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. 2016 Mar;93(3):033311. doi: 10.1103/PhysRevE.93.033311. Epub 2016 Mar 23.
In this work, a discrete gas-kinetic scheme (DGKS) is presented for simulation of two-dimensional viscous incompressible and compressible flows. This scheme is developed from the circular function-based GKS, which was recently proposed by Shu and his co-workers [L. M. Yang, C. Shu, and J. Wu, J. Comput. Phys. 274, 611 (2014)]. For the circular function-based GKS, the integrals for conservation forms of moments in the infinity domain for the Maxwellian function-based GKS are simplified to those integrals along the circle. As a result, the explicit formulations of conservative variables and fluxes are derived. However, these explicit formulations of circular function-based GKS for viscous flows are still complicated, which may not be easy for the application by new users. By using certain discrete points to represent the circle in the phase velocity space, the complicated formulations can be replaced by a simple solution process. The basic requirement is that the conservation forms of moments for the circular function-based GKS can be accurately satisfied by weighted summation of distribution functions at discrete points. In this work, it is shown that integral quadrature by four discrete points on the circle, which forms the D2Q4 discrete velocity model, can exactly match the integrals. Numerical results showed that the present scheme can provide accurate numerical results for incompressible and compressible viscous flows with roughly the same computational cost as that needed by the Roe scheme.
在这项工作中，提出了一种用于模拟二维粘性不可压缩和可压缩流动的离散气体动力学方法（DGKS）。该方法是从 Shu 及其同事最近提出的基于圆函数的 GKS 发展而来的[L. M. Yang, C. Shu, and J. Wu, J. Comput. Phys. 274, 611 (2014)]。对于基于圆函数的 GKS，基于 Maxwellian 函数的 GKS 的无穷域中矩的守恒形式的积分简化为沿着圆的积分。结果，导出了守恒变量和通量的显式公式。然而，对于粘性流动，基于圆函数的 GKS 的这些显式公式仍然很复杂，对于新用户的应用可能不容易。通过使用某些离散点来表示相速度空间中的圆，可以用简单的求解过程代替复杂的公式。基本要求是基于圆函数的 GKS 的矩守恒形式可以通过在离散点处对分布函数进行加权求和来准确满足。在这项工作中，表明在圆上的四个离散点进行积分求积，形成 D2Q4 离散速度模型，可以精确匹配积分。数值结果表明，该方案可以为不可压缩和可压缩粘性流动提供准确的数值结果，其计算成本与 Roe 方案大致相同。
