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Linear hydrodynamics and stability of the discrete velocity Boltzmann equations
Journal of Fluid Mechanics ( IF 3.6 ) Pub Date : 2020-06-17 , DOI: 10.1017/jfm.2020.374
P.-A. Masset , G. Wissocq

The discrete velocity Boltzmann equations (DVBE) underlie the attainable properties of all numerical lattice Boltzmann methods (LBM). To that regard, a thorough understanding of their intrinsic hydrodynamic limits and stability properties is mandatory. To achieve this, we propose an analytical study of the eigenvalues obtained by a von Neumann perturbative analysis. It is shown that the Knudsen number, naturally defined as a particular dimensionless wavenumber in the athermal case, is sufficient to expand rigorously the eigenvalues of the DVBE and other fluidic systems such as Euler, Navier–Stokes and all Burnett equations. These expansions are therefore compared directly to one another. With this methodology, the influences of the lattice closure and equilibrium on the hydrodynamic limits and Galilean invariance are pointed out for the D1Q3 and D1Q4 lattices, without any ansatz. An analytical study of multi-relaxation time (MRT) models warns us of the errors and instabilities associated with the choice of arbitrarily large ratios of relaxation frequencies. Importantly, the notion of the Knudsen–Shannon number is introduced to understand which physics can be solved by a given LBM numerical scheme. This number is also shown to drive the practical stability of MRT schemes. In the light of the proposed methodology, the meaning of the Chapman–Enskog expansion applied to the DVBE in the linear case is clarified.

中文翻译:

离散速度玻尔兹曼方程的线性流体动力学和稳定性

离散速度玻尔兹曼方程 (DVBE) 是所有数值晶格玻尔兹曼方法 (LBM) 可获得的特性的基础。在这方面,必须彻底了解它们的内在流体动力学限制和稳定性特性。为了实现这一点,我们建议对通过冯诺依曼微扰分析获得的特征值进行分析研究。结果表明,在无热情况下自然定义为特定无量纲波数的 Knudsen 数足以严格扩展 DVBE 和其他流体系统(如 Euler、Navier-Stokes 和所有 Burnett 方程)的特征值。因此,这些扩展直接相互比较。有了这个方法论,指出了晶格闭合和平衡对 D1Q3 和 D1Q4 晶格的流体动力学极限和伽利略不变性的影响,没有任何 ansatz。多松弛时间 (MRT) 模型的分析研究提醒我们注意与选择任意大松弛频率比率相关的错误和不稳定性。重要的是,引入了 Knudsen-Shannon 数的概念以了解给定 LBM 数值方案可以解决哪些物理问题。这个数字也显示出推动 MRT 计划的实际稳定性。根据所提出的方法,查普曼-恩斯科格展开式在线性情况下应用于 DVBE 的含义得到澄清。多松弛时间 (MRT) 模型的分析研究提醒我们注意与选择任意大松弛频率比率相关的错误和不稳定性。重要的是,引入了 Knudsen-Shannon 数的概念以了解给定 LBM 数值方案可以解决哪些物理问题。这个数字也显示出推动 MRT 计划的实际稳定性。根据所提出的方法,查普曼-恩斯科格展开式在线性情况下应用于 DVBE 的含义得到澄清。多松弛时间 (MRT) 模型的分析研究提醒我们注意与选择任意大松弛频率比率相关的错误和不稳定性。重要的是,引入了 Knudsen-Shannon 数的概念以了解给定 LBM 数值方案可以解决哪些物理问题。这个数字也显示出推动 MRT 计划的实际稳定性。根据所提出的方法,查普曼-恩斯科格展开式在线性情况下应用于 DVBE 的含义得到澄清。
更新日期:2020-06-17
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