Design and Time-Accuracy Analysis of ALE Schemes for Inviscid and Viscous Flow Computations on Moving Meshes Philippe Geuzaine * and Charbel Farhat † University of Colorado at Boulder Boulder, CO 80309-0429, U.S.A. We consider the solution of inviscid as well as viscous unsteady flow problems with moving boundaries by the arbitrary Lagrangian Eulerian (ALE) method. We present two computational approaches for achieving formal second-order time-accuracy on mov- ing grids. The first approach is based on flux time-averaging, and the second one on mesh configuration time-averaging. In both cases, we show that formally second-order time-accurate ALE schemes can be designed. We illustrate our theoretical findings and highlight their impact on practice with the solution of inviscid as well as viscous, un- steady, nonlinear flow problems associated with the AGARD Wing 445.6 and a complete F-16 configuration. 1 Introduction In many computational fluid dynamics (CFD) appli- cations, some or all of the boundaries delimiting the physical domain of the flow move in time. Examples include, among others, a large class of free-surface flow problems and a wide variety of fluid-structure interac- tion problems. When the moving boundaries undergo large displacements and/or rotations, or when they experience large deformations, the flow problem is of- ten formulated in an arbitrary Lagrangian Eulerian (ALE) 1, 2 frame and discretized on an unstructured moving grid. Such a discretization differs from that of the standard Eulerian formulation only in the in- troduction of some geometric quantities involving the positions and velocities of the moving grid points. For this reason, an ALE time-integrator is typically con- structed by combining a preferred time-integrator for fixed grids computations and an ad-hoc procedure for evaluating the geometric quantities arising from the ALE formulation. As noted in Refs. 3, 4, such an approach for designing an ALE time-integrator does not necessarily preserve the order of time-accuracy of its fixed grid counterpart. To address this issue, we present two different methods for extending to mov- ing grids a preferred time-integrator while preserving its order of time-accuracy established on fixed grids. These methods differ in the approaches they adopt for time-integrating between time t n and time t n+1 the convective, diffusive and source terms, when the grid moves from a position x n to a position x n+1 (here * Research Associate, Department of Aerospace Engineering Sciences, Center for Aerospace Structures. Currently Group Leader, CENAERO, Belgium. Member AIAA. † Professor, Department of Aerospace Engineering Sciences, Center for Aerospace Structures. Fellow AIAA. Copyright c  2003 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. and in the remainder of this paper, the superscript n designates the n-th time-instance t n ). For each time- interval [t n ,t n+1 ], the first method constructs a set of intermediate mesh configurations, evaluates the nu- merical fluxes and source terms on each one of them, then time-averages each set of these numerical quanti- ties. Alternatively, the second method defines a unique computational mesh configuration in the time-interval [t n , t n+1 ] by time-averaging the intermediate mesh configurations themselves, then computes the numeri- cal fluxes and source terms on this time-averaged mesh configuration. In both methods, the geometric quanti- ties arising from the ALE formulation, the parameters governing the construction of the intermediate mesh configurations and the time-averaging procedure are chosen as to guarantee that the resulting ALE time- integrator is formally p-order time-accurate on moving grids, where p characterizes the underlying fixed grid time-integrator. It turns out that for a given fixed grid time- integrator, each of the methods outlined above leads to multiple ALE versions which preserve its order of time-accuracy on moving grids. However, it is inter- esting to note that not all of these versions satisfy their discrete geometric conservation laws (DGCL). A DGCL 5, 6 states that the computation of the geo- metric parameters arising from an ALE formulation must be performed in such a way that, independently of the mesh motion, the ALE numerical scheme pre- serves the state of a uniform flow. Therefore, whereas in Ref. 6 the authors proved that satisfying the DGCL is a sufficient condition for an ALE numerical scheme to be consistent on moving grids, and in Refs. 7, 8 the authors proved that this law is a necessary and sufficient condition for some ALE numerical schemes to preserve on moving grids the nonlinear stability of 1 of 10 American Institute of Aeronautics and Astronautics Paper 2003–3694 16th AIAA Computational Fluid Dynamics Conference 23-26 June 2003, Orlando, Florida AIAA 2003-3694 Copyright © 2003 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Downloaded by STANFORD UNIVERSITY on December 10, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2003-3694