结构网格质量和差分格式误差的关系

Relationship Between Structured Mesh Quality and Error of Difference Scheme

网格生成所需人力资源和工作时间在整个流场数值模拟周期中占比较高,是计算流体力学应用软件的瓶颈。大量文献提供了结构网格质量特性对差分格式计算结果有影响的算例,但是很少给出正交性和光滑性影响误差的机理分析。分析了MUSCL和WENO差分格式在非均匀网格上产生的几何诱导误差,发现正交性和误差之间并无直接关联,有影响的几何参数是相邻网格的偏转角。理论推导证明,误差主要源自控制方程和差分格式,提高网格质量特性能够明显减小几何诱导误差,但是不能完全消除。近年来作者致力于改进算法,先后提出离散等价方程及其等价离散准则(discrete equivalence equation and its discrete rule,DEER)和非结构有限差分法。通过对自由流保持和线性流保持等算例的模拟,利用改进的算法在质量较差的网格上也能得到较好的计算结果。

The human resources and work time required for mesh generation are relatively high in the overall numerical simulation cycle of a flow field and represent a bottleneck in computational fluid dynamics applications.A large body of literature provides examples of calculations in which structured mesh quality has an impact on the results of difference scheme calculations,but few analyses are given of the mechanisms by which orthogonality and smoothness affect the errors.In this paper,the geometrically induced errors generated by the MUSCL and WENO difference schemes on non-uniform meshes were analysed,and it was found that there was no direct correlation between orthogonality and errors,and that the geometrical parameter that affects the errors was the deflection angle of the neighbouring grid points.The theoretical derivation proves that the error mainly originates from the governing equations and the difference schemes,and the improvement of the mesh quality can significantly reduce the geometrically induced error,but not completely eliminate it.In recent years,the algorithms were improved,and the discrete equivalent equations and their equivalent discretisation rule(DEER)and the unstructured finite difference methods were successively proposed.Through the simulations of free-stream preservation and linear-stream preservation cases,these improved algorithms can be used to obtain better computational results on poor quality meshes.