Volume 9, Issue 1
A Simplified Lattice Boltzmann Method Without Evolution of Distribution Function

Adv. Appl. Math. Mech., 9 (2017), pp. 1-22.

Published online: 2018-05

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• Abstract

In this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

• Keywords

Chapman-Enskog expansion analysis, lattice Boltzmann equation, Navier-Stokes equations, memory cost, stability.

76M28, 76P99

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@Article{AAMM-9-1, author = {Z. Chen , and C. Shu , and Y. Wang , and L. M. Yang , and D. Tan , }, title = {A Simplified Lattice Boltzmann Method Without Evolution of Distribution Function}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {1}, pages = {1--22}, abstract = {

In this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2016-0029}, url = {http://global-sci.org/intro/article_detail/aamm/12134.html} }
TY - JOUR T1 - A Simplified Lattice Boltzmann Method Without Evolution of Distribution Function AU - Z. Chen , AU - C. Shu , AU - Y. Wang , AU - L. M. Yang , AU - D. Tan , JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 1 EP - 22 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.OA-2016-0029 UR - https://global-sci.org/intro/article_detail/aamm/12134.html KW - Chapman-Enskog expansion analysis, lattice Boltzmann equation, Navier-Stokes equations, memory cost, stability. AB -

In this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

Z. Chen, C. Shu, Y. Wang, L. M. Yang & D. Tan. (2020). A Simplified Lattice Boltzmann Method Without Evolution of Distribution Function. Advances in Applied Mathematics and Mechanics. 9 (1). 1-22. doi:10.4208/aamm.OA-2016-0029
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