A Godunov steady-state Euler solver for structured grids

Abstract

We present an algorithm to solve the twodimensional Euler equations for steady-state flows in complex geometries using structured body-fitted grids. The solution is marched explicitly in time by a eel-centered second order multidimensional upwind method. Convergence to steady state is accelerated by local time stepping and a multigrid method. Cases are run for high subsonic and transonic internal and external flows, and a convergence study confirms that the scheme is second order accurate for smooth flows. Introduction The unsplit multidimensional upwind method of Colella [7] was originally developed to compute timeaccurate solutions to systems of hyperbolic conservation laws on structured grid systems. The algorithm is accurate, robust, and well-tested. It has been used extensively, in many contexts, to compute the nonlinear convection terms of the compressible and incompressible Euler and Navier-Stofces equations on structured grids [1, 3, 4, 8]. Furthermore, it has been used within the framework of adaptive mesh refinement to solve the equations of gas dynamics [5], and compute the nonlinear convection terms for incompressible, variable density flow [2] and compressible viscous flow on mapped grids [18]. Therefore, there is significant experience with and infrastructure for, the unsplit multidimensional upwind method within an adaptively refined structured mesh. In fact, a C++ library of constructs which are used in adaptive finite difference calculations has been developed and is extensively used [2, 9, 18]. The motivation for this research is to take this well-established multidimensional upwind method and adaptive mesh refinement machinery and implement 'Graduate Research Assistant, Member AIAA ^Professor, Member AIAA Copyright © 1997 by the American Institute of Aeronautics and Astronautics, Inc. AH rights reserved. the necessary modifications to produce an efficient and accurate steady-state adaptive refinement algorithm for structured, mapped grids. This paper presents the steady-state algorithm for a single, non-adapted, mapped grid. Developing the algorithm initially on a single grid allows issues regarding convergence of a solution to steady-state to be isolated from issues regarding adaptivity. The method presented here will be incorporated within an adaptive mesh refinement algorithm. Governing Equations The two-dimensional Euler equations for inviscid fluid flow in integral form for a control volume Q with boundary 50 are