用于Navier-Stokes和非线性对流扩散方程的矩形多重弛豫时间格子Boltzmann方法：一般平衡态及一些重要问题
Rectangular multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: General equilibrium and some important issues
AI总结：
本文扩展了统一框架，提出用于Navier-Stokes和非线性对流扩散方程的通用矩形多重弛豫时间格子Boltzmann方法，解决了矩形格子上三阶矩各向异性问题，并证明该方法可应用于多种格子结构及矩形式。
摘要：
在本文中，我们通过扩展我们最近提出的MRT-LB方法的统一框架[Z.H. Chai和B.C. Shi, Phys. Rev. E 102, 023306 (2020)]，开发了一种用于Navier-Stokes方程（NSEs）和非线性对流扩散方程（NCDE）的通用矩形多重弛豫时间格子Boltzmann（RMRT-LB）方法，其中使用了矩形格子上的矩形平衡分布函数（REDF）[J.H. Lu等人, Phil. Trans. R. Soc. A 369, 2311-2319 (2011)]。由于矩形格子上离散速度的各向异性，REDF的三阶矩与流行的LB方法不一致，因此先前MRT-LB方法的统一框架不能直接应用于在矩形rDdQq格子上使用REDF的NSEs。通过适当地选择与动力粘度和体积粘度相关的弛豫子矩阵$\mathbf{S}_2$，可以通过直接的泰勒展开方法从RMRT-LB方法中恢复出宏观NSEs。虽然矩形格子不会导致REDF的零阶、一阶和二阶矩发生变化，因此MRT-LB方法的统一框架可以直接应用于NCDE。需要注意的是，NSEs的RMRT-LB模型可以在rDdQq格子上推导出来，包括rD2Q9、rD3Q19和rD3Q27格子，并且可以获得先前MRT-LB模型的RMRT-LB版本（如果存在），包括分别基于原始（自然）矩、中心矩、Hermite矩和中心Hermite矩的模型。
英文摘要：
In this paper, we develop a general rectangular multiple-relaxation-time lattice Boltzmann (RMRT-LB) method for the Navier-Stokes equations (NSEs) and nonlinear convection-diffusion equation (NCDE) by extending our recent unified framework of MRT-LB method [Z.H. Chai and B.C. Shi, Phys. Rev. E 102, 023306 (2020)], where a rectangular equilibrium distribution function (REDF) [J.H. Lu et al, Phil. Trans. R. Soc. A 369, 2311-2319 (2011)] on a rectangular lattice is utilized. Due to the anisotropy of discrete velocities on a rectangular lattice, the third-order moment of REDF is inconsistent with that of popular LB method, and thus the previous unified framework of MRT-LB method cannot be directly applied to the NSEs using the REDF on the rectangular rDdQq lattice. The macroscopic NSEs can be recovered from the RMRT-LB method through the direct Taylor expansion method by properly selecting the relaxation sub-matrix $\mathbf{S}_2$ which is related to kinetic viscosity and bulk viscosity. While the rectangular lattice does not lead to the change of the zero-th, first and second-order moments of REDF, thus the unified framework of MRT-LB method can be directly applied to the NCDE. It should be noted that the RMRT-LB model for NSEs can be derived on the rDdQq lattice, including rD2Q9, rD3Q19, and rD3Q27 lattices, and the RMRT-LB versions (if existed) of the previous MRT-LB models can be obtained, including those based on raw (natural)-moment, central-moment, Hermite-moment, and central Hermite-moment, respectively.
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