基于混合网格的高阶精度DG及DG/FV格式曲边界处理方法研究

In the computational fluid dynamics (CFD) community, high-order accuracy methods on unstructured/hybrid grids have been paid more and more attention in the nearly ten or twenty years because of the need for high accuracy results. The treatment of curved boundaries - especially of curved walls is crucial for high-order accuracy method. Inappropriate treatment not only makes the error on the boundary polluting the entire flow field, resulting in order reduction of high-order accuracy method; but also leads to the deterioration of convergence of high-order accuracy method, so the meaningful high-order accuracy solution cannot be obtained. Therefore, the "true" high-order accuracy results can only be gotten by taking into account the consistence of the treatment of the curve boundaries with the high-order distribution of physical variable in each element. In this application, a curved boundary representation method using spline curves/surfaces and a computation method of curved element will be proposed and applied to high-order accurate DG/FV hybrid method on unstructured/hybrid grids, which will enhance their ability in the application of complex geometries. Because the DG/FV method is cheaper than the same order of DG method in computation and has other advantages, we expect that the method described above can lay the groundwork for the large-scale high-order accurate numerical simulations in practical engineering problems.

在计算流体力学中，由于对高精度计算结果的需求，基于非结构/混合网格的高阶精度方法在近一二十年受到广泛的关注。曲边界-尤其是曲物面边界的处理对于高阶精度方法来说至关重要。不恰当的处理不但使得边界上的误差污染整个流场，导致高阶精度方法的降阶；还可能导致高阶精度方法的收敛性变差，从而使其得不到有意义的高阶精度的解。因此为了取得“真正”的高阶精度计算结果，必须改进曲边界的处理方式以便与高阶精度方法内部单元的物理场的高阶分布相容。本项目拟发展基于样条曲线/曲面的曲边界描述方法及曲单元计算方法,将高阶精度曲边界处理方法与非结构/混合网格上的高阶精度DG/FV混合方法相结合，增强其在复杂问题应用中的能力，期望利用DG/FV方法较同阶精度DG方法计算量小等优势，为其在实际工程问题的大规模数值模拟中早日得到应用打下基础。

在计算流体力学中，由于对高精度计算结果的需求，基于非结构/混合网格的高阶精度方法在近一二十年受到广泛的关注。曲边界-尤其是曲物面边界的处理对于高阶精度方法来说至关重要。不恰当的处理不但使得边界上的误差污染整个流场，导致高阶精度方法的降阶；还可能导致高阶精度方法的收敛性变差，从而使其得不到有意义的高阶精度的解。因此为了取得“真正”的高阶精度计算结果，必须改进曲边界的处理方式以便与高阶精度方法内部单元的物理场的高阶分布相容。本项目针对非结构/混合网格的上高阶精度DGM和DG/FV混合格式发展了二维基于三次样条曲线的曲边界描述方法及边界条件简化处理方法，发展了二维/三维基于拉格朗日基函数的高阶曲单元描述方法，研究了常见高阶曲单元类型的坐标变换方法、层次标准正交基函数构造方法及高斯积分点取法。初步研究了基于RBF插值技术的曲网格生成方法。初步实现了高阶曲单元上三维全耦合RANS方程求解的高阶计算方法，为提高S-A湍流模型的数值计算鲁棒性，实现了湍流工作变量出负时的模型修正方法，实现了精确Jacobian矩阵计算方法。初步实现了高阶曲网格情况下的DG/FV混合格式，其较同阶精度DG方法计算量更小，为其在实际工程问题的大规模数值模拟中早日得到应用打下基础。

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