Compressible Navier-Stokes Equations

A hierarchical multigrid algorithm for e cient steady solutions to the two-dimensional compressible Navier-Stokes equations is developed and demonstrated. The algorithm applies multigrid in two ways: a Full Approximation Scheme (FAS) for a nonlinear residual equation and a Correction Scheme (CS) for a linearized defect correction implicit equation. Multigrid analyses which include the e ect of boundary conditions in one direction are used to estimate the convergence rate of the algorithm for a model convection equation. Three alternating-lineimplicit algorithms are compared in terms of e ciency. The analyses indicate that full multigrid e ciency is not attained in the general case; the number of cycles to attain convergence is dependent on the mesh density for high-frequency cross-stream variations. However, the dependence is reasonably small and fast convergence is eventually attained for any given frequency with either the FAS or the CS scheme alone. The paper summarizes numerical computations for which convergence has been attained to within truncation error in a few multigrid cycles for both inviscid and viscous ow simulations on highly stretched meshes. Head, Aerodynamic and Acoustic Methods Branch (AAMB), AIAA Fellow. Member AAMB, AIAA Senior Member. Member AAMB, AIAA Associate Fellow. Member AAMB, AIAA Senior Member. Member AAMB Copyright c 1999 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for government purposes. All other rights are reserved by the copyright owner. Introduction There has been an explosive growth in the use of computational uid dynamics methods in the aircraft design cycle over the past twentyve years. Recently there has been an emphasis on threedimensional Navier-Stokes simulations over complex con gurations; computations with 10{20 million grid points are commonplace in focused applications. Even with the advent of more powerful computers, algorithms that attain optimal convergence rates are important to enabling these computations to be accomplished in a reasonable wall-clock time. An optimal method is one in which the arithmetic operations to attain a solution to within truncation error scale as O(N ), where N is the number of equations to be solved. Here, N is the number of nite volumes in the solution (NFV ) times the number of conservation equations for each nite volume (m); i.e., N = mNFV . Generally, the total operation count can be expressed as cN WMWU where WMWU is the operation count corresponding to one minimal work unit (MWU), i.e., the simplest possible discretization of the equations to the order desired, and c is a constant that di erentiates one optimally converging method from another. One method of attaining optimal convergence rates is the multigrid method. For elliptic equations, textbook e ciencies, which attain convergence in four to ve residual evaluations, are possible. For hyperbolic equations, O(N ) methods have been developed for the incompressible Euler equations and for compressible Euler equations using either the Full Approximation Scheme (FAS) or the defect correction (DC) scheme. Multigrid solvers for viscous ows have also been developed using these approaches. For the compressible Navier-Stokes equations, textbook e ciencies have not been attained for general situations; the barriers which need to be over1 American Institute of Aeronautics and Astronautics