The Influence of Viscous Operator and Wall Boundary Conditions on the Accuracy of the Navier-Stokes Equations

The discretization of the viscous operator in an edge-based flow solver for unstructured grids has been investigated. A compact discretization of the Laplace and thin-layer operators in the viscous terms is used with two different wall boundary conditions. Furthermore, a wide discretization of the same operators is investigated. The resulting numerical operators are all formally second order accurate in space; the wide operator has higher truncation errors. The operators are implemented weakly using a penalty formulation and are shown to be stable for a scalar model problem with given constraints on the penalty coefficients. The different operators are applied to a set of grid convergence test cases for laminar flow in two dimensions up to a large-scale three dimensional turbulent flow problem. The operators converge to the same solutions as the grids are refined with one exception where the wide operator converges to a solution with higher drag. The 2 nd compact discretization, being locally more accurate at a wall boundary than the original 1 st compact operator, reduces the grid dependency somewhat for most test cases. The wide operator performs very well and leads for most test cases to results with minimum spread between coarsest and finest grids. For one test case though, the wide operator has a negative influence on the steady state convergence.