Steady-state solution-adaptive Euler computations on structured grids

Abstract

A local solution-adaptive mesh refinement algorithm is used to produce steady-state flow results on structured grids for the two-dimensional Euler equations. The solution is marched to steady state using an explicit, cell-centered, second-order unsplit multidimensional upwind method. Convergence is accelerated by local time stepping and a multigrid method. The flexibility and efficiency of the algorithm are shown by presenting three test cases, a variety of subsonic and transonic internal and external flows. In trod uction An algorithm has been developed to compute steady-state solutions to the Euler equations using a multidimensional upwind method and local solutionadaptive mesh refinement on structured grids. The second-order unsplit multidimensional upwind method of Colella [lJ, as modified by Dudek and Colella for steady-state flows [2), is used to calculate the convective fluxes of the Euler equations. This method has been used in many contexts [3, 4, 5, 6], so it is well-tested and robust. The adaptive mesh refinement (AMR) algorithm allows computational resources to be utilized efficiently by placing refined grids only in areas in which they are needed. Thus, excess computations and memory are not wasted. In addition, the AMR algorithm 'Research Engineer, Member AlAA. tGroup Leader, Member AlAA. Also: Professor in Residence, Department of Mechanical Engineering, The University of California, Berkeley. Copyright @ 1998 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 1 gives great flexibility in problem solving, since it is not necessary to know beforehand where resolution will be needed. For instance) refinement around shocks is handled automatically, and the shock locations do not need to be known a priori. Thus, a simple base grid can be created and the AMR algorithm will ensure proper refinement where necessary. Since the grids are structured, the data structures are simply ordered) and optimization is straightforward, particularly on vector machines. The local adaptive mesh refinement method developed by Berger and Oliger [7] uses a sequence of nested levels of refined structured grids. After a solution is computed on the hierarchy of meshes, an error estimate is calculated, and blocks of cells where this error is high are refined locally in an efficient manner to produce a new hierarchy of levels. The solution is then computed on this new hierarchy and the process continues. Originally developed in the context of hyperbolic conservation laws in two-dimensions [3, 7], the algorithm has been extended to three dimensions [8], viscous two-dimensional flows on mapped grids [4], and incompressible flows [5]. There is extensive infrastructure for and experience with adaptive mesh refinement in conjunction with multidimensional upwind methods. In fact, a C++ library of functions which are used in adaptive finite difference calculations has been developed and is widely used [4, 9]. The motivation for this research is to take this well-established multidimensional upwind method and adaptive mesh refinement machinery and to combine them into an algorithm which can produce efficient and accurate steady-state solutions on structured grids. This paper is a continuation of work begun by Berger and Jameson [10J, who were the first to implement block-structured local adaptive mesh refinement on body-fitted mapped grids. We have modified and extended their work in a number of areas. They solved the two-dimensional Euler equations to steady state using the centered space differencing, RungeKutta time stepping algorithm of Jameson, Schmidt, and Turkel [14]. We advance the solution using an unsplit multidimensional upwind method with local time stepping [2]. We also take advantage of multi grid convergence acceleration, both at each level of refinement and over the entire mesh hierarchy. In addition, the grid generation process has been simplified and automated. Berger and Jameson required a fully refined grid at the finest level of refinement, from which all coarser grids were produced. Not only does this require an a priori knowledge of the amount of refinement needed, but this procedure may be difficult and expensive for complex geometries. In our approach, creation of refined body-fitted grids only requnes the coarse base grid, from which the refined grids are interpolated only to the extent that they are needed, rather than over the entire domain. Finally, Berger and Jameson applied their algorithm to a limited number of non-lifting external flow test cases, using only refinement ratios of two. Our algorithm has been tested on a wide variety of internal and external (both lifting and non-lifting) flows, using a variety of refinement ratios between grid levels. Governing Equations The two-dimensional time-dependent Euler equations for inviscid fluid flow in integral form for a control volume n with boundary an are aa r U dx + 1 F (U) . n dS :::: 0, (1) t in Jan where n is an outward-pointing normal. The variable U is the array of conserved quantities (mass, momentum in x-direction, momentum in y-direction, and energy), and F(U) == (F2:(U), FY(U)) is the vector of inviscid fluxes in the xand y-directions: U(x,t) == (:U) :; F2:(U) == ( PU~P ), puE + up 2 FY(U) ( i:p ). pvE + vp The density is denoted by p, the velocity in the xand y-directions by u :::: (u, v), the total energy per unit mass by E, and the pressure p is given by the equation of state for a perfect gas, ( (u2+v2)) P == (, 1) pE p 2 ' where I is the ratio of specific heats. Multidimensional Upwind Method Consider a structured-grid cell (i, j) as shown in Figure 1. Let us define Ui~j to be a discrete approximation to the average of the vector of conserved variables U over the cell at time tn: where (Ti)j is the area of cell (i,j), and x == (x, y). Using this definition, we then discretize the integral equations (1) using a simple forward difference in time, to determine the value of the solution at time tn+l: + F i 1/'2,j' D.i-l/2,j + F i ,j+l/2 'lli)j+l/2 (2) where ~t :::: tn+1 tn, n is an outward-facing area normal at the cell edges, and F is an approximation to the flux at the cell edge. The convective fluxes F . n are calculated by a version of the multidimensional upwind method of Colella [1], modified for steady-state flows by Dudek and Colella [2]. In this second-order two-dimensional Godunov method, a pair of edge states are extrapolated from adjacent cell-centers, taking into account the multidimensional nature of the flow. A Riemann problem is then approximately solved to uniquely define a state at the cell edge, from which the fluxes are computed. To prevent oscillations in the solution, the slopes used in this extrapolation are limited using the van Albada limiter [11]. Since we are interested in the steady state solution, a local time step is used. Each cell uses the maximum allowable time step based on