Flux Limiter Lattice Boltzmann for Compressible Flows

Commun. Theor. Phys. 56 (2011 333 338 Vol. 56, No. 2, August 15, 2011 Flux Limiter Lattice Boltzmann for Compressible Flows CHEN Feng (íô, 1 XU Ai-Guo (ÆÇÁ, 2, ZHANG Guang-Cai (¾, 2 and LI Ying-Jun (Ó 1, 1 State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining and Technology (Beijing, Beijing 100083, China 2 National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-26, Beijing 100088, China (Received December 27, 2010; revised manuscript received May 11, 2011 Abstract In this paper, a new flux limiter scheme with the splitting technique is successfully incorporated into a multiple-relaxation-time lattice Boltzmann (LB model for shacked compressible flows. The proposed flux limiter scheme is efficient in decreasing the artificial oscillations and numerical diffusion around the interface. Due to the kinetic nature, some interface problems being difficult to handle at the macroscopic level can be modeled more naturally through the LB method. Numerical simulations for the Richtmyer Meshkov instability show that with the new model the computed interfaces are smoother and more consistent with physical analysis. The growth rates of bubble and spike present a satisfying agreement with the theoretical predictions and other numerical simulations. PACS numbers: 47.11.-j, 51.10.+y, 05.20.Dd Key words: lattice Boltzmann method, flux limiter technique, compressible flows, multiple-relaxation-time, Richtmyer Meshkov instability 1 Introduction In the past few decades, the lattice Boltzmann (LB method [1 2] has become a powerful tool for computational fluids. Due to the kinetic nature, some interface problems being difficult to handle at the macroscopic level can be modeled more naturally through the LB method. The starting point of LB method is the evolution equation for the distribution function f i, f i t + v f i iα = Ω i, (1 x α where v iα is the discrete particle velocity, Ω i is the collision operator. The equation (1 may be solved with an appropriate finite difference scheme defined on the lattice. Generally, the forward Euler method is used to compute the time derivative f i / t. Appropriate spatial difference scheme is essential to reduce spurious behaviour around the interface, and the choice requires more consideration. The upwind difference scheme leads to stable simulation, but it gives very smeared solution with excessive numerical diffusion; [3 4] the higher order schemes, such as Lax Wendroff or Beam Warming scheme, are more accurate, but the wiggle phenomenon introduces unphysical oscillations in the density profile. [5 6] One strategy to overcome these problems is the hybrid method, flux limiter scheme. [7 11] At discontinuities the flux limiter is close to 0, so the first order upwind scheme is applied; while for the smooth areas the flux limiter approaches 1, a second order or second order approximation term is used. The flux limiter technique has been introduced into the Single-Relaxation-Time (SRT LB simulations by Sofonea, et al. [7 8] They showed that such a scheme can successfully improve the numerical accuracy. However, to date, the flux limiter scheme has not been incorporated into the Multiple-Relaxation-Time (MRT LB method for compressible flows. In order to simulate the multiphase flows under shock wave action, in this work, we propose a flux limiter scheme with the splitting technique [12] for an MRT LB model. The rest of the paper is organized as follows: Details of our finite difference scheme based on the flux limiter and splitting techniques are given in Sec. 2. In Sec. 3 we present some simulation results with the new model, and compare the simulation results with the theoretical predictions. A brief conclusion is given in Sec. 4. 2 Numerical Schemes Recently, we proposed a new MRT Finite Difference (FD lattice Boltzmann model for compressible flows. [13 14] This MRT model has better numerical stability and accuracy than its single-relaxation-time counterpart. In this MRT model the evolution of distribution function f i is governed by the following equation: where f i t + v f i iα = M 1 il Ŝ lk ( x ˆf k α eq ˆf k, (2 Supported by the Science Foundation of Laboratory of Computational Physics, Science Foundation of China Academy of Engineering Physics under Grant Nos. 2009A0102005, 2009B0101012, National Basic Research Program of China under Grant No. 2007CB815105, National Natural Science Foundation of China under Grant Nos. 11074300, 11075021, and 11074303, and the Fundamental Research Funds for the Central Universities under Grant No. 2010YS03 Corresponding author, E-mail: Xu Aiguo@iapcm.ac.cn Corresponding author, E-mail: lyj@aphy.iphy.ac.cn c 2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

334 Communications in Theoretical Physics Vol. 56 cyc : (±1, 0, for 1 i 4, cyc : (±6, 0, for 5 i 8, (v i1, v i2 = 2(±1, ±1, for 9 i 12, 3 2 (±1, ±1, for 13 i 16. (3 Details of transformation matrix M and the equilibrium distribution function to the original publication. [13] ˆf eq i in the moment space are referred Fig. 1 Characteristic lines and corresponding projections in the x and y directions for the following distribution functions: (a is for f 1(x, t; (b is for f 9(x, t. In this paper, we propose a flux limiter scheme corresponding to the MRT model. The splitting technique is used in composing the flux limiter. Here, the splitting technique means that the one-dimensional flux limiter scheme is used in each direction (x or y for a multidimensional flux limiter scheme. Figure 1 shows the characteristic lines in the flux limiter scheme and corresponding projections in x and y directions for two distribution functions f 1 (x, t and f 9 (x, t, respectively. (J 1 x and (J 1 y are corresponding projections of node J 1 in the x and y directions. Let f n i,j be the value of the distribution function at time t at the node J along the direction i, we rewrite the evolution of f i at node J at time step t + dt as follows, where f n+1 i,j = fi,j n dt A i dx [F i,j+1/2 n x Fi,J 1/2 n x] dt A i dy [F i,j+1/2 n y Fi,J 1/2 n y] dtm 1 il Ŝ lk ( ˆf k,j n n,eq ˆf k,j, (4 1, for 1 i 4, 1/6, for 5 i 8, A i = 1/ (5 2, for 9 i 12, 2/3, for 13 i 16. Fi,J+1/2 n x (Fi,J 1/2 n x and Fi,J+1/2 n y (Fi,J 1/2 n y are x and y components of the outgoing (incoming flux at node J along the direction i, F n i,j+1/2 x = f n i (ix, iy + 1 2 ( A i dx F n i,j 1/2 x = f n i (ix A i v ix, iy + 1 2 Fi,J+1/2 n y = fi n (ix, iy + 1 ( 2 A i dy F n i,j 1/2 y = f n i (ix, iy A i v iy + 1 2 [f n i (ix + A iv ix, iy f n i (ix, iy]ψ x(ix, iy, (6a ( A i dx [f n i (ix, iy f n i (ix A i v ix, iy]ψ x (ix A i v ix, iy, (6b [f n i (ix, iy + A iv iy f n i (ix, iy]ψ y(ix, iy, (6c ( A i dy [f n i (ix, iy f n i (ix, iy A i v iy ]ψ y (ix, iy A i v iy. (6d The sufficient condition for the discrete space with second order accuracy and TVD property is that the limiter lies in the limited area, where the upper contour is the superbee limiter, and the lower contour is the minmod limiter. The

No. 2 Communications in Theoretical Physics 335 closer the chosen limiter to the upper contour, the smaller the dissipation caused by the limiter is, and the higher the resolution is, but the lower the stability and the convergence are. On the contrary, the closer the chosen limiter to the lower contour, the bigger the dissipation, and the lower the resolution, but the higher the stability and convergence. Based on an overall consideration of the resolution and stability, we adopt the Monitorized Central Difference (MCD flux limiter [15] in this paper. The flux limiters ψ x and ψ y are expressed as follows 0, θi n(ix, iy x 0, 2θi n ψ x (ix, iy = (ix, iy x, 0 θi n(ix, iy x 1 3, (1 + θi n (ix, iy 1 x/2, 3 θn i (ix, iy (7a x 3, 2, 3 θi n(ix, iy x, 0, θi n(ix, iy y 0, 2θi n ψ y (ix, iy = (ix, iy y, 0 θi n(ix, iy y 1 3, (1 + θi n (ix, iy 1 y/2, 3 θn i (ix, iy (7b y 3, 2, 3 θi n(ix, iy y, where the smoothness functions are θi n (ix, iy x = fn i (ix, iy fn i (ix A iv ix, iy fi n(ix + A iv ix, iy fi n, (ix, iy (8a θi n (ix, iy y = fn i (ix, iy fn i (ix, iy A iv iy fi n(ix, iy + A iv iy fi n. (ix, iy (8b The Lax Wendroff scheme is recovered for the flux limiter ψ x = ψ y = 1. The first order upwind scheme is recovered when ψ x = ψ y = 0. 3 Simulation Results The Richtmyer Meshkov (RM instability has been studied extensively in the linear and nonlinear regimes. Before going to LB simulations, we first summarize briefly some relevant results. For the model summary given below, it is useful to define some key quantities appearing in the models: a is the mixing layer amplitude, a b and a s are the bubble and spike amplitudes, k = 2π/λ is the wave number, λ is the initial perturbation wavelength, u is the velocity change across the interface, A 1 is the post-shock Atwood number, a 1 represents the post-shock initial perturbation amplitude. Zhang and Sohn [16] developed a model for the growth of RM unstable interface in the case of light-heavy transition. They proposed da dt = v 0 1 + k 2 v 0 a 1 t + max[0, (ka 1 2 (A 1 2 + 0.5](kv 0 t 2 (9a da b dt = da dt A 1 kv0t 2 1 + 2k 2 v 0 a 1 t + 4[(ka 1 2 + 1/3(1 A 2 1 ](kv 0t 2, (9b da s dt = da dt + A 1 kv0 2t 1 + 2k 2 v 0 a 1 t + 4[(ka 1 2 + 1/3(1 A 2 1 ](kv 0t 2, (9c for the overall growth rate and the growth rates of bubble and spike, respectively. Here v 0 = k ua 1 a 1. Goncharov [17] extended the two-dimensional Layzer model to the A 1 1 case and obtained Sadot et al. [18] proposed the growth rate of the bubble and spike, da b dt = 3 + A 1 3(1 + A 1 kt. (10 da b/s dt = (1 + kv 0 tv 0 1 + (1 ± A 1 kv 0 t + [(1 ± A 1 /(1 + A 1 ][(kv 0 t 2 /2πC], (11 where C = 1/(3π for A 1 0.5, C = 1/(2π for A 1 0, and the upper (+ and lower ( signs in ± denote the bubble and spike, respectively. Sohn [9] extended the Zufiria model to arbitrary Atwood numbers, and the bubble velocity is da b dt = [ 3 + A1 3(1 + A 1 1 q + 2A 1 3(1 + A 1 q 2 ] 1 kt, (12 where q = q(a 1 is the root of the cubic polynomial (3 A 1 q 3 (21 + 9A 1 q 2 + (3 + 15A 1 q 4A 1 = 0.

336 Communications in Theoretical Physics Vol. 56 Here, the simulation results with the new model are compared to the results from Lax Wendroff scheme simulations, and to the predictions of models mentioned above. The following is the initial condition of our simulations: An incident shock wave, with the Mach number 1.2, traveling from the left side, hits an interface with sinusoidal perturbation. The initial macroscopic quantities are as follows: (ρ, u 1, u 2, p l = (1.341 61, 0.361 538, 0, 1.513 32, (ρ, u 1, u 2, p m = (1, 0, 0, 1, (ρ, u 1, u 2, p r = (5.04, 0, 0, 1, (13 where ρ is the density, u 1 and u 2 are the x and y components of fluid velocity, p is pressure, and the subscripts l, m, r indicate the left, middle, right regions of the whole domain. The initial sinusoidal perturbation at the interface is: x = 0.25 Nx dx+0.008 cos(20πy, where the cycle of initial perturbation is 0.1, the amplitude is 0.008, Nx is grid number in the x direction, and dx is grid size. The following boundary conditions are imposed: (i inflow boundary is applied at the left side, (ii periodic boundary conditions are applied at the top and bottom boundaries, and (iii extrapolation technique is adopted at the right boundary. γ = 1.4 in the whole domain. Fig. 2 Comparison of the simulation results with flux limiter scheme and Lax Wendroff scheme at t = 0.1 and t = 0.3, respectively: dotted lines are for flux limiter scheme, solid lines are for Lax Wendroff scheme. Fig. 3 Comparison of the simulation results with flux limiter scheme and Lax Wendroff scheme: amplitude and growth rate changes with time. Figure 2 shows a comparison of the simulation results with flux limiter scheme and Lax Wendroff scheme at t = 0.1 and t = 0.3, respectively. The dotted lines correspond to simulation results with the flux limiter scheme, and solid lines represent the simulation results with Lax Wendroff scheme. The abscissa is for ix, and the vertical axis is for iy, where ix and iy are the indexes of lattice node in the x- and y-directions. The parameters are dx = dy = 0.001, dt = 10 5, s 5 = 10 3, and 10 5 for the other relaxation parameters. We adopt the same parameters in the following simulations. Figure 3 shows the changes of perturbation amplitudes and growth rates with time simulated with the two different schemes. From these figures, we can find that the flux limiter scheme is efficient in decreasing the oscillations, and the computed interfaces are more smooth and consistent with physical analysis. The smooth interface is beneficial to accurately

No. 2 Communications in Theoretical Physics 337 estimate the disturbance amplitude. Fig. 4 Snapshots of RM instability: density contours at t = 0, t = 0.05, t = 0.3, t = 0.5, t = 0.7, t = 1.1, respectively. From black to white the gray level corresponds to the increase of values. In Fig. 4, we show the density contours simulated with the flux limiter scheme at t = 0, t = 0.05, t = 0.3, t = 0.5, t = 0.7, t = 1.1, respectively. In the first snapshot corresponding to the initial condition (13, the gray levels on the left side of the interface disturbance are not obvious, because the density difference of these regions is too small compared with the value of right region. From the density distribution, we can clearly find the locations of the interface and transmission wave. At t = 0.7, the transmission wave has been out of the computational domain. In the last snapshot (t = 1.1, we also find a discontinuity near ix = 470, which is a reflected shock wave from the right boundary. It should be noted that the extrapolation technique is very difficult to ensure complete nonreflective outflow. But, since the reflection wave has not reach the location of interface, so the calculation results are still valid and impervious. How to construct an appropriate and complete non-reflective outflow boundary condition for this model is still an open problem for us. In the impact process, the heavy and light fluids gradually penetrate into each other, the light fluid rises to form a bubble and the heavy fluid falls to generate a spike. At time t = 1.1, the interface disturbance has obtained full development. At the top of the spike, due to the Kelvin Helmholtz instability, a mushroom shape is formed eventually. Fig. 5 Simulation results of flux limiter scheme: bubble and spike. (a, (b, (c, (d are the bubble amplitude, spike amplitude, bubble and spike growth rates, respectively. In Fig. 5, we present the amplitudes and growth rates of bubble and spike, where Figs. 5(a and 5(b describe the changes of bubble and spike amplitudes with time, Figs. 5(c and 5(d show the comparisons of bubble and spike growth rates with theoretical predictions, respectively. The location of the shocked, unperturbed interface is used as a reference for the determination of the bubble and spike amplitudes. The distance from the unperturbed interface to the bubble tip is the bubble amplitude a b, while the distance from the unperturbed interface to the spike tip is the spike amplitude a s. When the shock wave passes through the initial interface disturbance area, due

338 Communications in Theoretical Physics Vol. 56 to the compression, the interface produces a small deformation. The bubble and spike amplitudes reduce slightly, as shown in Figs. 5(a and 5(b. In Fig. 5(c, we compare our simulated bubble growth rate with the predictions of Zhang Sohn model (Eq. (9b, Goncharov model (Eq. (10, model by Sadot et al. (Eq. (11 and Sohn Zufiria model (Eq. (12. The bubble growth rates predicted by the model of Sadot et al. and the Sohn-Zufiria model are in very close agreement in the late time and are closest to the simulation bubble growth rate. The Goncharov model overpredicts the bubble growth rate, and the Zhang Sohn model underpredicts the bubble growth rate. In Figs. 5(d, we compare the simulated spike growth rate with the predictions of Zhang Sohn model (Eq. (11 and the model by Sadot et al. (Eq. (11. The model by Sadot et al. overpredicts the spike growth rate, and the Zhang Sohn model underpredicts the spike growth rate. The agreement between the theories and the simulations for the bubble is better than that for the spike. And it is consistent with a comment proposed by Zhang Sohn, the prediction for the spike growth rate becomes less accurate than that for the bubble at later times. [20] The simulation results are also accordant with those by other numerical methods. [21] 4 Conclusions We propose a flux limiter scheme with splitting technique for a multiple-relaxation-time lattice Boltzmann method. The new model works for both high speed and low speed thermal compressible flows under shack wave reaction. It is shown that, the new scheme can efficiently decrease the unphysical oscillations and numerical diffusion around the interfaces. Some interface problems which are difficult to handle at the macroscopic level can be modeled more naturally in the LB method. The new model was applied to simulate a traditional difficult problem, the Richtmyer Meshkov instability. The obtained growth rates of bubble and spike are compared with predictions of the Zhang-Sohn model, Goncharov model, Sohn Zufiria model and the model by Sadot et al. The numerical results are shown to be in good agreement with those from analytic calculations and other macroscopic approaches. Since the modeling of LB method includes intrinsically non-equilibrium effects, it has more potential advantages than the macroscopic methods in studying the non-equilibrium behaviors of fluid instability problems, such as the Richtmyer Meshkov mixing. References [1] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University Press, New York (2001. [2] Ai-Guo Xu, G. Gonnella, and A. Lamura, Phys. Rev. E 67 (2003 056105; Phys. Rev. E 74 (2006 011505; Physica A 331 (2004 10; Physica A 344 (2004 750. [3] V. Sofonea and R.F. Sekerka, J. Comp. Phys. 184 (2003 422; Int. J. Mod. Phys. C 16 (2005 1075. [4] A. Cristea and V. Sofonea, Int. J. Mod. Phys. C 14 (2003 1251. [5] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer Verlag, Berlin (1999. [6] S. Teng, Y. Chen, and H. Ohashi, Int. J. Heat and Fluid Flow 21 (2000 112. [7] A. Cristea and V. Sofonea, Proceedings of the Romanian Academy, Series A 4 (2003 59; Central European J. Phys. 2 (2004 382. [8] V. Sofonea, A. Lamura, G. Gonnella, and A. Cristea, Phys. Rev. E 70 (2004 046702. [9] V. Sofonea, J. Comput. Phys. 228 (2009 6107. [10] G. Gonnella, A. Lamura, and V. Sofonea, Eur. Phys. J. Special Topics 171 (2009 181. [11] A. Cristea, Int. J. Mod. Phys. C 17 (2006 1191. [12] S.P. Spekreijse, Math. Comput. 49 (1987 135. [13] F. Chen, A.G. Xu, G.C. Zhang, Y.J. Li, and S. Succi, Europhys. Lett. 90 (2010 54003. [14] F. Chen, A.G. Xu, G.C. Zhang, and Y.J. Li, Commun. Theor. Phys. 55 (2011 325. [15] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Second Edition, Springer Verlag, Berlin (1999. [16] Q. Zhang and S. Sohn, Phys. Fluids 9 (1997 1106. [17] V.N. Goncharov, Phys. Rev. Lett. 88 (2002 134502. [18] O. Sadot, L. Erez, U. Alon, D. Oron, and L.A. Levin, Phys. Rev. Lett. 80 (1998 1654. [19] S. Sohn, Phys. Rev. E 67 (2003 026301. [20] Q. Zhang and S. Sohn, Appl. Math. Lett. 10 (1997 121. [21] M. Latini, O. Schilling, and W.S. Don, Phys. Fluids 19 (2007 024104.