CPR方法与高分辨率二阶格式的混合算法

Research on hybrid method based on CPR and high-resolution second order scheme

重构修正方法(correction procedure via reconstruction,CPR)具有紧致高效的优点,但对较强激波的捕捉能力还相对较弱,而加权紧致非线性格式(weighted compact nonlinear scheme,WCNS)具有很强的激波捕捉能力。将基于高阶WCNS插值的二阶格式引入到高阶CPR方法中,构造了一种高效高分辨率的混合激波捕捉格式。首先,基于非线性权偏离线性权的程度的激波侦测器侦测出问题单元,并在问题单元附近引入缓冲单元,其余单元则标记为光滑单元。然后,针对问题单元和缓冲单元采用二阶格式计算,光滑单元采用CPR方法计算,构造混合格式。通过对等熵涡问题、含激波的问题以及激波旋涡干扰问题的数值模拟,测试了混合格式的精度、激波捕捉能力和计算效率。数值模拟结果充分说明了该混合格式具有很强的激波捕捉能力,同时在光滑区具有高分辨特性,可以应用于高超声速流动问题的高效数值模拟中。相比于基于高阶WCNS插值的二阶格式,此格式具有更高的计算效率和更高的分辨率。

Correction Procedure via Reconstruction(CPR) is a compact and efficient numerical method, but it still has drawbacks in capturing strong shocks. However, Weighted Compact Nonlinear Scheme(WCNS) can capture shocks well. By introducing second-order scheme based on high-order interpolation of WCNS into highorder CPR method, an efficient hybrid shock-capturing method with high resolution property is proposed. Firstly,shock detection indicator based on the deviation of nonlinear weights from linear weights is used to judge troubled cells. Buffer cells are introduced in the neighbors of troubled cells, and other cells are marked as smooth cells. Therefore we use the second-order scheme to calculate in troubled cells and buffer cells, and CPR to calculate in smooth cells, which is the main idea of the proposed hybrid method. By calculating several numerical problems including isentropic vortex, shock and shock-vortex interaction, the accuracy, shock capturing ability and efficiency of hybrid method are validated with reference results. Numerical results show that the proposed hybrid method has satisfying shock-capturing ability, and has high-resolution property in smooth areas, which can be efficiently applied to numerical simulations of supersonic flows. Moreover, the method is more efficient compared with the second-order scheme based on high-order interpolation of WCNS.