AIAA JOURNAL Vol. 40, No. 12, December 2002 Large Eddy Simulation of a Road Vehicle with Drag-Reduction Devices R. Verzicco, ¤ M. Fatica, † G. Iaccarino, ‡ and P. Moin § Stanford University, Stanford, California 94305-3030 and B. Khalighi ¶ General Motors Corporation, Warren, Michigan 48090-9055 The ow around an idealized road vehicle at Reynolds numbers up to 10 5 has been simulated using large eddy simulation; the numerical technique is based on the immersed boundary approach, which allows efcient cal- culations to be carried out on a Cartesian grid. The effect of the Reynolds number and the wake modications produced by drag reduction devices attached to the base have been analyzed and compared with available experi- mental data. Averaged drag coefcient and mean velocity proles are in good agreement with measurement. The effect of subgrid-scale modeling (Smagorinsky and dynamic model) has also been studied. Introduction V EHICLE aerodynamic performance is mainly determined by drag coefcient, which directly affects engine requirements and fuel consumption. Drag reduction, however, is not the only concern;the soiling of the vehiclerear surfaceand the aerodynamic noisegeneratedby unsteadyow separationdecreasevehiclesafety andoperatingcomfort.The studyofunsteadyaerodynamiceffectsis alsorelevantfortheinducedunsteadyforces,which againcontribute to decreasing comfort and safety. Reynolds-averaged Navier–Stokes (RANS) techniques are usu- ally adopted to study the aerodynamics of road-vehicle; the main drawbackisthatonlylimitedinformationcanbeobtainedontheow and vortexdynamics.In addition,the resultsare usuallystronglyde- pendenton the turbulencemodeladoptedin the simulations. 1 On the other hand, large eddy simulation (LES), even if computationally very expensive,has been successfullyused to simulate the unsteady separated ow around a bluff body 2 and in an asymmetric diffuser 3 yielding accurate results in terms of time-averaged and instanta- neous quantities. In thisworkLES of theow arounda roadvehicleis carriedout.In particular, the Reynolds-number effect and the wake modications produced by two drag-reduction devices (a cavity and a boat-tail attached to the base of the vehicle) are analyzed and compared with the available experiments. Numerical simulations are carried out on Cartesian grids that allow the use of simple energy-conservative nite difference discretization schemes, which are required for re- alistic LES simulations.To treat arbitrarygeometriccongurations, we apply the immersed boundarymethod in which boundarycondi- tions are assignedindependentlyof the grid, by prescribingsuitable body forces. 4 These forces yield the desired velocity value on a given surface, which does not coincide with the coordinate lines. This simplication makes the cost of numerical simulation of the ow around complex, three-dimensionalgeometry similar to that in a rectangular domain discretized by a Cartesian mesh. Received 5 April 2001; revision received 2 May 2002; accepted for pub- lication 8 July 2002. Copyright c ° 2002 by the American Institute of Aero- nautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/02 $10.00 in cor- respondence with the CCC. ¤ Visiting Fellow, Center for Turbulence Research; currently Professor, DiMeG, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy. † Senior Research Associate, Center for Turbulence Research; fatica@ ctr.stanford.edu. ‡ Research Associate, Center for Turbulence Research. Member AIAA. § Professor, Center for Turbulence Research. Associate Fellow AIAA. ¶ Staff Research Engineer, Research and Development Center. Taking advantage of this technique, we were able to simulate the ow around a road vehicle with up to 10 million grid points using less than 1 GB of memory. Details of the numericalmethod are in the secondsection.In the thirdand fourth sectionsthe physicalproblem and the computational setup are described. Finally, the results are presented and discussed in the fth section. Numerical Method The present LES technique is based on the solution of the three- dimensional unsteady ltered Navier–Stokes equations with an ad- ditionalbody-forceterm f toenforcetheno-slipboundarycondition on solid surfaces. The anisotropic part of the subgrid-scale stress (SGS) is modeled using the Smagorinsky subgrid-scalemodel. The value of the model coefcient in the subgrid-scale turbulent vis- cosity is determined by a dynamic procedure and does not require a priori specication of any model constants. 5;6 Provided that grid resolution is adequate in the vicinity of solid walls, the dynamic model properly accounts for wall proximity without explicit damp- ing functions (for example, the van Driest function necessary in the classical Smagorinsky model). This implies that enough grid points must be clustered near the immersed boundary. The boundary body force is prescribed at each time step to establish the desired veloc- ity on an arbitrary surface inside the computational domain. 7 This forcing is active only in the ow region where we account for the presence of the solid body and it is set to zero elsewhere. In general, the surface of the immersed body does not coincide with the grid; therefore, the value of the forcing at the node closest to the surface but outsidethe solid body is linearlyinterpolated.This interpolation procedure is consistent with a centered second-order nite differ- enceapproximation,and the overallaccuracyof the schemeremains second order. 7 The ltered Navier–Stokes equations have been spatially dis- cretized in a Cartesian coordinate system using a staggered cen- tered second-order nite difference approximation. Details of the numerical method are given in Ref. 8; only the main features are summarized here. In a three-dimensional inviscid ow kinetic en- ergy is conserved, and this feature is retained in the discretized equations.It has been shown that nondissipativenumericalschemes are superior to upwind-biased schemes for LES. 9 The discretized system is integrated in time using a fractional-step method, where the viscous terms are advanced in time implicitly and the convec- tive terms explicitly. The large sparse matrix resulting from the im- plicit terms is inverted by an approximate factorization technique. At each time step the momentum equations are provisionally ad- vanced using the pressure at the previous time step, giving an in- termediate nonsolenoidal velocity eld. A scalar quantity is intro- duced to project the nonsolenoidaleld onto a solenoidal one. The large-banded matrix associated with the elliptic equation for this 2447