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Extension of lattice Boltzmann flux solver for simulation of 3D viscous compressible flows. (English) Zbl 1443.65170
Summary: The lattice Boltzmann flux solver (LBFS), which was presented by Shu and his coworkers [L. M. Yang et al., Comput. Fluids 79, 190–199 (2013; Zbl 1284.76313); C. Z. Ji et al., Mod. Phys. Lett. B 23, No. 3, 313–316 (2009; Zbl 1419.76520); C. Shu et al., “Lattice Boltzmann flux solver: an efficient approach for numerical simulation of fluid flows”, Trans. Nanjing Univ. Aeronaut. Astronaut. 31, No. 1, 1–15 (2014)] for simulation of inviscid compressible flows, is extended to simulate 3D viscous compressible flows in this work. In the solver, the inviscid flux at the cell interface is evaluated by local reconstruction of one-dimensional lattice Boltzmann solution through the application of non-free parameter D1Q4 model to the Riemann problem, while the viscous flux is evaluated by conventional smooth function approximation. In the existing LBFS [Yang et al., loc. cit.; Ji et al., loc. cit.; Shu et al., loc. cit.], the distribution functions at the cell interface streamed from neighboring points are directly used to compute the inviscid flux, which contains superabundant numerical dissipation for simulation of viscous flows. In the present work, we start from the Chapman-Enskog analysis [Z. Guo and C. Shu, Lattice Boltzmann method and its applications in engineering. Hackensack, NJ: World Scientific (2013; Zbl 1278.76001)] and consider both the equilibrium part and non-equilibrium part of the distribution function at the cell interface. It is well known that the inviscid flux can be fully determined by the equilibrium part and the non-equilibrium part can be viewed as numerical dissipation for the calculation of inviscid flux. The drawback of the existing LBFS is removed by introducing a switch function which ranges from 0 to 1 in order to control the numerical dissipation. In the smooth region such as in boundary layer, the switch function takes a value close to zero, while around the strong shock wave, it tends to one. Through test cases with complex geometry, it has been demonstrated that the present solver can work very well for simulation of 3D viscous compressible flows.
MSC:
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76M28 Particle methods and lattice-gas methods
76N06 Compressible Navier-Stokes equations
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[1] Guo, Z. L.; Shu, C., Lattice Boltzmann Method and its Applications in Engineering (2013), World Scientific Publishing
[2] Hu, Y.; Li, D.; Shu, S.; Niu, X. D., Simulation of steady fluid-solid conjugate heat transfer problems via immersed boundary-lattice Boltzmann method, Comput. Math. Appl., 70, 2227-2237 (2015)
[3] Touil, H.; Ricot, D.; Leveque, E., Direct and large-eddy simulation of turbulent flows on composite multi-resolution grids by the lattice Boltzmann method, J. Comput. Phys., 256, 220-233 (2014) · Zbl 1349.76141
[4] Yuan, H. Z.; Niu, X. D.; Shu, S.; Li, M.; Yamaguchi, H., A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating a flexible filament in an incompressible flow, Comput. Math. Appl., 67, 1039-1056 (2014) · Zbl 1381.74085
[5] Qian, Y. H.; Humieres, D.; Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett., 17, 479-484 (1992) · Zbl 1116.76419
[6] Chen, S.; Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30, 329 (1998)
[7] Deng, L.; Zhang, Y.; Wen, Y.; Shan, B.; Zhou, H., A fractional-step thermal lattice Boltzmann model for high Peclet number flow, Comput. Math. Appl., 70, 1152-1161 (2015)
[8] Xi, H. W.; Peng, G. W.; Chou, S. H., Finite-volume lattice Boltzmann method, Phys. Rev. E, 59, 6202-6205 (1999)
[9] Shu, C.; Chew, Y. T.; Niu, X. D., Least-squares-based lattice Boltzmann method: a meshless approach for simulation of flows with complex geometry, Phys. Rev. E, 64, Article 045701 pp. (2001)
[10] Kataoka, T.; Tsutahara, M., Lattice Boltzmann method for the compressible Euler equations, Phys. Rev. E, 69, Article 056702 pp. (2004)
[11] Qu, K.; Shu, C.; Chew, Y. T., Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number, Phys. Rev. E, 75, Article 036706 pp. (2007)
[12] Qu, K.; Shu, C.; Chew, Y. T., Simulation of shock-wave propagation with finite volume lattice Boltzmann method, Internat. J. Modern Phys. C, 18, 447-454 (2007) · Zbl 1137.76463
[13] Li, Q.; He, Y. L.; Wang, Y.; Tao, W. Q., Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations, Phys. Rev. E, 76, Article 056705 pp. (2007)
[14] Yang, L. M.; Shu, C.; Wu, J., Development and comparative studies of three non-free parameter lattice Boltzmann models for simulation of compressible flows, Adv. Appl. Math. Mech., 4, 454-472 (2012)
[15] Yang, L. M.; Shu, C.; Wu, J., A moment conservation-based non-free parameter compressible lattice Boltzmann model and its application for flux evaluation at cell interface, Comput. & Fluids, 79, 190-199 (2013) · Zbl 1284.76313
[16] Kataoka, T.; Tsutahara, M., Lattice Boltzmann method for the compressible Navier-Stokes equations with flexible specific-heat ratio, Phys. Rev. E, 69, R035701 (2004)
[17] Ubertini, S.; Bella, G.; Succi, S., Lattice Boltzmann method on unstructured grids: Further developments, Phys. Rev. E, 68, Article 016701 pp. (2003)
[18] Stiebler, M.; Tölke, J.; Krafczyk, M., An upwind discretization scheme for the finite volume lattice Boltzmann method, Comput. & Fluids, 35, 814-819 (2006) · Zbl 1177.76329
[19] Ji, C. Z.; Shu, C.; Zhao, N., A lattice Boltzmann method-based flux solver and its application to solve shock tube problem, Modern Phys. Lett. B, 23, 313-316 (2009) · Zbl 1419.76520
[20] Shu, C.; Wang, Y.; Yang, L. M.; Wu, J., Lattice Boltzmann flux solver: An efficient approach for numerical simulation of fluid flows, Trans. Nanjing Univ. Aeronaut. Astronaut., 31, 1-15 (2014)
[21] Swanson, R. C.; Radespiel, R., Cell centered and cell vertex multigrid schemes for the Navier-Stokes equations, AIAA J., 29, 697-703 (1991)
[22] Blazek, J., Computation Fluid Dynamics: Principle and Application (2001), Elsevier · Zbl 0995.76001
[23] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066
[24] Einfeldt, B., On Godunov-Type methods for gas dynamics, SIAM J. Numer. Anal., 25, 294-318 (1988) · Zbl 0642.76088
[25] van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136 (1979) · Zbl 1364.65223
[26] van Albada, G. D.; van Leer, B.; Roberts, W. W., A comparative study of computational methods in cosmic gas dynamics, Astron. Atrophys., 108, 76-84 (1982) · Zbl 0492.76117
[27] Yang, L. M.; Shu, C.; Wu, J., A three-dimensional explicit sphere function-based gas-kinetic flux solver for simulation of inviscid compressible flows, J. Comput. Phys., 295, 322-339 (2015) · Zbl 1349.76751
[28] Yoon, S.; Jameson, A., Lower-upper Symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations, AIAA J., 26, 1025-1026 (1988)
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