边界层内WENO格式特征重构精度的理论分析

WENO格式因其高精度以及良好的激波捕捉能力而被广泛地用于可压缩湍流的数值模拟过程中。为了在有强间断的流场中仍然能保证数值稳定，通常将流动物理量投影到特征空间后再进行WENO重构。然而这种特征重构方法在边界层内的精度问题仍需要进一步分析与研究。通过可压缩边界层自相似解构造平板边界层内法向的流动参数，并在该方向上进行WENO重构，得到求解半点数值通量的WENO非线性权重。通过分析实际得到的权重与理想权重的偏差大小发现在边界层流动中特征重构的精度明显低于分量重构。为了更深入地分析特征重构精度降低的原因，通过对曲线坐标系下的流动方程进行理论分析，推导得到特征重构时的特征变量形式。根据特征变量表达式以及自相似解中权重误差分布发现引起特征重构精度降低的主要原因是特征投影过程在半点处出现了额外的极值点，同时这种因特征投影过程而产生极值的特性将不会随着左右特征矩阵的不同选取而发生改变。进一步地，基于理论分析的普适性，对于任何光滑流场，只要使用特征重构的投影过程，都会在半点处产生极值从而使得WENO系列格式产生精度降低的现象。

Theoretical analysis of reconstruction accuracy of characteristic-wise WENO scheme within boundary layer

Weighted Essentially Non-Oscillatory (WENO) scheme has been widely used in the numerical simulation of compressible turbulent flows due to its high accuracy and excellent shock-capture capability. In order to ensure numerical stability in flow fields with strong discontinuities, the physical quantities of the flow are usually projected into the characteristic space before the WENO reconstruction. In this paper, we construct the flow parameters normal to the plate boundary layer by compressible boundary layer self-similar solutions and perform WENO reconstruction in this direction to obtain the nonlinear weights for solving the half-point numerical fluxes. By analyzing the magnitude of deviation of the actual obtained weights from the ideal weights, it is found that the accuracy of the characteristic-wise reconstruction in the boundary layer flow is significantly lower than that of the component-wise reconstruction. In order to analyze more deeply the reasons for the reduced accuracy of the characteristic reconstruction, the form of the characteristic variables in the characteristic reconstruction process has been derived by theoretical analysis of the flow equations in the curved coordinate system. Based on the expressions of the characteristic variables and the weight error distribution in the norm direction, it is found that the main cause for the degradation of the accuracy is the appearance of additional extreme points at the half-point of the projection process, and this characteristic of extreme values due to the projection process will not change with the different selection of the left and right eigenmatrices. Further, based on the generality of the theoretical analysis, for any smooth flow field, as long as the projection process of characteristic reconstruction is used, the extreme points will cause the accuracy of WENO scheme.