刘君, 韩芳, 夏冰. 有限差分法中几何守恒律的机理及算法[J]. 空气动力学学报, 2018, 36(6): 917-926. DOI: 10.7638/kqdlxxb-2018.0034
引用本文: 刘君, 韩芳, 夏冰. 有限差分法中几何守恒律的机理及算法[J]. 空气动力学学报, 2018, 36(6): 917-926. DOI: 10.7638/kqdlxxb-2018.0034
LIU Jun, HAN Fang, XIA Bing. Mechanism and algorithm for geometric conservation law in finite difference method[J]. ACTA AERODYNAMICA SINICA, 2018, 36(6): 917-926. DOI: 10.7638/kqdlxxb-2018.0034
Citation: LIU Jun, HAN Fang, XIA Bing. Mechanism and algorithm for geometric conservation law in finite difference method[J]. ACTA AERODYNAMICA SINICA, 2018, 36(6): 917-926. DOI: 10.7638/kqdlxxb-2018.0034

有限差分法中几何守恒律的机理及算法

Mechanism and algorithm for geometric conservation law in finite difference method

  • 摘要: 采用有限差分法求解复杂外形物体绕流场时经常进行坐标变换,由此会引入坐标变换系数等几何参数,采用不同的差分格式离散坐标变换系数得到的结果不同,导致在计算过程中可能出现均匀流场不能保持均匀的现象,消除这种误差需要研究几何守恒律。本文对坐标变换过程进行理论分析,发现坐标变换过程中采用的数学恒等式在离散条件下不再成立,这是引起物理量不守恒的本质机理,认为增加坐标变换系数恒等式作为源项的方程形式才是曲线贴体坐标系下的离散等价方程,提出只要源项和对流项的离散格式相同就能满足几何守恒律的构造准则。按照上述理论准则建立了基于离散等价方程的几何守恒律算法,通过AUSM、HLLC、Roe、VanLeer四种分裂格式的算例,表明这种新的几何守恒律算法适用于通量差分裂格式(Flux-Difference Splitting,FDS)和矢通量分裂格式(Flux-Vector Splitting,FVS),且均能消除由坐标变换(包括网格运动)引起的误差,保持流场的均匀特性。

     

    Abstract: Coordinate transformation is often used in solving CFD problems on structured meshes, including curvilinear meshes and moving and deforming meshes, where the coordinate-transformation metrics and Jacobian are introduced. The differential forms of the metrics and Jacobian's discretization may cause different errors, which may lead to wrong flow distribution. The investigation of geometric conservation law (GCL) is needed to reduce or eliminate such errors. In this paper, we discuss the GCL for finite difference method(FDM), and draw a conclusion that the main error source is the inaccurate identities derived from coordinate transformation under different discrete operation. The GCL is satisfied under following conditions:the governing equations of discretized grid meshes with identities as source term is equal to that of uniform grids; and the discrete operators of the flux term and source term are the same. In view of the aforementioned facts, we present a new algorithm to satisfy the GCL based on the discrete and equivalent equations. The new algorithm is verified by an example of uniform flow with the numerical flux schemes of AUSM, HLLC, Roe, and VanLeer, and the simulation result has shown that it can eliminate errors caused by the GCL errors effectively either for FDS or FVS.

     

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