A high‐order accurate method for two‐dimensional incompressible viscous flows

Abstract

SUMMARYA high-order accurate solution method for complex geometries is developed for two-dimensional flows using the stream function-vorticity formulation. High-order accurate spectrally optimized compact schemes along with appropriate boundary schemes are used for spatial discretization while a two-level backward Euler implicit scheme is used for the time integration. The linear system of equations for stream function and vorticity are solved by an inner iteration while contravariant velocities constitute outer iterations. The effect of curvilinear grids on the solution accuracy is studied. The method is used to compute Cartesian and inclined driven cavity, flow in a triangular cavity and viscous flow in constricted channel. Benchmark-like accuracy is obtained in all the problems with fewer grid points compared to reported studies.

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

“…Nevertheless, for Re = 100, the FOU and SOU results obtained for a 129×129 mesh agree well with published results for this mesh size [37]. Our results for Re = 400 on a 257×257 grid, with comparison to Ghia et al [37], Ben-Artzi et al [5] and De and Eswaran [6], are shown in Figures 11 and 12. It should also be noted that these author have used second-and fourth-order schemes for the convective terms.…”

“…Since Equations (6) and (7) are linear, u , v and p also satisfy Equation (6). These equations represent the dependence of the velocity corrections u and v on the pressure correction p .…”

“…Suppose u * and v * are the discrete u and v fields resulting from a solution of the discrete u-and v-momentum equations, i.e. Equations (6), and p * represents the discrete pressure field which is used to obtain this solution of the momentum equations. Then u * , v * and p * satisfy the equations…”

“…e.g. [1][2][3][4][5][6][7]). While finite difference, finite element and finite volume methodologies have been extensively developed, the majority of the early fundamental research work in this field was based on finite difference formulations, whereas the finite volume method dominates more recent activity and the commercial codes used in industry.…”

“…e.g. [4][5][6]). Generally these authors, and other papers using vorticity-streamfunction formulations, do not discuss how to calculate the pressure field.…”