A local Navier-Stokes preconditioner for all Mach and cell Reynolds numbers

Abstract

A new family of optimal Euler preconditioners is presented that may resolve the stability problem associated with stagnation points. These preconditioners produce a close-to-orthogonal eigenvector system and are less sensitive to flow-angle variation across cells. Singleand multi-grid benefits of Euler preconditioning are presented and illustrated by some numerical results. Navier-Stokes preconditioning can be extended from the Euler technique with two methods: (a) by using block-Jacobi preconditioning for the viscous terms; (b) by introducing analytic dependence on the cellReynolds number in the preconditioner. Block-Jacobi preconditioning can also be applied to turbulent viscous flow calculations for a proper rescaling of the source terms. With these techniques it is possible to produce a local Navier-Stokes preconditioner effective for all Mach and cell-Reynolds numbers. Both two methods are illustrated with numerical tests. 1 Why use local preconditioning? For a Navier-Stokes code to converge fast it is necessary (but not sufficient) that the embedded Euler code converges fast, too. In working toward more efficient Navier-Stokes codes it therefore is useful to stay informed on Euler methods, and exploit the latest advances in that area. 'Postdoctoral Fellow, Member AIAA * Professor, Fellow AIAA ' Researcher Copyright ©1997 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. During the 1990s our understanding of the Euler convergence process has significantly increased. There is mounting evidence that the treatment of the true advection equations hidden in the Euler system (e.g. advection of entropy) has to be different from that of the coupled acoustic equations arising in subsonic flow. Whether starting from the equations in conservation form or primitive variables, this calls for a suitable decomposition of the system, in which the embedded acoustic equations are isolated from the advection equations. Ta'asan [22] suggests using a set of canonical equations based on the steady form of the Euler equations; this has its roots in vintage rules prescribed by Brandt in his 1984 Multi-Grid Guide [3]. In Brandt's work the emphasis is on achieving the theoretical limit of multigrid convergence, and not necessarily on the accuracy of the spatial operator. In contrast, Roe and Mesaros [18, 17] searched for a genuinely multi-dimensional splitting of the spatial Euler operator (fluctuation splitting) that would match accuracy to compactness. They ended up discretizing of the system that results from applying the local preconditioning of Van Leer, Lee and Roe [28]. The most advanced applications of these preconditioned equations are by Deconinck et al. [7], and use the so-called matrix form of the PSI advection scheme, which automatically distinguishes between omnidirectional acoustic wave propagation and unidirectional advection. The combination of the preconditioned equations and the fluctuation-split scheme, formulated on unstructured triangular or tetrahedral grids, differs in essence from standard Euler schemes. Such a nu-