Multigrid solution of the steady compressible Navier-Stokes equations coupled to the k-epsilon turbulence equations

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A relaxation method for the steady turbulent compressible Navier-Stokes equations is developed. The flow equations are fully coupled to the turbulence equations. The principles of the method are illustrated for three different k-E models. The relaxation method can be used in multigrid form. Multigrid results are given for one turbulence model on a flat plate test case. Two variants of the multigrid formulation are considered. In the first, the turbulence equations are not used on the coarse grids. In the second, the turbulence equations are used on all grids. In this last variant, without damping of the coarse grid corrections for the turbulence quantities, negative values for k and E are always encountered leading to a breakdown of the calculation. Therefore, damping of the coarse grid corrections for k and E is introduced. Introduction in the turbslence equations. This handicaps the optimization of the multigrid method. Properly defined implicit time stepping schemes overcome this difficulty. Examples of methods of implicit type are the methods of Vandromme and Ha ~ i n h ~ and orris on^. The first method uses the central discretization, the last uses the upwind discretization. These methods are not used in multigrid form. Some methods are of hybrid type like the method of Mavriplis and ~ a r t i n e l l i ~ , where the Navier-Stokes part is solved explicitly and turbulence part implicitly and where the Navier-Stokes equations are discritized in the central way and the turbulence equations in the upwind way. Implicit time stepping methods generally allow very large time steps. This observation indicates that it is possible to develop methods that solve directly the steady equations without the use of time stepping. Such a method is then equivalent to an implicit time stepping method with infinite time step. In this paper we develop a relaxation method for the steady equations and employ it in multigrid form. To ensure solvability by a relaxation method, the system of discrete equations has to be so-called positive. To obtain positiveness, the different parts of Basically, two classes of methods for the turbuthe equations have to be treated carefully: the convective part, the diffusive part and the source part. lent compressible Navier-Stokes equations are used nowadays. A first class of methods consists of the exThese three parts have to be split into positive and non-positive contributions. The positive contribuplicit time-stepping schemes. In the multigrid contions form the left hand side of the system; the nontext, we cite as examples the method of Dimitriadis and Leschzinerl using explicit Lax-Wendroff steppositive contributions form the right hand side. ping and the Low-Reynolds number Chien model In multigrid methods employing the turbulence and the method of ~ ~ ~ ~ l ~ ~ ~ ~ 2 using also Laxequations on the coarse grids1,215, often the coarse Wendroff stepping but the Launder-Sharma model. grid corrections cause the turbulence quantities k These methods suffer from a severe time step restricand & to b e ~ ~ m e negativeThis leads to breakdown tion due to the presence of the source terms of the calculations. So it is necessary to damp the coarse grid corrections. This can be done by limit*Associate Professor, Mechanical Engineering, member AIAA t Senior Research Associate, Mechanical Engineering ing the coarse grid corrections to some upper values Copyright 01995 by the American Institute of Aeronautics [2,5]. Here we follow a more sophisticated approach, and Astronautics, Inc. All rights reserved explained later, developed by Lien and ~ e s c h z i n e r ~ for incompressible Navier-Stokes equations. In this paper, we use the low Reynolds number form of the turbulence equations. We consider two variants of the multigrid formulation. In the first variant the k -e equations are not used on the coarse grids. In the second variant, the k E equations are solved on the coarse grids, but the coarse grid corrections are damped. We compare the efficiency of both methods and come to the conclusion that it is advantageous to employ the turbulence equations on the coarse grids. The best multigrid performance is obtained with the second variant for the largest possible value of the damping factor.