一种分裂形式CPR格式在欠解析流动中的稳定性研究  

On the stability of a split form CPR scheme in under-resolved flows

作　　者：贾斐然 朱华君 严红[1] 燕振国 石京昶 JIA Feiran;ZHU Huajun;YAN Hong;YAN Zhenguo;SHI Jingchang(School of Power and Energy,Northwestern Polytechnical University,Xian 710000,China;State Key Laboratory of Aerodynamics,Mianyang 621000,China)

机构地区：[1]西北工业大学动力与能源学院,西安710000 [2]空气动力学国家重点实验室,绵阳621000

出　　处：《空气动力学学报》2023年第11期103-115,I0002,共14页Acta Aerodynamica Sinica

基　　金：国家数值风洞工程(NNW);国家自然科学基金(12172375,11902344);空气动力学国家重点实验室基金(SKLA2019010101)。

摘　　要：基于重构的修正过程(correction procedure via reconstruction,CPR)方法是一种适用于非结构网格、紧致高效的高阶方法,但在离散非线性对流项时容易因为混淆误差的累积而出现数值不稳定。本文研究了基于LG(Legendre-Gauss)点分裂形式的CPR方法在欠解析流动中的稳定性,并将其与子单元限制技术结合,求解含激波的欠解析流动问题。结果表明:经过边界通量修正的LG点分裂形式CPR格式能够满足离散守恒律,且与子单元限制相结合后守恒律依然成立;在无激波的欠解析流动问题中,相比于散度形式CPR格式,分裂形式明显提高了计算的稳定性,而且比使用LGL(Legendre-Gauss-Lobatto)点的分裂形式CPR格式具有更小的数值误差;在含激波的欠解析流动求解中,相比基于LGL点的间断伽辽金谱元法子单元限制策略,基于LG点的分裂形式CPR格式子单元限制策略具有更高的分辨率和更小的振荡。The correction procedure via reconstruction(CPR)method is a compact and efficient high-order method suitable for unstructured grids.However,when discretizing the nonlinear convection term,it is easy to cause numerical instability due to the accumulation of aliasing errors.In the present work,we study the stability of the split form CPR method based on LG(Legendre-Gauss)points in under-resolved flows,and combine the method with the subcell limiting technique to solve under-resolved flows with shock waves.First,numerical tests are carried out to verify that the split form CPR method based on LG points with the boundary flux correction can satisfy the discrete conservation law,and such conservation is still preserved under subcell limiting.In the simulation of under-resolved flows without shock waves,compared to the divergence form CPR method,the split form significantly improves the stability of the calculation,and has smaller numerical errors than the split form CPR method using LGL(Legendre-Gauss-Lobatto)points.When solving under-resolved flows with shock waves,compared to the subcell limiting strategy of discontinuous Galerkin spectral element method based on LGL points,the subcell limiting strategy of split form CPR scheme based on LG points developed in this paper has higher resolution and less oscillation.

关 键 词：CPR格式 分裂形式 非线性稳定性 混淆误差 欠解析流动 

分 类 号：V211.3[航空宇航科学与技术—航空宇航推进理论与工程]