Comparison of Implicit Multigrid Schemes forThree-Dimensional Incompressible Flows

To develop a robust and efficient computational flow simulation tool for incompressible flow applications, a number of different implicit multigrid schemes for solving the three-dimensional incompressible Navier–Stokes equations are compared in the current study. These schemes consist of a common full approximation storage (FAS) multigrid algorithm implemented in conjunction with three different implicit schemes, which include a modified point Gauss relaxation, a standard Gauss–Seidel line relaxation, and the Beam–Warming alternating direction implicit (ADI) scheme. The flow solver used in the study is based on artificial compressibility and uses a third-order upwind difference for the convective terms and a second-order central difference for the viscous terms. The efficiency of each implicit multigrid scheme is assessed in terms of the computing time required for two laminar flow problems: the entry flow through a 90◦ bent square duct, and the steady-state and unsteady flows past a prolate spheroid at incidence with an axis ratio of 4 : 1. It is found that implementation of Neumann boundary conditions on the coarse grid in terms of the flow variable correction rather than the flow variable itself is essential for obtaining good convergence in the collocated finite difference discretization. The results of steady-state flow computations show that all the implicit multigrid schemes yield more than 50% computational time savings over their single grid counterparts, and the point or line relaxation multigrid scheme outperforms the ADI multigrid scheme by at least a factor of 2. However, in unsteady flow computations, the computational time saving of the multigrid scheme is less than that in steady-state cases. The current study concludes that the FAS multigrid algorithm implemented with the modified point Gauss relaxation scheme is preferable for simulating both steady-state and time-dependent incompressible flows. c © 2002 Elsevier Science (USA)