气体动理学统一算法中相容性条件不满足引起的数值误差及其影响研究

求解玻尔兹曼(Boltzmann) 模型方程的气体动理学统一算法(unified gas kinetic scheme,UGKS) 是为模拟存在显著稀薄气体效应流动而建立的. 在该方法中,如果速度空间离散采用传统的离散速度坐标法(discreteordinate method,DOM),将会导致相容性条件得不到严格满足,从而引入数值误差. 本文从理论分析及数值试验两方面说明了该数值误差,正比于来流马赫数,反比于来流努森数. 引入了守恒型的离散速度坐标法(conservativediscrete ordinate method,CDOM),在离散层面上确保了相容性条件得到严格满足. 圆柱绕流计算结果表明,来流马赫数较高、努森数较小时,相容性条件满足与否对计算结果影响较大,采用CDOM 可以在较稀的速度空间网格上得到网格无关解,缩减计算量最大可达2/3.

STUDY ON THE NUMERICAL ERROR INTRODUCED BY DISSATISFYING THE CONSERVATION CONSTRAINT IN UGKS AND ITS EFFECTS

The Unified Gas Kinetic Scheme (UGKS) was developed by Xu Kun especially for simulating rarefied flows. In UGKS, the conservation constraint can not be satisfied with conventional Discrete Ordinate Method (DOM) and numerical error would be introduced. We found theoretically and numerical experimentally that the error was proportional to the freestream Mach number and inversely proportional to the freestream Knudsen number. Conservative discrete ordinate method (CDOM) was introduced with which the conservation constraint was satisfied at discrete level. The supersonic and hypersonic flows around a cylinder at different Knudsen numbers were simulated and results showed that whether the conservative constraint was satisfied would have a big effect on the variable distributions and the drag coefficients when the Mach number was high or the Knudsen number was low. With CDOM, the grid-free solution could be obtained with a relative coarse velocity grid, resulting in a maximum cost reduction of 2/3.