Multigrid for the 2-D compressible Navier-Stokes equations

Abstract

Multigrid methods are presented for the solution of the steady-state 2-D compressible Navier-Stokes equations discretized by an upwind finite-volume scheme on structured grids. A one-equation turbulence model is included in the set of equations. This paper describes the appropriate multigrid ingredients necessary to achieve effective convergence rates for-high Reynolds number turbulent transonic flows. Trades are described for the formulation of the equations to .be relaxed, the multigrid coarsening strategy, the choice of relaxation on each grid, ‘ad the degree of coupling between relaxation of the conservation ,equations and the turbulence model equation. In addition, modifications are proposed for the one-equation turbulence model to alterits non-physical transient behavior as an aid to achieve steady-state solutions; these modific$ions, however, do not change the steady-state formulation of the model. The resulting multigrid methods are tested on two cases: an inviscid transonic channel flow and a turbulent transonic airfoil. Achieved convergence rates are substantial improvements over current technology. Introduction and Motivation We address the subject of steady-state solution of the 2-D compressible Reynolds-Averaged Navier-Stokes equations on single-block structured grids through the use of multigrid methods. Calculations involving high Reynolds number flows have long been plagued by slow convergence. Typical multigrid convergence rates tend to hover around 0.99, and even the best attempts usually achieve no better than 0.95 without some form of global acceleration wrapped around the multigrid method. In stark contrast, multigrid has proven incredible convergence rates approaching 0.1 for certain model problems. This suggests a potential for dramatic *Principal Engineer Copyright @American Institute of Aerhnautics and Astronautics, Inc., l’999. AU rights reserved. improvements in convergence rates and overall solution time for Navier-Stokes flow solvers. The sources for the slow Navier-Stokes multigrid convergence abound. The equations admit solutions with complex features including shocks and thin shear layers; these features provide a wide range of characteristic scales that must be all effectively ‘addressed by the discretization and solver. High Reynolds number Navier-Stokes calculations must be performed on highly stretched anisotropic grids. Procedures for dealing with anisotropy are well known, but the multigrid literature has not fully explored construction of restriction, prolongation and coarse-grid operators in the presence of grid stretching and skewness. An additional source of difficulty is the presence of a turbulence model which can dramatically alter the character of the governing equations. Apart from additional unknowns that need to be solved, turbulence models tend to increase the nonlinearity and stiffness of the governing equations. Multigrid has iis own idiosyncrasies that make its application to Navier-Stokes calculations difficult. Effective multigrid convergence requires relaxation of the discretized equations on extremely coarse grids, where grid stretching can become excessive and grid smoothness disappears. On these coarse @ids the discretization can break down, and nonlinearity and stiffness present in the equations can become accentuated. All these lead to robustness problems for the solution process and can adversely affect convergence. In an alternate approach where approximate equation sets are solved on coarse grids, these approximations tend-to become increasingly poor representations of the goyerning equations; and this in turn, leads to degraded multigrid convergence rates. Reduced (or nonexistent) resolution of flow features on coarse grids adds an additional complexity to the problem. The complications of extremely coarse grids cannot be easily skirted by-using only a few multigrid levels. Effective multigrid convergence requires (in theory) an exact sdlve on thi: coarsest grid level, and this can lead to excessive computational effort if the coarsest grid constitutes a rel+tively large number of grid points. Also,