Comparison of multigrid methods for neutral and stably stratified flows over two-dimensional obstacles

Coupled and decoupled methods for solving the Navier-Stokes equations are compared as underlying smoothers for a multigrid algorithm. Numerical results for the benchmark problem of the lid-driven cavity confirm that residual reduction factors per multigrid cycle for the coupled method are superior to those of the decoupled method. The line-wise implementation of the coupled method is shown to be more efficient than the cell-wise while retaining good convergence rates and is the,fastest method for this problem. Both approaches are applied to the more challenging problem of homogeneous and inhomogeneous viscous flow past obstacles (a vertical barrier and a cosine-shaped bump), where the flow is largely unidirectional and good convergence rates for the coupled method can now only be achieved by solving the coupled equations in vertical lines. The convergence rates of both methods are shown to deteriorate for these flows, compared with those for the lid-driven cavity, but the deterioration is generally less for the decoupled method, however, and the relative efficiency of the decoupled method means that execution times are significantly less than those required for the coupled method. In the case of stratified flows convergence difficulties are found for the coupled approach when a high order discretisation is used for the density transport equation. Strategies developed to overcome this, based on the use of double discretisation techniques, are described. (C) 1998 Academic Press.