网格与高精度差分计算问题

PROBLEMS ABOUT GIRD AND HIGH ORDER SCHEMES

研究NS方程差分求解时来流雷诺数、计算格式精度和计算网格之间的关系．给出了判定空间三个方向上的粘性贡献在给定雷诺数、格式精度和网格下是否能够正确计入的估计方法．指出在NS方程的二阶差分方法的数值模拟中，由于物面法向采用了压缩网格技术，物面附近的网格间距很小，该方向上的粘性贡献可被计入．但是如果流向和周向的网格较粗，相应的差分方程中的粘性贡献可能落入截断误差相同的量级，因此在精度上等于仍是求解略去流向和周向粘性项的薄层近似方程．指出，高阶精度的差分计算格式，可以避免对网格要求苛刻的困难．并进一步讨论了建立高阶精度格式的问题，提出了建立高阶精度格式应该满足的原则：耗散控制原则以及色散控制原则．为了避免激波附近可能出现的微小非物理振荡，建议发展混合高阶精度格式，即在激波区，采用网格自适应的NND格式，在激波以外的区域，采用按上述原则发展的高阶格式．

In this paper, the relation between the difference scheme and grid system is studied for solving Navier-Stokes equations with given Reynolds number. Only if this relation is satisfied in the directions x, y, z respectively, the Navier-Stokes equations can be simulated properly. In many references solving full Navier-Stokes equations with second order difference scheme, the grids in the direction z normal to the wall airs fine enough since the clustering grid technique is used. The proposed relation is satisfied in this direction. However, the grids along the circumferential and main flow directions are not. Then the calculated viscous terms in these two directions will have the magnitude in the same order as the truncation error of the difference equations. In this case,it seems to solve full Navier-Stokes equations, in fact it only equivalent to solving the thin-layer approximation equations.From the criteria of grid interval, less grid points are needed when a higher order difference scheme is used. Then this paper discussed further how to establish a higher order difference scheme.The principles are proposed to construct high order schemes for solving Navier-Stokes equations,which are related to suppressing non-physical oscillations, maintaining computational stability and capturing the shock wave narrowly and sharply. Based on these principles, hybrid schemes are presented, which are NND schemes in the shock wave region and higher order difference schemes established according to the above principles in the whole region except the shocks.