Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations
Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations
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DOI：
10.1103/physreve.100.033305
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发表时间：
2019-09-10
影响因子：
2.4
通讯作者：
Latt, Jonas
中科院分区：
文献类型：
--
作者：
Coreixas, Christophe;Chopard, Bastien;Latt, Jonas
在过去的几十年里，已经提出了几种类型的碰撞模型来扩展格子玻尔兹曼方法（LBM）的有效域，每种模型都以自己的形式介绍。本文提出了一种形式主义，在一个共同的数学框架内描述所有这些方法，并以这种方式使我们能够绘制它们之间的直接联系。在这里，重点放在单一和多弛豫时间碰撞模型的原始时刻，中心时刻，累积量，或正规化的形式。与此同时，几个基地（非正交的，正交的，埃尔米特）被认为是人口的多项式扩展。首先推导出矩之间的一般关系，以了解矩空间如何相互关联。此外，碰撞模型的审查进一步揭示了碰撞模型，可以改写成线性矩阵的形式。通过比较碰撞后种群的显式表达式，进行了更多的定量数学研究。由于这一点，可以推导出多项式基（原始，Hermite，中心，中心Hermite，累积量）和正则化步骤对等温LBM的影响。广泛的结果提供了D1Q3，D2Q9，和D3Q27晶格，后者被进一步扩展到D3Q19速度离散化。最常见的两个和多弛豫时间碰撞模型的链接也提供了完整性的缘故。这项工作结束时强调的重要性，一个准确的表示的平衡状态，独立的时刻空间的选择。除了本文的理论目的之外，还提供了一般说明，以帮助读者实现最复杂的碰撞模型。
Over the last decades, several types of collision models have been proposed to extend the validity domain of the lattice Boltzmann method (LBM), each of them being introduced in its own formalism. This article proposes a formalism that describes all these methods within a common mathematical framework, and in this way allows us to draw direct links between them. Here, the focus is put on single and multirelaxation time collision models in either their raw moment, central moment, cumulant, or regularized form. In parallel with that, several bases (nonorthogonal, orthogonal, Hermite) are considered for the polynomial expansion of populations. General relationships between moments are first derived to understand how moment spaces are related to each other. In addition, a review of collision models further sheds light on collision models that can be rewritten in a linear matrix form. More quantitative mathematical studies are then carried out by comparing explicit expressions for the post-collision populations. Thanks to this, it is possible to deduce the impact of both the polynomial basis (raw, Hermite, central, central Hermite, cumulant) and the inclusion of regularization steps on isothermal LBMs. Extensive results are provided for the D1Q3, D2Q9, and D3Q27 lattices, the latter being further extended to the D3Q19 velocity discretization. Links with the most common two and multirelaxation time collision models are also provided for the sake of completeness. This work ends by emphasizing the importance of an accurate representation of the equilibrium state, independently of the choice of moment space. As an addition to the theoretical purpose of this article, general instructions are provided to help the reader with the implementation of the most complicated collision models.