High aspect ratio grid effects on the accuracy of Navier–Stokes solutions on unstructured meshes

The method of manufactured solutions is used to evaluate the stability and accuracy of several unstructured discretization schemes in the presence of high-aspect ratio grids. Both inviscid and viscous discretizations are studied by systematically varying parameters such as the aspect ratio, mesh stretching, curvature, skewness, and non-planar faces. For the inviscid terms, gradient reconstruction based on cell least squares and nodal Green–Gauss are considered. In particular, high curvature grids are shown to lead to instabilities with both classes of schemes and a new Green–Gauss scheme based on nodal projection is developed that retains stability and formal second-order accuracy under all conditions. A further advantage of the nodal Green–Gauss schemes is that the nodal values can be used in the viscous discretization as well. All the viscous schemes tested here are demonstrated to preserve second-order accuracy. Finally, for three-dimensional meshes, triangulation of non-planar faces is found to be necessary to preserve second-order accuracy.