Computation of unsteady viscous ¯ows around moving bodies using the k ±e turbulence model on unstructured dynamic grids Bruno Koobus, Charbel Farhat * , Hai Tran Department of Aerospace Engineering Sciences, Center for Aerospace Structuring, University of Colorado, Campus Box 429, Boulder, CO 80309-0429, USA Received 8 December 1997; received in revised form 23 September 1999 Abstract We consider the numerical solution on unstructured dynamic meshes of the averaged Navier±Stokes equations equipped with the k±e turbulence model and a wall function. We discuss discretization issues pertaining to conservation laws, moving grids, and nu- merical dissipation. We also present a robust spring analogy method for constructing dynamic meshes. We validate our implemen- tation of this two-equation turbulence model and justify its usage for a class of vortex shedding problems by correlating our computational results with experimental data obtained for a ¯ow past a square cylinder. We also apply our solution methodology to the two-dimensional aerodynamic stability analysis of the Tacoma Narrows Bridge, and report numerical results that are in good agreement with observed data. Ó 2000 Elsevier Science S.A. All rights reserved. 1. Introduction The numerical simulation of vortex shedding ¯ows past oscillating obstacles is important for many engineering applications, including the bueting analysis of aircraft [1], and the aerodynamic stability analysis of suspension bridges [2,3]. It requires the solution of the Navier±Stokes equations on moving and possibly deforming grids. Direct numerical simulations (DNS) have made signi®cant progress in recent years, as three-dimensional vortex shedding computations with Reynolds numbers as large as Re  5000 are now possible on large-scale massively parallel processors [5]. Large-eddy simulations (LES) at Reynolds numbers as large as Re  10000 are also frequent nowadays [4]. However, even though they are not perfect, two-equation turbulence models are still widely used for the simulation of viscous ¯ow problems at higher Reynolds numbers, or when the computational resources required by DNS, LES, and high-order closure models such as the Reynolds stress equations (RSE) models are not available. Whether DNS or LES are preferred, or only a turbulence model can be aorded, the solution of the governing Navier±Stokes equations on unstructured moving and deforming grids calls for careful spatial and temporal discretization methods, and a robust strategy for constructing dynamic meshes. For this reason, our ®rst objective in this paper is to present a computational methodology for the simulation of vortex dominated ¯ow problems past moving bodies, and to illustrate it with the aeroelastic stability analysis of a suspension bridge. Our second objective is to show that when coupled with a suitable wall law and carefully discretized, the k±e turbulence model is suitable for the prediction of the unsteady mean characteristics of a class of low-speed vortex dominated ¯ows. In any case, many of the algorithmic and www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 190 (2000) 1441±1466 * Corresponding author. Tel.: +1-303-492-3992; fax: +1-303-492-4929. E-mail address: charbel@alexandra.Colorado.edu (C. Farhat). 0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 0 0 ) 0 0 1 7 2 - 9