Yang L M, Shu C, Chen Z, Liu Y Y, Wu J, Shen X
Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street, Nanjing 210016, Jiangsu, China.
Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260.
Phys Rev E. 2021 Jul;104(1-2):015305. doi: 10.1103/PhysRevE.104.015305.
In this work, a high-order gas kinetic flux solver (GKFS) is developed for simulation of two-dimensional (2D) compressible flows. Different from the conventional gas kinetic scheme, which uses the local integral solution to the Boltzmann equation to reconstruct the numerical fluxes of macroscopic governing equations, the GKFS evaluates the numerical fluxes by the local asymptotic solution to the Boltzmann equation. This local asymptotic solution consists of the equilibrium distribution function and its substantial derivative at the cell interface. To achieve high-order accuracy in the simulation, the substantial derivative is discretized by a difference scheme with second-order accuracy in time and fourth-order accuracy in space, which results in a polynomial of the equilibrium distribution function at different locations and time levels. The Taylor series expansion is then introduced to simplify this polynomial. As a result, a simple high-order accurate local asymptotic solution to the Boltzmann equation is obtained and the numerical fluxes of macroscopic governing equations are given explicitly. A series of numerical examples are presented to validate the accuracy and capability of the developed high-order GKFS. Numerical results demonstrate that the high-order GKFS can achieve the desired accuracy on both the quadrilateral mesh and the triangular mesh and it outperforms the second-order counterpart.
在这项工作中，开发了一种高阶气体动力学通量求解器（GKFS），用于模拟二维（2D）可压缩流动。与传统气体动力学格式不同，传统格式使用玻尔兹曼方程的局部积分解来重构宏观控制方程的数值通量，而GKFS通过玻尔兹曼方程的局部渐近解来评估数值通量。这种局部渐近解由平衡分布函数及其在单元界面处的实质导数组成。为了在模拟中实现高阶精度，实质导数通过时间上二阶精度和空间上四阶精度的差分格式进行离散，这导致了平衡分布函数在不同位置和时间层的多项式。然后引入泰勒级数展开来简化这个多项式。结果，得到了一个简单的玻尔兹曼方程高阶精确局部渐近解，并明确给出了宏观控制方程的数值通量。给出了一系列数值例子来验证所开发的高阶GKFS的精度和性能。数值结果表明，高阶GKFS在四边形网格和三角形网格上都能达到所需的精度，并且优于二阶格式。
