3 u y + a L 4 u z + 2 al 5 (u 2 x + u 2 y + u 2 z + ξ 2 ) a R = a R + a R 2 u x + a R 3 u y + a R 4 u z + 2 ar 5 (u 2 x + u 2 y + u 2 z + ξ 2 ) A = A + A 2 u x + A 3 u y + A 4 u z + 2 A 5 (u 2 x + u 2 y + u 2 z + ξ 2 ). (6) Here u x, u y and u z are the local particle velocities normal and tangent to the cell interface, respectively. The equilibrium distribution functions to the left and right of the cell interface, and their slopes are obtained as g L a L ψ ci dξ = ρ L ci ρu L g L x ci ρu R ψ ci dξ = ρu y L ci, g R x ci ψ ci dξ = ρu R y ci ρuz L ci ρuz R ci ρeci L ρeci R (ρ L ci ρl cc (ρ R cc ρ R ci (ρux L ci ρux L cc (ρu (ρu y L ci ρuy L cc, g R a R x R cc ρux R ci ψ ci dξ = (ρu y R cc ρuy R ci (ρuz L ci ρuz L cc (ρuz R cc ρuz R ci (ρeci L ρel cc (ρecc R ρeci R where the subscript ci corresponds to the reconstructed values at the cell interface, the subscript cc refers to the cell center values, s is the distance from the cell interface to the cell center and ψ ci stands for the vector of moments at the cell interface u x ψ ci = u y (9) u z 2 (u2 x + u 2 y + u 2 z + ξ 2 ) The variables ρ, U x, U y, U z and λ in g L and g R (Eq. (5)) can be uniquely determined from Eq. (7). Once g L and g R are obtained, the slopes a L and a R can be computed from Eq. (8). ρ R ci (7) (8) 4 of 3

5 The equilibrium state g can be evaluated using the compatibility condition g ψ ci dξ = g L ψ ci dξ + g R ψ ci dξ (2) u> from which the values of ρ, U x, U y, U z and λ in g are determined. The slopes a L and a R can then be obtained through the relations g a L ψ ci dξ = (ρ ρ L cc (ρu x ρu L x cc (ρu y ρu L y cc (ρu z ρu L z cc (ρe ρe L cc, u< g a R ψ ci dξ = (ρ R cc ρ (ρu R x cc ρu x (ρu R y cc ρu y (ρu R z cc ρu z (ρe R cc ρe Having determined the equilibrium states to the left, to the right and at the centroid of the cell interface, and the corresponding spatial slopes, the conservation constraint (Eq. ()) at the cell interface ci can be applied and integrated over the time step t t (2) (f g) ψ ci dξ dt =, (22) to evaluate the time slope term A within f and g. Substituting Eqs. (3) and (4) into Eq. (), the final gas distribution function at the cell interface ci is expressed as f(x ci, t, u, v, w, 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 ] (23) where H(u) is the Heaviside function H(u) = {, if u <, if u >. (24) The local numerical fluxes for the mass, momentum and total energy across the cell interface ci can then be computed as F ρ F ρux F ρuy F ρuz F ρe The collision time in Eq. (23) is taken as 3 = t t u x f xci ψ ci dξ dt. (25) τ = µ + pl p R p p L t. (26) + pr Here the first term refers to the collision time related to the physical viscosity while the non-dimensional pressure jump in the second term corresponds to a numerical collision time. The second term is necessary especially in the under-resolved discontinuous regions and diminishes in smooth regions. The fluxes at the cell faces are first computed and the total flux for all types of cells is then obtained by adding the flux contributions. 5 of 3

6 C. Parallel Processing Parallel processing is based on domain decomposition. The unstructured hybrid grid is partitioned using METIS software package. METIS needs a graph file for the unstructured hybrid mesh, which is actually the neighbor connectivity of the cells. Partitioning of the graph is then performed using kmetis program. During the partitioning, each cell is weighted by its number of faces so that each partition has about the same number of total faces to improve the load balancing in parallel computations. 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. IV. Results and Discussions The gas-kinetic BGK scheme developed is validated through two viscous flow cases; laminar flow over a flat plate and a transonic flow over a wing. The parallel solutions, which significantly improve the computation time, are performed on an Itanium2 Linux cluster. Dual Itanium2 processors operate at.3ghz with 3MB L2 cache and 2GB of memory for each. The present solutions are compared to the analytical solutions and the numerical solutions available in the literature. A. Case : Laminar Flow Over a Flat Plate A flat plate at zero angle of attack is selected as a first validation case for which the analytical Blasius solution is available. The freestream conditions and the computational mesh with partitions for this case are given in Table and Fig. 2, respectively. The computational mesh consists of 43 nodes and 25 cells. The flat plate is placed between x = and x = locations and the boundary layer region is meshed with hexahedron cells while pyramidal and tetrahedron cells are used at the farfield. Table. Freestream conditions for Case. Mach Number Angle of Attack, deg Reynolds Number Figure 3 shows the Mach contours and velocity vectors inside the boundary layer on the flat plate at y = plane. The comparisons of u-velocity and v-velocity profiles at three different y-planes for two different x-locations are given in Figs. 4 and 5, respectively, along with the analytical Blasius solutions. The results of the gas-kinetic BGK scheme compare quite well with the Blasius solutions at different y-planes and x- locations. It is observed that the use of hexahedron cells near the wall region leads to the better resolution of the boundary layer while the use of tetrahedron cells outside reduces the number of cells in the viscous flow region. The parallel efficiency of the computations is given in Table 2 and Fig. 6. It is observed that the high parallel efficiency of the gas-kinetic BGK scheme is maintained as the number of processors increases, which is attributed to the high computing to communication ratio in the gas-kinetic BGK methods. Although the gas-kinetic BGK methods are computationally expensive, computational requirements may not be an issue in parallel computations. 6 of 3

7 Z Y X Figure 2. Computational mesh with 6 partitions for Case..2.9 z x Figure 3. Mach number contours and velocity vectors at y = plane for Case (zoomed view). 7 of 3

8 Blasius y=-.5 y= y=.5 5 Blasius y=-.5 y= y=.5 z/x..5 Re 4 3 z/x..5 Re u/u w/u. 2Re. x (a) u-velocity (b) v-velocity Figure 4. Velocity profiles in the boundary layer at x =.44 for Case Blasius y=-.5 y= y=.5 5 Blasius y=-.5 y= y=.5 z/x..5 Re 4 3 z/x..5 Re u/u w/u. 2Re. x (a) u-velocity (b) v-velocity Figure 5. Velocity profiles in the boundary layer at x =.66 for Case. 8 of 3

9 Table 2. Parallel efficiency of computations for Case. Number of Nodes CPU Time, sec/iter Computational Efficiency, % Ideal Gas-Kinetic BGK Efficiency 6 Speed Up Ideal Gas-Kinetic BGK # of Processors # of Processors (a) Efficiency (b) Speed up Figure 6. Parallel efficiency of computations for Case. B. Case 2: Laminar Flow Over a Transonic Wing The laminar flow over transonic wing is selected as a second validation case. A NACA2 airfoil section is used by extruding it in the spanwise direction, which has been used as a benchmark study for 2-D viscous flows. 22 The freestream conditions and the computational mesh with partitions are given in Table 3 and Fig. 7, respectively. The close-up view of the computational mesh near the wall region is also shown in Fig. 8. The computational mesh consists of 7668 nodes and cells. Table 3. Freestream conditions for Case 2. Mach Number Angle of Attack, deg Reynolds Number.8. 5 The Mach contours obtained from the gas-kinetic BGK scheme at y = is shown in Fig. 9a. The weak shock wave on the upper surface of the wing interacts with the boundary layer and causes the flow separation. The recirculating flow region is clearly seen with the streamlines in Fig. 9b. Recently, May et al. 23 studied the same problem with a different gas-kinetic schemes on 2-D unstructured grids. The surface pressure distribution at y = plane is presented in Fig. along with the results of May et al. 23 As seen, both predictions agree well. 9 of 3

10 Z Y X Figure 7. Computational mesh with 6 partitions for Case 2. Figure 8. Computational mesh near the wall region for Case 2 (zoomed view). of 3

11 Mach (a) Mach number contours (b) Streamlines and separation Figure 9. Mach number contours, streamlines and separation on the suction side for Case p/p May et al. [23] Gas-Kinetic BGK x Figure. Nondimensional pressure distribution on the wing surface at y = plane for Case 2. of 3

12 V. Conclusion In this paper, a gas-kinetic BGK scheme for 3-D viscous flows on unstructured hybrid grids are presented. The second order finite volume formulations are given and the solutions are obtained in parallel. The laminar flow solutions based on the gas-kinetic BGK scheme on unstructured hybrid grids agree well with the experimental data as well as available numerical studies. It is shown that the use of unstructured hybrid grids remove the difficulties faced in viscous flow computations on tetrahedral grids only. It also leads to the better resolution of boundary layers in viscous flows while reducing the total number of cells. In addition, it is shown that the high parallel efficiency of the gas-kinetic BGK scheme is maintained as the number of processors increases. Thus, the parallel computations significantly improve the computation time of the gaskinetic BGK scheme which, in turn, enable the method to be used for the solution of practical aerodynamic flow problems. 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, pp , Mandal, J. C., and Deshpande, S. M., Kinetic Flux Vector Splitting for Euler Equations, Comp. Fluids, Vol. 23-2, p.447, Chou, S. Y., and Baganoff, D., Kinetic Flux Vector Splitting for the Navier-Stokes Equations, J. Comp. Phys., Vol. 3, pp.27-23, Prendergast, K. H., and Xu, K., Numerical Hydrodynamics from Gas-Kinetic Theory, J. Comp. Phys., Vol. 9, pp.53-66, Prendergast, K. H., and Xu, K., Numerical Navier-Stokes Solutions from Gas-Kinetic Theory, J. Comp. Phys., Vol. 4, pp.9-7, Xu, K., and Jameson, A., Gas-Kinetic Relaxation (BGK-Type) Schemes for the Compressible Euler Equations, AIAA Paper , Xu, K., Martinelli, L., and Jameson, A., Gas-Kinetic Finite Volume Methods, Flux Vector Splitting and Artificial Diffusion, J. Comp. Phys., Vol. 2, pp.48-65, Xu, K., BGK-Based Scheme for Multicomponent Flow Calculations, J. Comp. Phys., Vol. 34, pp.22-33, Xu, K., A Gas-Kinetic Scheme for the Euler Equations with Heat Transfer, SIAM J. Sci. Comp., Vol. 2-4, pp , 997. Xu, K., and Hu, J., Projection Dynamics in Godunov-Type Schemes, J. Comp. Phys., Vol. 42, pp , 998. 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, pp.-27, 2. 2 Lian, Y. S., Xu, K., A Gas-Kinetic Scheme for Multimaterial Flows and Its Application in Chemical Reactions, J. Comp. Phys., Vol. 63, pp , 2. 3 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, pp , 2. 4 Kim, C., and Jameson, A., A Robust and Accurate LED-BGK Solver on Unstructured Adaptive Meshes, J. Comp. Phys., Vol. 43, pp , May, G., Srinivasan, B., and Jameson, A., Three Dimensional Flow Calculations on Arbitrary Meshes Using a Gas- Kinetic BGK Finite-Volume Method, AIAA Paper , May, G., and Jameson, A., Improved Gaskinetic Multigrid Method for Three-Dimensional Computation of Viscous Flows, AIAA Paper 25-56, Ilgaz, M., and Tuncer, I., H., Parallel Implementation of Gas-Kinetic Schemes for 2-D Flows on Unstructured Grids, 3rd Ankara International Aerospace Conference, AIAC-25-8, Ankara, Turkey, Ilgaz, M., and Tuncer, I., H., Parallel Application of Gas-Kinetic BGK Method on Unstructured Grids for 3-D Inviscid Missile Flows, Parallel CFD Conference 27, May 2-24, Antalya, Turkey, Ilgaz, M., and Tuncer, I., H., Parallel Implementation of Gas-Kinetic BGK Scheme on Unstructured Hybrid Grids, 36th AIAA Fluid Dynamics Conference and Exhibit, AIAA , San Francisco, 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, Ilgaz, M., Gas-Kinetic Methods for 3-D Inviscid and Viscous Flow Solution on Unstructured/Hybrid Grids, Ph.D. Thesis, Middle East Technical University, Ankara, Turkey, INRIA, Gamm Workshop, December 46, 985, Nice, France, Numerical Simulation of Compressible Navier Stokes Equations-External 2D Flows Around a NACA2 Airfoil, Centre de Rocquefort, de Rennes et de Sophia-Antipolis, of 3

13 23 May, G., Srinivasan, B., and Jameson, A., An Improved Gas-Kinetic BGK Finite-Volume Method For Three-Dimensional Transonic Flow, J. Comp. Phys., Vol. 22, pp , of 3