An improved global-direction stencil based on the face-area-weighted centroid for the gradient reconstruction of unstructured finite volume methods

The accuracy of unstructured finite volume methods is greatly influenced by the gradient reconstruction, for which the stencil selection plays a critical role. Compared with the commonly used face-neighbor and vertex-neighbor stencils, the global-direction stencil is independent of the mesh topology, and characteristics of the flow field can be well reflected by this novel stencil. However, for a high-aspect-ratio triangular grid, the grid skewness is evident, which is one of the most important grid-quality measures known to affect the accuracy and stability of finite volume solvers. On this basis and inspired by an approach of using face-area-weighted centroid to reduce the grid skewness, we explore a method by combining the global-direction stencil and face-area-weighted centroid on high-aspect-ratio triangular grids, so as to improve the computational accuracy. Four representative numerical cases are simulated on high-aspect-ratio triangular grids to examine the validity of the improved global-direction stencil. Results illustrate that errors of this improved methods are the lowest among all methods we tested, and in high-mach-number flow, with the increase of cell aspect ratio, the improved global-direction stencil always has a better stability than commonly used face-neighbor and vertex-neighbor stencils. Therefore, the computational accuracy as well as stability is greatly improved, and superiorities of this novel method are verified.

Extending the global-direction stencil with“face-area-weighted centroid”to unstructured finite volume discretization from integral formfinitevolumediscretization from integral form

Accuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction.For the commonly used k-exact reconstruction method,the cell centroid is always chosen as the reference point to formulate the reconstructed function.But in some practical problems,such as the boundary layer,cells in this area are always set with high aspect ratio to improve the local field resolution,and if geometric centroid is still utilized for the spatial discretization,the severe grid skewness cannot be avoided,which is adverse to the numerical performance of unstructured finite volume solver.In previous work[Kong,et al.Chin Phys B 29(10):100203,2020],we explored a novel global-direction stencil and combined it with the face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy.Greatly inspired by the differential form,in this research,we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization from integral form on both second and third-order finite volume solver.Numerical examples governed by linear convective,Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension.Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid,the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid.As a result,on unstructured finite volume discretization from integral form,the method also has superiorities on both computational accuracy and convergence rate.