Pressure-based multigrid algorithm for flow at all speeds

Abstract

A full multigrid/full approximation storage (FMG/FAS) method is developed along with a unified pressurebased algorithm to handle the complex fluid flow problems encompassing both compressible and incompressible regimes. Besides highlighting the key elements of the present algorithm on a curvilinear staggered grid system, several flow problems, ranging from incompressible to hypersonic speeds, are computed, in conjunction with an adaptive grid method, to demonstrate its computational capability. With wide variations in geometry, fluid physics, and grid distribution, speedup between the single- and multigrid procedures can be obtained. Because different pressure boundary conditions are needed for incompressible, subsonic, and supersonic inlet and outlet, the number of grid levels that can be effectively used by the present multigrid method appears flow dependent; the implications of this observation are discussed. I. Introduction U SE of pressure correction methods for solving incompressible viscous flow problems in Cartesian or cylindrical coordinates is well established, and a number of methods of this type such as SIMPLE,1'2 SIMPLER,2 and PISO3 have been developed. The basic similarity of these methods is that the momentum equations are first solved using a guessed pressure field, resulting in a tentative velocity field. An equation for the pressure corrections is obtained via manipulation of the momentum and continuity equations. This equation is solved to obtain the pressure corrections, and the velocities are then corrected to satisfy the continuity equation. The extension of the SIMPLE algorithm to a general nonorthogonal curvilinear coordinate system has been described in Refs. 4-6 and has been successfully applied to various problems as documented in Ref. 6. When solving a viscous flow problem using a pressure-based method, the computer time required to solve the pressure (correction) equation is often a sizable fraction of the total computational effort.7'8 Van Doormaal and Raithby7 report that this fraction can be as high as 80% of the total CPU time. This observation is also qualitatively confirmed in Ref. 8, which finds that, for high Reynolds number flows, the pressure-correction equation requires many more sweeps of a point-symmetric successive over-relaxation (point-SSOR) or line-SSOR method to converge than either the momentum equation or other scalar transport equations. This is because the pressure-correction equation, which has the form of an anisotropic elliptic equation, is diffusion dominated, whereas the other equations are convection dominated. It is well known that point- and line-iterative methods converge rather slowly on elliptic problems such as the Poisson equation, particularly as the number of mesh points becomes large. To expedite the convergence rates, the multigrid (MG) method9'10 has been found to be very useful for improving the performance of these single-grid (SG) solvers for elliptic equations.