Unsteady three-dimensional incompressible Euler and Navier-Stokes solver for stationary and dynamic grids

Abstract

An implicit finite-volume numerical approach based on artificial compressibility is presented for solving the unsteady three-dimensional incompressible Euler and Navier-Stokes equations. The numerical flux at cell faces is calculated using the Roe approximate Riemann solver. A nonsingular eigensystem for the transformed flux Jacobians is derived. Newton's method is the iterative process used to obtain both steady and unsteady solutions; however, the solution matrix operator is new in that it consists of a numerical approximation of the Jacobian ' of the Roe flux vector. Navier-Stokes solutions are simulated through an explicit treatment of the diffusive fluxes, which have been simplified through the thin-layer approximation. The turbulence model used is the Baldwin-Lomax algebraic model. Several check cases have been investigated to test thevalidity of this approach. Computed results for stationary and dynamicgrids have shown favorable agreement with either theoretical or experimental data for both inviscid and viscous flows. Introduction Numerous solution methodologies have evolved for the incompressible Navier-Stokes equations. Each one manages to model the relationship between velocity and pressure in such a way that the divergence free constraint imposed by the continuity *Graduate Student, Department of Aerospace Engineering, Member AIAA Professor, Department of Aerospace Engineering, Member AlAA Copyright 1991 by the Amerlcan Institute of Aeronautics and Astronautics, Inc. All rights reserved. v equation is satisfied. The approach taken in this work was firstsuggested by Chorin [I] and termed artificial compressibility In this procedure, a fictitious time derivative of pmsure is added to the mass conservation equation. This formulation not only directly couples pressure and velocity, but also allows the modified incompressibleequations to besolved in the framework of a timemarching compmsible flow algorithm. The initial use of artificial compressibility was restricted to steadystate problems [Z, 3, 4,5, 61; however, its use has been extended to include unsteady flow calculations as well [7,8,9,1O, 111. Various researchers have incorporated advances in compressible flow algorithms into artificial compressibility algorithms. Many of the techniques used in this work have already been utilized in other codes,although in differentcombinations. As pointed out by Pan and Chakravarthy [71 and Rogers and Kwak [IO], artificial compressibility must be intmduced after the time-dependent curvilinear transformation to insure that a divergence free solution is obtained for a dynamic grid. A high-resolution upwind flux-difference split scheme based on the Roe approximate solver [I21 is used in this work. Hsu, et al. [131, Rogers and Kwak [lo], Rogers, et al. [lll , and Pan and Chakravarthy [7] employ the same flux-difference split scheme and all but the last reference use a non MUSCL 1141 type approach. A different formulation for the numerical fluxvector,aspresented byWhitfield,etal. [I51 isused in the current work. Also, a new nonsingular eigensystem is derived for the transformed Jacobian matrices. Newton's method is the iterative method utilized to obtain both steady and unsteady solutions as in Reference 171. However, the solution matrix operator is new in that it is a discretized Jacobian, whose elements are obtained by using simple