GMRES PHYSICS-BASED PRECONDITIONER FOR ALL REYNOLDS AND MACH NUMBERS - NUMERICAL EXAMPLES

This paper presents several numerical results using a vectorized versi on of a 3D finite element compressible and nearly incompressible Euler and Navier-Stokes code. The assumptions were set on laminar flows and Newtonian fluids. The goal of this research is to show the capabiliti es of the present code to treat a wide range of problems appearing in laminar fluid dynamics towards the unification from incompressible to compressible and from inviscid to viscous flow codes. Several authors with different approaches have tried to attain this target in CFD with relative success. At the beginning the methods based on operator spli tting and perturbation were preferred, but lately, with the wide usage of time-marching algorithms, the preconditioning mass matrix (PMM) ha s become very popular. With this kind of relaxation scheme it is possi ble to accelerate the rate of convergence to steady state solutions wi th the modification of the mass matrix under certain restrictions. The selection of the mass matrix is not an easy task, but we have certain freedom to define it in order to improve the condition number of the system. In this paper we have used a physics-based preconditioner for the GMRES implicit solver developed previously by us and an SUPG formu lation for the semidiscretization of the spatial operator. In sections 2 and 3 we present some theoretical aspects related to the physical p roblem and the mathematical model, showing the inviscid and viscous fl ow equations to be solved and the variational formulation involved in the finite element analysis. Section 4 deals with the numerical soluti on of non-linear systems of equations, with some emphasis on the preco nditioned matrix-free GMRES solver. Section 5 shows how boundary condi tions were treated for both Euler and Navier-Stokes problems. Section 6 contains some aspects about vectorization on the Gray C90. The perfo rmance reached by this implementation is close to 1 Gflop using multit asking. Section 7 presents several numerical examples for both models covering a wide range of interesting problems, such as inviscid low su bsonic, transonic and supersonic regimes and viscous problems with int eraction between boundary layers and shock waves in either attached or separated flows. (C) 1997 John Wiley & Sons, Ltd.