Abstract
This study considers the accuracy of cell-to-face centre interpolation of convected quantities in unstructured finite volume meshes with cell-centred storage of variables. The accuracy of the interpolation algorithms were tested in isolation using ideal data to determine their numerical accuracy on both standard and artificially distorted meshes. It was found that the formally second- and third-order accurate interpolations based on one-dimensional interpolation along the line connecting the cells to the right and left of the face under consideration only have first-order accuracy in standard unstructured mesh, and less than first-order accuracy in distorted unstructured mesh. L1 interpolation errors in the distorted unstructured mesh are greater than in standard unstructured mesh. The order of accuracy and L1 errors can be improved by applying spatial corrections. The formally second-order accurate multi-dimensional interpolations tested in this study that are not based on one-dimensional interpolation along lines connecting the neighbour cells have first-order accuracy in both standard and distorted unstructured mesh. Linear interpolation between end vertices produces greatest L1 error in standard mesh; polynomial interpolation, linear interpolation between cell centres and standard QUICK produce the greatest L1 error in distorted mesh. Spatially correct QUICK, spatially correct linear interpolation between cell centres, Laplacian interpolation to face centres, and Taylor series expansion about an upstream cell produce the smallest L1 error in both standard and distorted mesh. Based on accuracy and the simplicity of implementation, Taylor series expansion about an upstream cell is the best choice for use in unstructured mesh.
Highlights
The finite volume solvers that used unstructured and cell centre storage mesh have gained high popularity in the solution of heat and fluids flow problems because the methods can efficiently model real engineering fluids flow problems that have complex geometric boundaries
These auxiliary points are related to cell centres L and R, but are located on the normal bisector of the face, equidistant from the face centre, such that the distance between L0 and R0 is equal to the distance between L and R; φ at the auxiliary points L0 and R0 is determined by Taylor series expansions about cell centres L and R, respectively
After testing a range of compact-stencil interpolation algorithms in this study, the following conclusions can be drawn: 1. Formally second- and third-order accurate interpolations based on one-dimensional interpolation along the line connecting cell centres to the left and the right of the face under consideration only have firstorder accuracy on standard unstructured mesh
Summary
The finite volume solvers that used unstructured and cell centre storage mesh have gained high popularity in the solution of heat and fluids flow problems because the methods can efficiently model real engineering fluids flow problems that have complex geometric boundaries. Unstructured meshes typically use triangular cells for twodimensional problems, allowing great flexibility in efficiently modelling complex boundaries, and in enabling localized grid refinement The cost of this flexibility is that the familiar locally one-dimensional operators for differentiation and interpolation are no longer applicable and must be replaced by new multi-dimensional algorithms, the properties of which are less well-established, the algorithms have been widely reported in the literature. Vakilipour et al [12] developed physical influence upwind interpolation schemes for linking pressure and velocity fields in incompressible flow solutions. These interpolation algorithms have advantages and disadvantages; a comparison is necessary to determine the best algorithm. The Kovasznay's model of viscouse flow [16] and potential flow past a circular cylinder [17] were used as analytical solution benchmarks
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