A Fourth-Order Gas-Kinetic CPR Method for the Navier-Stokes Equations on Unstructured Meshes

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1 Tenth International Conference on Computational Fluid Dnamics (ICCFD), Barcelona,Spain, Jul 9-, 8 ICCFD-8- A Fourth-Order Gas-Kinetic CPR Method for the Navier-Stokes Equations on Unstructured Meshes Chao Zhang and Qibing Li Corresponding author: lqb@tsinghua.edu.cn AML, Department of Engineering Mechanics, Tsinghua Universit, Beijing 84, China. Abstract: A fourth-order accurate gas-kinetic CPR method is developed for the Navier-Stokes equations on triangular meshes. Different from the previous single-stage third-order gas-kinetic CPR method, the current method adopts the second-order gas-kinetic flu solver, which is simpler and less epensive, to reduce the computational cost of flu. Meanwhile, the temporal accurac is improved b using the two-stage temporal discretization, which is more efficient than the four-stage Runge-Kutta method usuall adopted in eisting fourth-order CPR. Numerical tests are presented to demonstrate the accurac and efficienc of the current method. Kewords: Gas-Kinetic Scheme, Correction Procedure via Reconstruction, Two-stage Temporal Discretization. Introduction During the past decades, high-order CFD methods have attracted man researchers due to their advantages on accurac and efficienc. There are man popular high-order methods, such as the Discontinuous Galerkin (DG) method, the P N P M and so on. In recent ears, the correction procedure via reconstruction (CPR) method has been developed rapidl, which provides a unified framework for man high-order methods []. The CPR method has been shown to be ver competitive in terms of accurac and efficienc, compared to the original DG method. Most of high-order methods mainl concentrate on the high-order reconstruction. As for the flu evolution, the Riemann solution of the Euler equations is the foundation of compressible flu solvers. However, developing a genuinel multidimensional Riemann solution is ver difficult. For solving viscous flows, the viscous terms in the Navier-Stokes equations require special treatment. Alternativel, the gas-kinetic scheme (GKS) offers a different wa to recover the N-S solutions, which has shown good performance in a wide range of flow problems. Based on the local integral solution of the BGK model, the time evolving kinetic flu function can be constructed, in which the inviscid and viscous terms are coupled and obtained simultaneousl. Through the high-order epansion of the gas distribution function, a high-order gas evolution model has been constructed successfull, which provides more abundant information to describe the flu evolution process []. Accordingl, a series of high-order gas-kinetic schemes have been developed. Recentl, a third-order gas-kinetic CPR method, denoted as CPR-GKS(P), has been developed for the Navier-Stokes equations on unstructured meshes []. It combines the high-efficient CPR framework with the third-order single-stage time stepping gas-kinetic flu solver, which shows high accurac and efficienc in man tpical flow problems. However, the third-order gas-kinetic flu solver is more complicated and time-consuming than its second-order counterpart. In this stud the fourth-order CPR framework is adopted to enhance the spatial accurac. The secondorder time-dependent gas-kinetic flu solver [4] is adopted for less computational cost, combined with the two-stage time-stepping method [5, 6] to improve the temporal accurac. In comparison with the four-stage Runge-Kutta time-stepping method, the temporal discretization is more efficient due to less intermediate stages. The developed scheme, denoted as CPR-GKS(P) is then validated through tpical numerical tests.

2 Fourth-Order Gas-Kinetic CPR Method. CPR framework B dividing the computational domain into N non-overlapping triangular cells Ω i, a solution polnomial Q i of degree k is constructed in each cell b using the Lagrange interpolation based on a set of solution points (SPs). For fourth-order CPR, i.e., k =, solution points are needed in each cell. Additionall, to compute the common flu, k + flu points (FPs) are set at each cell interface. For efficienc, the solution points are chosen the same as the flu points, as shown in Fig.. Figure. Solution points (circles) and flu points (squares) for k=. Then the CPR framework can be epressed as Q i,j t + Π j ( F ) (Q i ) + Ω i s Ω i l= 4 α j,s,l [F n ] s,l Γ s =, () where Q i,j is the conservative variable at each solution point, j =. The second term is the projection of the flu divergence onto P k. The last one is the correction term where [F n ] s,l is the normal flu difference at each flu point, i.e., [F n com(q i, Q i+, n) F n (Q i )] s,l. α j,s,l is the lifting coefficient, Ω i is the area of Ω i, Γ s is the length of triangular edge. More details of the flu divergence term and correction term can be found in [].. Gas-kinetic scheme In the present stud, the flu in the CPR framework is computed b the gas-kinetic flu solver. For completeness, the gas-kinetic scheme is briefl reviewed. For convenience, the summation convention is adopted in the following, where = (, ) = (, ), u = (u, u ) = (u, v), macroscopic velocit vector U = (U, U ) = (U, V ). The scheme is based on the BGK equation, f t + u f i = g f, () i τ where f is the gas distribution function, u i is the particle velocit, and τ = µ/p is the particle collision time related to the viscosit and p is the pressure. The equilibrium state g is the Mawellian distribution g = ρ ( ) K+ λ e λ( u U +ξ ), () π where ρ is the densit. λ is equal to /(RT ), where R is the gas constant and T is the temperature. Since mass, momentum, and energ are conserved during particle collisions, the collision term in Eq.() satisfies the compatibilit condition g f ψdξ =, (4) τ

3 where ψ is the the vector of moments ψ = (, u, ( u + ξ )) T, (5) and dξ = du du dξ dξ dξ K is the element of the phase space and ξ = ξ + ξ ξk. The total number of degrees of freedom K is equal to (5 γ)/(γ ) + for two-dimensional flow, in which γ is the specific heat ratio. The relations between macroscopic variables and the distribution function are Q = fψdξ, F = ufψdξ. (6) Taking moments of BGK equation, the N-S equations can be recovered based on the first-order Chapman- Enskog epansion f NS = g τ( g t + u g i ). (7) i In order to compute the flu at each solution point and flu point, the gas distribution function is constructed locall at these points. Since the flow distribution inside each cell is continuous, Eq.(7) can be directl adopted for computing the flu at each solution point (SP). The corresponding gas distribution function can be epressed as f SP (t, u, ξ) =g ( τa i u i + (t τ)a). (8) B introducing the notation = g ( )ψdξ, (9) the coefficients a i and A can be determined b the spatial derivatives of conservative variables and the compatibilit condition [] a i = Q i a i, a i u i + A = A. () To compute the common flu at each flu point (FP) where the discontinuit eists, the gas distribution function is constructed based on the local analtical solution of BGK equation, f(, t, u, ξ) = τ t g(, t, u, ξ)e (t t )/τ dt + e t/τ f ( ut,, u, ξ), () where f is the piece-wise discontinuous initial distribution function at the beginning of each time step (t = ), = u(t t ) is the particle trajector. Note that the construction is performed in a local coordinate sstem = (, ), in which the -ais is perpendicular to the cell interface. Based on Eq.(), the second-order time-dependent gas distribution function can be epressed as ( f F P (t, ũ, ) ) ( ξ) =g e t/τ + g ((t ) + τ)e t/τ τ a i ũ i + g t τ + τe t/τ A + e t/τ g R ( (τ + t)a R i ũ i τa R) H(u) () + e t/τ g L ( (τ + t)a L i ũ i τa L) ( H(u)), where H(u) is the Heaviside function. The equilibrium states g L, g R correspond to the flow states at the left and right sides of the cell interface. The coefficients a i, A, a L i, AL and a R i, AR come from derivatives of g, g L and g R respectivel, which can be determined b Eq.() as well. Besides, the equilibrium state g is

4 determined b the compatibilit condition g ψdξ = Q = u > g L ψdξ + u < g R ψdξ, () where Q are the conservative variables corresponding the equilibrium state g. In order to full determine g, a continuous flow distribution across the cell interface is constructed b using the least-square method, with the stencil shown in Fig.. Figure. The stencil for constructing the equilibrium state in the local coordinate.. Two-stage temporal discretization Based on the above gas-kinetic flu solver, the solution at each solution point inside a cell can be updated b adopting the following two-stage temporal discretization, ( t Q i,j = Q n i,j Π j ( ) Q n+ i,j = Q n i,j Π j ( F )) n i, t)dt Ω i Ω i s Ω i ) α j,s,l (ˆF l s Ω i s,l ( t α j,s,l l Γ s, ) ˆF (Q n i, t)dt Γ s, s,l where t = + t/ is the intermediate stage, ˆF (Q n i, t) indicates the normal flu difference in Eq.(), and F can be epressed as 8 t F = n i, t)dt tn+ n i, t)dt + 4 t + t i, t)dt 8 tn+ t i, t)dt, (5) t and ˆF has the same form as Eq.(5) b replacing F with ˆF. It is not eas to construct a fourth-order gas-kinetic flu solver with one-step time discretization as the corresponding gas distribution function is complicated and the computational cost is far more epensive. Fortunatel, with the above two-step time discretization, the second-order gas-kinetic flu solver can be used, which onl takes about one-sith the computational cost of its third-order counterpart. Therefore, although the current method computes the flu pointwisel and has two stages, the total cost of flu is at the same level as the single-stage third-order gas-kinetic CPR, however, both spatial and temporal accurac are improved. Furthermore, the current temporal discretization is more efficient than the fourth-order Runge-Kutta method due to less intermediate stages. As a result, although the gas-kinetic flu solver is more epensive than traditional flu solvers, the current scheme can achieve nearl the same efficienc as the original fourth-order CPR. What s more, when simulating high-speed compressible viscous flows, the slope limiting procedure is necessar to capture flow discontinuities, which takes a large part of computational time. Therefore, it can be epected that the current scheme can be more efficient than the original CPR in compressible viscous flows. (4) 4

5 Numerical Results. Compressible Couette flow In order to verif the accurac of the current scheme in viscous flows, the compressible Couette flow is simulated. The traditional CPR is also tested b using the HLLC and BR scheme with the fourth-stage Runge-Kutta method, denoted as CPR-BR(P). The lower wall with = is stationar and adiabatic. The upper wall at = is moving with a constant speed U and temperature T =. There is an analtical solution for this stead flow, i.e., ( + Pr γ ) ( Ma H = U + Pr γ Ma U ( ) ) U, U U U T V =, = + Pr γ ( ) ) (6) U Ma (, p = /γ, T U where the Prandtl number is Pr =, the Mach number is Ma =.5. The viscosit is determined b the linear law µ = µ T/T. The Renolds number is Re = 5. The CFL number to compute time steps is set as. in all test cases Figure. Computational mesh for compressible Couette flow. - CPR-GKS (P) CPR-BR (P) -5 4th order rd order - CPR-GKS (P) CPR-BR (P) -5 L error -7-9 L error Mesh size CPU time Figure 4. Temperature error vs. mesh size (left) and CPU time (right) for the compressible Couette flow. The coarsest mesh used in this case is shown in Fig.. The temperature error vs. mesh size and CPU time are presented in figure 4, in which the CPU time is the computational cost to achieve a stead state when the temperature residual less than 5. It shows that CPR-GKS(P) can achieve the fourth-order accurac. To achieve the same level of absolute error, CPR-GKS(P) is more efficient than CPR-GKS(P) and has nearl the same efficienc as CPR-BR(P), demonstrating the high accurac and efficienc of the current scheme. Nevertheless, further tests are necessar, especiall for supersonic flows, to evaluate the 5

6 robustness of the current scheme. It can be anticipated that, with less intermediate stages, CPR-GKS(P) can be more efficient than the CPR-BR(P) in supersonic flows when the slope limiting is included which has a considerable computational cost.. Lid-driven cavit flow The second case is the lid-driven cavit flow problem. The top boundar of the unit square [, ] [, ] is a moving plate at a speed U =. The Mach number is set as Ma =.5. The Renolds number is Re =. The non-slip and isothermal boundar conditions are imposed on all boundaries with temperature T w =. The initial flow is stationar with densit ρ = and tempture T = T w. As presented in Fig.5, the mesh contains elements with the minimum mesh size h = Figure 5. Computational mesh (left) and streamlines (right) in lid-driven cavit flow. The streamlines are also shown in Fig.5, in which the flow structure, including the primar and secondar vortices, are well captured. The results of U-velocities along the vertical centerline and V -velocities along the horizontal centerline are shown in Fig.6. It can be observed thaumerical results obtained b both CPR-GKS (P) and CPR-BR (P) match ver well with the benchmark data. CPR-GKS (P) is more accurate than with such a coarse mesh, which also demonstrates the high accurac and efficienc of the current scheme...8 CPR-GKS (P) CPR-BR (P) Ghia.6.4. CPR-GKS (P) CPR-BR (P) Ghia U.4 V Figure 6. U-velocities along the vertical centerline and V -velocities along the horizontal centerline with Re= in the lid-driven cavit flow. 6

7 4 Conclusion In this stud, based on the second-order gas-kinetic flu solver and the two-stage temporal discretization, a fourth-order gas-kinetic CPR method is developed for the Navier-Stokes equations on unstructured meshes. Compared with the previous third-order gas-kinetic CPR method, the second-order gas-kinetic flu solver is adopted to reduce the computational cost of flu evaluations, while the temporal accurac can be improved b using the two-stage temporal discretization. Numerical tests show that the current scheme is more efficient than its third-order counterpart and achieves nearl the same efficienc as the original CPR method. Further investigations is required to solve high-speed flows where a more complicated reconstruction with slope limiters is needed to capture flow discontinuities, which takes much more CPU time. Accordingl, it can be epected that, with less intermediate stages, the current scheme can be more efficient than the original CPR method with the four-stage Runge-Kutta method. 5 Acknowledgments This work is supported b the NSFC (6758), National Ke Basic Research and Development Program (4CB744) and Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) (U55). References [] Z.J. Wang and H.Y. Gao. A unifing lifting collocation penalt formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mied grids. J. Comput. Phs., 8:86 886, 9. [] Q.B. Li, K. Xu, and S. Fu. A high-order gas-kinetic Navier-Stokes flow solver. J. Comput. Phs., 9:675 67,. [] C. Zhang, Q.B. Li, S. Fu, and Z.J. Wang. A third-order gas-kinetic CPR method for the Euler and NavierĺCStokes equations on triangular meshes. J. Comput. Phs., 6:9 5, 8. [4] K. Xu. A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phs., 7:89 5,. [5] J. Li and Z. Du. A two-stage fourth order time-accurate discretization for La-Wendroff tpe flow solvers, I: hperbolic conservation laws. SIAM J. Sci. Comput., 8:A46 A69., 6. [6] L. Pan, K. Xu, Q.B. Li, and J. Li. An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations. J. Comput. Phs., 6:97, 6. 7