气体动理学格式研究进展

介绍了近年来气体动理学格式(gas-kinetic scheme, GKS, 亦简称BGK 格式) 的主要研究进展, 重点是高阶精度动理学格式及适合从连续流到稀薄流全流域的统一动理学格式. 通过对速度分布函数的高阶展开和对初值的高阶重构, 构造了时间和空间均为三阶精度的气体动理学格式. 研究表明, 相比于传统的基于Riemann 解的高阶格式, 新格式不仅考虑了网格单元界面上物理量的高阶重构, 而且在初始场的演化阶段耦合了流体的对流和黏性扩散, 也能够保证解的高阶精度. 该研究为高精度计算流体力学(computatial uiddymamics, CFD) 格式的建立提供了一条新的途径. 通过分子离散速度空间直接求解Boltzmann 模型方程,在每个时间步长内将宏观量的更新和微观气体分布函数的更新紧密地耦合在一起, 建立了适合任意Knudsen(kn) 数的统一格式, 相比于已有的直接离散格式具有更高的求解效率. 最后, 本文还讨论了合理的物理模型对数值方法的重要性. 气体动理学方法的良好性能来自于Boltzmann 模型方程对计算网格单元界面上初始间断的时间演化的准确描述. 气体自由运动与碰撞过程的耦合是十分必要的. 通过分析数值激波层内的耗散机制,我们认识到采用Euler 方程的精确Riemann 解作为现代可压缩CFD 方法的基础具有根本的缺陷, 高马赫数下的激波失稳现象不可避免. 气体动理学格式为构造数值激波结构提供了一个重要的可供参考的物理机制.

PROGRESS IN GAS-KINETIC SCHEME

Recent progress in the development of the gas-kinetic scheme is reviewed in this article, with emphasis laid on the construction of high-order-accurate gas-kinetic flux function for the Navier-Stokes equations and the unified gas-kinetic scheme for flow simulations in the entire Knudsen number regimes. A third-orderaccurate gaskinetic scheme is presented through the high-order reconstruction of the initial data and the high-order gas evolution model of the gas distribution function. Different from traditional high-order schemes based on Riemann solution, the new scheme not only takes into account the high-order initial reconstruction at a cell interface, but also follows its time evolution, which ensures a high-order time accurate flux function. This study pioneers a new way to construct high accurate time-space coupling CFD method. The unified gas- kinetic scheme for arbitrary Knudsen number is developed by direct solving the Boltzmann model equation in the discrete velocity space, where the update of both macroscopic conservative variables and microscopic gas distribution function takes place simultaneously within a time step. The newly developed method is more efficient than existing DVMs, where the continuum flow limit can be easily obtained in the unified scheme due to its hydrodynamic scale part of the flux function. The importance of using a valid physical evolution model in the construction of a numerical method is also discussed. The good performance of gas-kinetic scheme comes mainly from its capability of capturing a rational gas evolution process from an initial discontinuity using gas-kinetic model. The coupling of particle free transport and collision plays an important role here. Through the analysis of dissipative mechanism inside a numerical shock layer, it is realized that the adoption of exact Riemann solution of the Euler equations as a foundation of modern compressible CFD methods has fundamental flaws, and the shock instability at high Mach number simulation is unavoidable. The gas-kinetic scheme follows a valid physical process in the construction of numerical shock structure.