An upwind discretization scheme for the finite volume lattice Boltzmann method
An upwind discretization scheme for the finite volume lattice Boltzmann method
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DOI：
10.1016/j.compfluid.2005.09.002
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发表时间：
2006-09
影响因子：
2.8
通讯作者：
M. Stiebler;J. Tölke;M. Krafczyk
中科院分区：
文献类型：
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作者：
M. Stiebler;J. Tölke;M. Krafczyk
经典的格子Boltzmann方法仅限于笛卡尔网格，这一事实启发了一些研究人员将有限体积[Nannelli F，Succi S]格子Boltzmann方程应用于不规则格子。统计物理杂志1992；68：401-7；彭刚，奚红，邓肯C，周世华。非结构网格上格子Boltzmann方法的有限体积格式。物理出版社，E 1999；59：4675-82；Chen H.流体动力学格子Boltzmann方法的体积公式：基本概念。PHYS Rev E 1998；58：3955-63]或有限元[Lee T，Lin CL.离散Boltzmann方程的特征Galerkin方法。J Comp Phys 2001；171：336-56；石旭，林军，余正。三角形单元的间断Galerkin谱单元格子Boltzmann方法。InJ Numer方法流体2003；42：1249-61]离散Boltzmann方程的方法。有限体积法是由彭等人提出的。适用于非结构化栅格，因此允许增加几何灵活性。然而，与标准的LBE模型相比，该方法存在很大的数值不稳定性。该格式的计算效率与标准方法相比并不具有竞争力。与Peng等人描述的中心格式不同，我们提出了一种使用迎风格式离散对流算子的替代方法。我们将我们的方法应用于两个空间维度上的一些测试问题，证明了新格式的稳定性和计算效率的显著提高。与在分层网格上工作的格子Boltzmann求解器进行了比较，我们发现目前离散Boltzmann方程的有限体积方法还不能作为独立的流体求解器。
The fact that the classic lattice Boltzmann method is restricted to Cartesian Grids has inspired several researchers to apply Finite Volume [Nannelli F, Succi S. The lattice Boltzmann equation on irregular lattices. J Stat Phys 1992;68:401–7; Peng G, Xi H, Duncan C, Chou SH. Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys Rev E 1999;59:4675–82; Chen H. Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept. Phys Rev E 1998;58:3955–63] or Finite Element [Lee T, Lin CL. A characteristic Galerkin method for discrete Boltzmann equation. J Comp Phys 2001;171:336–56; Shi X, Lin J, Yu Z. Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Int J Numer Methods Fluids 2003;42:1249–61] methods to the Discrete Boltzmann equation. The finite volume method proposed by Peng et al. works on unstructured grids, thus allowing an increased geometrical flexibility. However, the method suffers from substantial numerical instability compared to the standard LBE models. The computational efficiency of the scheme is not competitive with standard methods. We propose an alternative way of discretizing the convection operator using an upwind scheme, as opposed to the central scheme described by Peng et al. We apply our method to some test problems in two spatial dimensions to demonstrate the improved stability of the new scheme and the significant improvement in computational efficiency. Comparisons with a lattice Boltzmann solver working on a hierarchical grid were done and we found that currently finite volume methods for the discrete Boltzmann equation are not yet competitive as stand alone fluid solvers.