HIGHER-ORDER DIFFERENCING METHOD WITH A MULTIGRID APPROACH FOR THE SOLUTION OF THE INCOMPRESSIBLE FLOW EQUATIONS AT HIGH REYNOLDS NUMBERS

Abstract

A higher-order differencing method was recently proposed for the convection-diffusion equation, which even with a coarse mesh gives oscillation-free solutions that are far more accurate than those of the upwind scheme. In this subsequent work, the performance of this method was investigated in conjunction with the performance of different iterative solvers for the solution of the Navier-Stokes equations in the vorticUy-streamfunetion formulation for incompressible flow at high Reynolds numbers. Flow in a square cavity with a moving lid was chosen as a model problem. Solvers that performed well at low Re numbers either failed to converge or had a computationally prohibitive convergence rate at high Re numbers. The additive correction method of Settari and Aziz and an iterative incomplete lower and upper (ILU) solver were used in a multigrid approach that performed well in the whole range of Re numbers considered (from 1000 to 10,000) and for uniform as well as nonuniform grids. At high Re numbers, point or line Gauss-Seidel solvers converged with uniform grids, but failed to converge with nonuniform grids. The predictions of the higher-order differencing scheme used in this work agree very well with those of Gkia et al., which were used as a benchmark. The computation time of the higher-order method was only 15% longer than that of the first-order upwind scheme. This work demonstrates the effectiveness of the combination of the higher-order differencing scheme with a multigrid approach to give accurate solutions of the Navier-Stokes equations at high Re numbers.