Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured Meshes

The accuracy of the least-squares technique for gradient reconstruction on unstructured meshes is examined. While least-squares techniques produce accurate results on arbitrary isotropic unstructured meshes, serious di culties exist for highly stretched meshes in the presence of surface curvature. In these situations, gradients are typically under-estimated by up to an order of magnitude. For vertex-based discretizations on triangular and quadrilateral meshes, and cell-centered discretizations on quadrilateral meshes, accuracy can be recovered using an inverse distance weighting in the least-squares construction. For cell-centered discretizations on triangles, both the unweighted and weighted least-squares constructions fail to provide suitable gradient estimates for highly stretched curved meshes. Good overall ow solution accuracy can be retained in spite of poor gradient estimates, due to the presence of ow alignment in exactly the same regions where the poor gradient accuracy is observed. However, the use of entropy xes, or the discretization of physical viscous terms based on these gradients has the potential for generating large but subtle discretization errors, which vanish in regions with no appreciable surface curvature.