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A moment conservation-based non-free parameter compressible lattice Boltzmann model and its application for flux evaluation at cell interface. (English) Zbl 1284.76313
Summary: Based on the idea of constructing equilibrium distribution functions directly from the conservation forms of moments, a platform for developing non-free parameter lattice Boltzmann models is presented in this work. It is found that the existing compressible lattice Boltzmann models such as D1Q4L2, D1Q5L2 and D1Q5 models can be derived by the platform. This paper goes further to determine the lattice velocities of a non-free parameter D1Q4 model by incorporating two additional higher order conservation forms of moments. Since the lattice velocities are determined physically rather than specified artificially, the non-free parameter D1Q4 model can be applied to simulate compressible flows with a wide range of Mach numbers. The developed non-free parameter D1Q4 model is then applied to a local Riemann problem at the cell interface to establish a new Riemann flux solver for the solution of Euler equations by the finite volume method (FVM). Some test problems, such as Sod shock tube, shock reflection, compressible flows around NACA0012 airfoil, hypersonic flows around a blunt body and double Mach reflection, are simulated to illustrate the capability of present solver. Numerical results show that the present non-free parameter D1Q4 model can provide accurate results with faster convergence rate.

76M28 Particle methods and lattice-gas methods
76N15 Gas dynamics (general theory)
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[1] Ginzburg, I., Truncation errors, exact and heuristic stability analysis of two-relaxation-times lattice Boltzmann schemes for anisotropic advection-diffusion equation, Commun Comput Phys, 11, 1439-1502, (2012) · Zbl 1373.76241
[2] Zhang, T.; Shi, B. C.; Chai, Z. H.; Rong, F. M., Lattice BGK model for incompressible axisymmetric flows, Commun Comput Phys, 11, 1569-1590, (2012) · Zbl 1373.76282
[3] Singh, S.; Krithivasan, S.; Karlin, I. V.; Succi, S.; Ansumali, S., Energy conserving lattice Boltzmann models for incompressible flow simulations, Commun Comput Phys, 13, 603-613, (2013)
[4] Swift, M. R.; Orlandini, E.; Osborn, W. R.; Yeomans, J. M., Lattice Boltzmann simulations of liquid-gas and binary fluid systems, Phys Rev E, 54, 5041-5052, (1996)
[5] He, X. Y.; Chen, S. Y.; Zhang, R. Y., A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability, J Comput Phys, 152, 642-663, (1999) · Zbl 0954.76076
[6] Luo, L. S., Theory of the lattice Boltzmann method: lattice Boltzmann models for nonideal gases, Phys Rev E, 62, 4982-4996, (2000)
[7] Inamuro, T.; Ogata, T.; Tajima, S.; Konishi, N., A lattice Boltzmann method for incompressible two-phase flows with large density differences, J Comput Phys, 198, 628-644, (2004) · Zbl 1116.76415
[8] Huang, H. B.; Krafczyk, M.; Lu, X. Y., Forcing term in single-phase and shan-Chen-type multiphase lattice Boltzmann models, Phys Rev E, 84, 046710, (2011)
[9] Srivastava, S.; Perlekar, P.; Biferale, L.; Sbragaglia, M.; ten Thije Boonkkamp, J. H.M.; Toschi, F., A study of fluid interfaces and moving contact lines using the lattice Boltzmann method, Commun Comput Phys, 13, 725-740, (2013)
[10] Guo, Z. L.; Zhao, T. S., Lattice Boltzmann model for incompressible flows through porous media, Phys Rev E, 66, 036304, (2002)
[11] Tang, G. H.; Tao, W. Q.; He, Y. L., Gas slippage effect on microscale porous flow using the lattice Boltzmann method, Phys Rev E, 72, 056301, (2005)
[12] Ladd, A. J.C.; Verberg, R., Lattice-Boltzmann simulations of particle-fluid suspensions, J Stat Phys, 104, 1191-1251, (2001) · Zbl 1046.76037
[13] Chen, S. Y.; Chen, H. D.; Martinez, D.; Matthaeus, W., Lattice Boltzmann model for simulation of magnetohydrodynamics, Phys Rev Lett, 67, 3776-3779, (1991)
[14] Chen, S.; Dawson, S. P.; Doolen, G. D.; Janecky, D. R.; Lawniczak, A., Lattice methods and their applications to reacting systems, Comput Chem Eng, 19, 617-646, (1995)
[15] Qian, Y. H., Simulating thermohydrodynamics with lattice BGK models, J Sci Comput, 8, 231-242, (1993) · Zbl 0783.76004
[16] Alexander, F. J.; Chen, S.; Sterling, J. D., Lattice Boltzmann thermohydrodynamics, Phys Rev E, 47, R2249-R2252, (1993)
[17] Chen, Y.; Ohashi, H.; Akiyama, M., Thermal lattice Bhatnagar-Gross-Krook model without nonlinear deviations in macrodynamic equations, Phys Rev E, 50, 2776-2783, (1994)
[18] Watari, M.; Tsutahara, M., Two-dimensional thermal model of the finite-difference lattice Boltzmann method with high spatial isotropy, Phys Rev E, 67, 036306, (2003)
[19] Watari, M.; Tsutahara, M., Possibility of constructing a multispeed Bhatnagar-Gross-Krook thermal model of the lattice Boltzmann method, Phys Rev E, 70, 016703, (2004)
[20] Hu, S. X.; Yan, G. W.; Shi, W. P., A lattice Boltzmann model for compressible perfect gas, Acta Mech Sin - PRC, 13, 218-226, (1997)
[21] Yan, G. W.; Chen, Y. S.; Hu, S. X., Simple lattice Boltzmann model for simulating flows with shock wave, Phys Rev E, 59, 454-459, (1999)
[22] Kataoka, T.; Tsutahara, M., Lattice Boltzmann method for the compressible Euler equations, Phys Rev E, 69, 056702, (2004)
[23] Kataoka, T.; Tsutahara, M., Lattice Boltzmann method for the compressible Navier-Stokes equations with flexible specific-heat ratio, Phys Rev E, 69, R035701, (2004)
[24] Sun, C. H., Lattice-Boltzmann models for high speed flows, Phys Rev E, 58, 7283-7287, (1998)
[25] Sun, C. H., Adaptive lattice Boltzmann model for compressible flows: viscous and conductive properties, Phys Rev E, 61, 2645-2653, (2000)
[26] Sun, C. H., Simulations of compressible flows with strong shocks by adaptive lattice Boltzmann model, J Comput Phys, 161, 70-84, (2000) · Zbl 0971.76074
[27] Sun, C. H.; Hsu, A. T., Three-dimensional lattice Boltzmann model for compressible flows, Phys Rev E, 68, 016303, (2003)
[28] 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, 036706, (2007)
[29] Qu, K.; Shu, C.; Chew, Y. T., Simulation of shock-wave propagation with finite volume lattice Boltzmann method, Int J Mod Phys C, 18, 447-454, (2007) · Zbl 1137.76463
[30] McNamara, G. R.; Garcia, A. L.; Alder, B. J., Stabilization of thermal lattice Boltzmann models, J Stat Phys, 81, 395-408, (1995) · Zbl 1106.82353
[31] Dellar, P. J., Two routes from the Boltzmann equation to compressible flow of polyatomic gases, Prog Comput Fluid Dyn, 8, 84-96, (2008) · Zbl 1187.76725
[32] 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)
[33] Xu K. Gas-kinetic schemes for unsteady compressible flow simulations. VKI for fluid dynamics lecture series; 1998.
[34] Xu, K., A gas-kinetic BGK scheme for the Navier-stocks equations and its connection with artificial dissipation and Godunov method, J Comput Phys, 171, 289-335, (2001) · Zbl 1058.76056
[35] Ji, C. Z.; Shu, C.; Zhao, N., A lattice Boltzmann method-based flux solver and its application to solve shock tube problem, Mod Phys Lett B, 23, 313-316, (2009) · Zbl 1419.76520
[36] Chen, H. D., Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept, Phys Rev, 58, 3955-3963, (1998)
[37] Ubertini, S.; Succi, S.; Bellal, G., Lattice Boltzmann schemes without coordinates, Philos Trans R Soc Lond A, 362, 1763-1771, (2004) · Zbl 1205.76222
[38] Mandal, J. C.; Deshpande, S. M., Kinetic flux vector spitting for Euler equations, Comput Fluids, 23, 447-478, (1994) · Zbl 0811.76047
[39] Coquel, F.; Helluy, P.; Schneider, J., Second-order entropy diminishing scheme for the Euler equations, Int J Numer Meth Fluids, 50, 1029-1061, (2006) · Zbl 1138.76416
[40] Chou, S. Y.; Baganoff, D., Kinetic flux-vector splitting for the Navier-Stokes equations, J Comput Phys, 130, 217-230, (1997) · Zbl 0873.76057
[41] Liu, S. H.; Xu, K., Entropy analysis of kinetic flux vector splitting schemes for the compressible Euler equations, Z Angew Math Phys, 52, 62-78, (2001) · Zbl 0989.76052
[42] Venkatakrishnan V. On the accuracy of limiters and convergence to steady state solutions. AIAA paper 93-0880; 1993.
[43] Venkatakrishnan, V., Convergence to steady-state solutions of the Euler equations on unstructured grids with limiters, J Comput Phys, 118, 120-130, (1995) · Zbl 0858.76058
[44] 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
[45] Stolcis, L.; Johnston, L. J., Solution of the Euler equations on unstructured grids for two-dimensional compressible flow, Aeronaut J, 94, 181-195, (1990)
[46] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J Comput Phys, 54, 115-173, (1984) · Zbl 0573.76057
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