A Comparison Between Colocated Mesh and Staggered Mesh Solutions for the Steady State Simple Driven Cavity Problem

Abstract

This paper presents some sample computations that employ three different schemes for the discretization of the incompressible Navier-Stokes equations: colocated mesh (CM) with basic second order finite difference approximations for the interior nodes, with two different implementations of the pressure boundary condition, and the conventional staggered mesh (SM). The specific goal is to better appreciate the well known spatial oscillation, or “pressure wiggle”, phenomenon usually attributed to the use of colocated mesh. A modified artificial compressibility method (ACM) and the MAC method were used for the colocated and staggered mesh calculations, respectively, but the focus of our findings is on the converged steady state results which pertain more to the asymptotic steady state discretization scheme (i.e. SM or CM) than the pseudo-time iteration method for obtaining these asymptotic solutions. Two different implementations of the pressure boundary condition were employed in conjunction with the ACM: 1) the requirement that the boundary pressure acts so that the continuity equation is satisfied at the boundary or 2) the requirement that the normal pressure gradient on the boundary satisfies the Navier-Stokes equation. Sample 2D and 3D calculations are performed on the driven cavity problem using these three techniques for a Reynolds number of 100. The results of these sample calculations are analyzed based on solutions available in the literature, and a comparison is made between the various methods and boundary condition implementations. The colocated mesh results indicate that the spatial oscillations, when present, were never greater than the overall accuracy, which is judged to be consistent with expected truncation errors of the various methods. The major objections of the oscillations are thus cosmetic rather than substantive. Furthermore, when the normal pressure gradient condition from the Navier-Stokes equation is used in conjunction with a colocated mesh, the spatial oscillations in the computations are significantly reduced for the pressure and are essentially non-existent for the velocities. These results suggest that the colocated mesh, with artificial compressibility or with other methods of computation, is a viable discretization scheme without the use of complex interpolation schemes to simulate a staggered mesh.