Abstract
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.
Highlights
Compared with the block-structured grid, unstructured grid is highly automated in grid generation [1,2,3] and convenient for the grid adaptation [4, 5], while the computational accuracy and stabilities are hard to be guaranteed [6], especially on high-aspect-ratio triangular grids [7,8,9]
3.2.2 Combination of global-direction stencil and face-area-weighted centroid In Section 3.1, we have demonstrated that compared with commonly used vertexneighbor and face-neighbor stencils, global-direction stencil cells are along the normal and tangential directions of the wall, the only data obtained by k-exact reconstruction and required for the flux evaluation are solution vectors at cells local origins, rather than stencil cells themselves, and if we still use the geometric centroid, grid skewness is unable to be eliminated
4.3.2 Results on regular and randomly perturbed C-type grid From the results shown in Figs. 29 and 30, we can find on C-type grid, more accurate lift and drag coefficients are obtained by the global-direction stencil with face-areaweighted centroid as well
Summary
Compared with the block-structured grid, unstructured grid is highly automated in grid generation [1,2,3] and convenient for the grid adaptation [4, 5], while the computational accuracy and stabilities are hard to be guaranteed [6], especially on high-aspect-ratio triangular grids [7,8,9]. Compared with the traditional geometric centroid, the distribution of this novel reference point is more regular, and the connection of that is almost parallel to the boundary normal vector Based on this phenomenon, in previous work, we combined this novel centroid and global-direction stencil for the second-order unstructured finite volume method [23], and it is verified that the global-direction stencil with face-area-weighted centroid has a lower discretization error than stencils with the geometric centroid including face-neighbor and vertex-neighbor stencils as well as the global-direction stencil. The current situation related to accuracy loss and stability deterioration on highaspect-ratio triangular grids is greatly ameliorated by the employment of this improved global-direction stencil This method is designed for the second-order differential finite volume solver [28], where both solution and source term vectors are evaluated as point values, and the source term integration is totally avoided.
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