边界层内WENO格式特征重构精度的理论分析

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.