PROBLEMS ABOUT GIRD AND HIGH ORDER SCHEMES
网格与高精度差分计算问题

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.