Chai Zhenhua, Yuan Xiaolei, Shi Baochang
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.
Institute of Interdisciplinary Research for Mathematics and Applied Science, Huazhong University of Science and Technology, Wuhan 430074, China.
Phys Rev E. 2023 Jul;108(1-2):015304. doi: 10.1103/PhysRevE.108.015304.
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 the multiple-relaxation-time lattice Boltzmann (MRT-LB) method [Chai and Shi, Phys. Rev. E 102, 023306 (2020)10.1103/PhysRevE.102.023306], where an equilibrium distribution function (EDF) [Lu et al., Philos. Trans. R. Soc. A 369, 2311 (2011)10.1098/rsta.2011.0022] on a rectangular lattice is utilized. The anisotropy of the lattice tensor on a rectangular lattice leads to anisotropy of the third-order moment of the EDF, which is inconsistent with the isotropy of the viscous stress tensor of the NSEs. To eliminate this inconsistency, we extend the relaxation matrix related to the dynamic and bulk viscosities. As a result, the macroscopic NSEs can be recovered from the RMRT-LB method through the direct Taylor expansion method. Whereas the rectangular lattice does not lead to the change of the zero-, first- and second-order moments of the EDF, the unified framework of the 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 (q discrete velocities in d-dimensional space, d≥1) lattice, including rD2Q9, rD3Q19, and rD3Q27 lattices, while there are no rectangular D3Q13 and D3Q15 lattices within this framework of the RMRT-LB method. Thanks to the block-lower triangular relaxation matrix introduced in the unified framework, the RMRT-LB versions (if existing) of the previous MRT-LB models can be obtained, including those based on raw (natural) moment, central moment, Hermite moment, and central Hermite moment. It is also found that when the parameter c_{s} is an adjustable parameter in the standard or rectangular lattice, the present RMRT-LB method becomes a kind of MRT-LB method for the NSEs and NCDE, and the commonly used MRT-LB models on the DdQq lattice are only its special cases. We also perform some numerical simulations, and the results show that the present RMRT-LB method can give accurate results and also have a good numerical stability.
在本文中，我们通过扩展我们最近的多松弛时间格子玻尔兹曼（MRT-LB）方法的统一框架[Chai和Shi，《物理评论E》102，023306（2020）10.1103/PhysRevE.102.023306]，为纳维-斯托克斯方程（NSEs）和非线性对流扩散方程（NCDE）开发了一种通用的矩形多松弛时间格子玻尔兹曼（RMRT-LB）方法，其中利用了矩形格子上的平衡分布函数（EDF）[Lu等人，《皇家学会哲学学报A》369，2311（2011）10.1098/rsta.2011.0022]。矩形格子上格子张量的各向异性导致EDF三阶矩的各向异性，这与NSEs粘性应力张量的各向同性不一致。为消除这种不一致性，我们扩展了与动力粘度和体粘度相关的松弛矩阵。结果，通过直接泰勒展开法可从RMRT-LB方法恢复宏观NSEs。虽然矩形格子不会导致EDF零阶、一阶和二阶矩的变化，但MRT-LB方法的统一框架可直接应用于NCDE。应当指出，NSEs的RMRT-LB模型可在rDdQq（d维空间中q个离散速度，d≥1）格子上推导得出，包括rD2Q9、rD3Q19和rD3Q27格子，而在RMRT-LB方法的这个框架内不存在矩形D3Q13和D3Q15格子。得益于统一框架中引入的块下三角松弛矩阵，可得到先前MRT-LB模型的RMRT-LB版本（如果存在），包括基于原始（自然）矩、中心矩、埃尔米特矩和中心埃尔米特矩的版本。还发现当参数(c_{s})在标准或矩形格子中为可调参数时，当前的RMRT-LB方法成为一种用于NSEs和NCDE的MRT-LB方法，并且DdQq格子上常用的MRT-LB模型只是其特殊情况。我们还进行了一些数值模拟，结果表明当前的RMRT-LB方法能给出准确结果且具有良好的数值稳定性。
