Liu Z J, Shu C, Chen S Y, Liu W, Yuan Z Y, Yang L M
Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260.
Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China.
Phys Rev E. 2022 Apr;105(4-2):045302. doi: 10.1103/PhysRevE.105.045302.
In this work, the explicit formulations of the Grad's distribution function for the 45 moments (G45)-based gas kinetic scheme (GKS) are presented. Similar to the G13 function-based gas kinetic scheme (G13-GKS), G45-GKS simulates flows from the continuum regime to the rarefied regime by solving the macroscopic governing equations based on the conservation laws, which are widely used in conventional Navier-Stokes solver. These macroscopic governing equations are discretized by the finite volume method, where the numerical fluxes are evaluated by the local solution to the Boltzmann equation. The initial distribution function is reconstructed by the G45 distribution function, which is a higher order truncation of the Hermite expansion of distribution function compared with the G13 distribution function. Such high order truncation of Hermite expansion helps the present solver to achieve a better accuracy than G13-GKS. Moreover, the reconstruction of distribution function makes the development of explicit formulations of numerical fluxes feasible, and the evolution of the distribution function, which is the main reason why the discrete velocity method is expensive, is avoided. Several numerical experiments are performed to examine the accuracy of G45-GKS. Results show that the accuracy of the present solver for almost all flow problems is much better than G13-GKS. Moreover, some typical rarefied effects, such as the direction of heat flux without temperature gradients and thermal creep flow, can be well captured by the present solver.
在这项工作中，给出了基于45矩（G45）的气体动理学格式（GKS）的Grad分布函数的显式表达式。与基于G13函数的气体动理学格式（G13 - GKS）类似，G45 - GKS通过求解基于守恒定律的宏观控制方程来模拟从连续介质区域到稀薄区域的流动，这些守恒定律在传统的Navier - Stokes求解器中广泛使用。这些宏观控制方程通过有限体积法离散化，其中数值通量通过玻尔兹曼方程的局部解来评估。初始分布函数由G45分布函数重构，与G13分布函数相比，G45分布函数是分布函数的Hermite展开的高阶截断。Hermite展开的这种高阶截断有助于本求解器比G13 - GKS获得更好的精度。此外，分布函数的重构使得数值通量显式表达式的开发可行，并且避免了分布函数的演化，而分布函数的演化是离散速度方法计算成本高的主要原因。进行了几个数值实验来检验G45 - GKS的精度。结果表明，本求解器对于几乎所有流动问题的精度都比G13 - GKS好得多。此外，本求解器能够很好地捕捉一些典型的稀薄效应，如无温度梯度时的热流方向和热蠕变流。
