peter s. bernard, pat collins and mark potts- vortex method simulation of ground vehicle...

8/3/2019 Peter S. Bernard, Pat Collins and Mark Potts- Vortex Method Simulation of Ground Vehicle Aerodynamics

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2005-01-0549

Vortex Method Simulation of Ground Vehicle Aerodynamics

Peter S. BernardUniversity of Maryland and VorCat, Inc.

Pat Collins and Mark PottsVorCat, Inc.

Copyright 2004 SAE International

ABSTRACT

Recent results are presented from the application of agrid-free vortex method to the prediction of turbulentaerodynamic flows produced by road vehicles. The

approach has the character of a large eddy simulationand incorporates a fast technique for evaluating themutual interactions of several million vortex elementsneeded in describing the dynamically significant structureof the turbulent field. The simulations provide informationabout the unsteady evolution of the flow including forces,vortical structure and separation regions. Predictedmean velocity statistics are shown to compare favorablywith physical experiments.

INTRODUCTION

Advances in supercomputing have brought Large Eddy

Simulation (LES) techniques to the point where theyrepresent a viable alternative to more traditionalReynolds-averaged Navier-Stokes (RANS) closures inmodeling flows past ground vehicles [9,16]. Not only doLES schemes provide detailed transient informationabout forces, separation and wake structure that areabsent in RANS methods [2,5,7,17], but they have thepotential to provide substantially greater accuracy if forno other reason than that they make less demands onmodeling per se. To a significant extent the advantagesof LES can be further enhanced if a gridfree formulationbased on the vortex method [12] is used in place oftraditional gridbased schemes. In fact, vortex methodsallow for a number of efficiencies and accuracies that aredifficult to replicate within the confines of mesh basedapproaches. This paper describes some of the essentialaspects of performing simulations of road vehicle flowsusing a gridfree scheme and then presents some resultsfrom computations of flows past generic automobileshapes that illustrate its capabilities.

The specific approach to be described is that developedby VorCat, Inc. 1 [3,4] in which convecting and stretching

1 The VorCat software is protected under U.S. PatentNo. 6512999.

vortex tubes are used as the principal computationalelement in conjunction with a thin mesh of triangularprism sheets, several layers thick, next to solidboundaries upon which the 3D vorticity equation issolved via a finite volume scheme. The wall calculation is

highly resolved approaching that typically seen in directnumerical simulations (DNS) since this determines thelocation and strength of vorticity produced at the wallsurface and then ultimately the vorticity field held on thegridfree elements.

Similar considerations in modeling the near wall flowapply in the case of traditional gridbased LES schemes.In fact, since all scales of motion next to boundaries areboth energetic and subject to the effects of viscosity it isnot evident that subgrid scale models are appropriate inthis region. Rather, direct computation may be necessaryif predictions are to be successful. For increasingReynolds number the difficulty of employing adequatewall meshes is exacerbated. This often leads to thepractice of imposing wall function boundary conditionsthat tend to be accurate under very limitedcircumstances. In the present case the boundaryresolution problem is ameliorated to some extentbecause the thickness of the gridded region scales inwall units so it of necessity only increases with Reynoldsnumber in two dimensions and not three. Secondly, thetriangular prism elements are sheet-like and thus areefficient at providing coverage of the largely two-dimensional region of high vorticity next to boundaries.

There are several other features of the vortex method

that provide advantages beyond that of gridbasedmethods. Thus, vortex tubes offer a substantially moreefficient means of numerically representing physicalvortices that are a dominant part of the dynamics ofturbulent flow. Moreover, the convection and stretchingof vortex elements can be accomplished withoutsignificant numerical diffusion so that sharp features ofthe flow, such as detached vortices and shear layers,remain sharp. Another useful aspect of vortices is thatthey provide an opportunity for imposing de factosubgrid scale models (that limit the resolution ofvortices in the simulation) that are neither diffusive norinterfere with the natural process of backscatter by which


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energy flows from small to large scales. Vortex methodsare also naturally self-adaptive in the sense that theysupply enhanced resolution to local areas of the flow asdetermined by the physics.

The next section describes some of the principal aspectsof the gridfree methodology based on the vortex method.Following this is a discussion of some results taken fromcomputations of the flow past the Morel [11] and Ahmed[1,10] bodies.

GRIDFREE ALGORITHM

Vortex methods are gridfree numerical schemes forsolving the 3D vorticity equation [3]

,1

)()( 2++=

e R

U U t

(1)

where U and are velocity and vorticity vectors,respectively, and R e is the Reynolds number. The termson the right hand side represent, in order, convection,vortex stretching and diffusion. Some of the particularconsiderations needed in solving (1) for a collection oftube elements and in a fixed mesh of triangular prismsadjacent to boundaries is now considered.

TUBE ELEMENTS

Vortex tubes in the form of straight line segments formthe gridfree computational element used in the VorCatcode. Each tube is defined by its position as determinedby its end points and its circulation . The evolution of atube is determined by the motion of its end points. In thisway both the length and orientation of the tubes change

in time. According to Kelvin's theorem, the circulation ofa material vortex tube in an inviscid flow remainsconstant in time. This provides some justification -particularly in high Reynolds number flows away fromboundaries - for assuming that remains constant foreach tube during its motion. Taken together these stepsprovide an approximation to the convection andstretching terms in (1) and it is thus seen that thedynamics of tubes is easily accommodated in thenumerical scheme.

Associated with each vortex tube is the velocity fieldgiven by the approximate relation

) / (||4 3

r r

sr U

= (2)

that is derived from the Biot-Savart law. Here r = x - x c isthe distance from a point x to the center of the tube, x c ,and s is an axial vector along the tube axis. Following thecommon procedure for vortex methods, (r / ) is asmoothing function that is used to desingularize the Biot-Savart kernel appearing in this relation. is a parameterthat determines the size of the region over which thesmoothing takes place. Beyond a distance proportionalto , = 1 and there is no smoothing.

As a flow evolves in time, there is a tendency for thetubes to increase in length as part of the natural processof vortex stretching. A maximum length of tubes isallowed and whenever tubes are longer than this they aresubdivided. In this way, tubes that start out as singlestraight segments are likely to become chains of tubesforming filaments as time goes on. Note that suchconfigurations may be viewed as discreteapproximations to smooth physical tubes, so the shorterthe tubes are allowed to be the more closely the discreteapproximation follows the behavior of real tubes. Thereis a trade off between numerical efficiency and accuracy.Longer tubes are more computationally efficient but alsoless accurate. Generally, the maximum tube length is setas the average edge length of the triangles used inrepresenting the boundary.

It may be seen in (2) that the velocity field associatedwith a vortex tube is proportional to the product |s |. Thevalue of associated with a tube is determined at thetime of its creation near the top of the wall meshdepending on local flow conditions and varies from tubeto tube. It has been determined that discretization error is

reduced if a bound is placed on the product |s | sotubes are also subdivided according to this condition.This allows the strongest vortices to more naturallyevolve in time (i.e., fold and stretch more responsively tolocal flow conditions).

Vortex tubes as computational elements have asignificant capacity to mimic the true physical process bywhich energy cascades to small scales by vortex foldingand stretching in spatially intermittent regions. For avortex method to be practical it must restrain the growthin the number of tubes by modeling dissipation withouttrying to actually compute the complete folding and

stretching process. The ways in which this is done in thecode represent a de facto subgrid-scale model in VorCat.In essence, the idea is to eliminate those vortices whosefuture evolution can be expected to involve dissipation.Folding vortex tubes with the appearance of hairpins areassociated with a primarily local velocity field [6] due tocancellation of the far field velocities produced by theoppositely rotating legs of the hairpin. If it can beassumed that the lengths of the tube segments are smallenough - generally in the inertial subrange - then it isreasonable to identify the local energy associated withthe hairpins as energy that is destined to be lost viaviscosity after subsequent stretching and folding takes itto the dissipation range. Removing the hairpinseliminates the associated energy without the expense ofa detailed calculation. This is similar to the effect thatsubgrid scale models have in mesh based schemes inremoving energy traveling to unresolved scales.

Another aspect of the present subgrid scale modeling iscontained in the idea that the vortices, even if they do notform tight hairpins, tend to stretch rapidly in manyinstances. As they stretch so too does the materialvolume that they occupy. For volume to be conservedthe thickness of the tubes must shrink as they get longer.It should be noted that the tube area in this instance is


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understood implicitly and is not an actual parameter inthe computations. After an individual tube has stretchedand been subdivided many times, it is to be expectedthat its associated thickness is reduced to the scale ofthe dissipation range, and if so, the corresponding tubecan be eliminated on the belief that its coherent vorticitywould have been smoothed away by viscosity. Betweenhairpin removal and the elimination of highly subdividedtubes, the population of vortices is prevented fromincreasing so rapidly as to render the methodimpractical.

VORTEX PRISMS

Solid surfaces are represented in VorCat throughtriangularizations. The triangular mesh used in thecomputation of the wall flow is grown out from theboundary triangles by erecting perpendiculars at thenode points. The number of layers is usually taken to be11. The first layer is of half-thickness, say y /2, where y is the direction normal to the boundary. Each of theremaining layers are of thickness y . The vorticity isassumed to be constant over each triangular prism.

The vorticity components at the wall surface representboundary conditions for the computation of vorticity in thesheets. These are evaluated via finite differenceapproximations using the no-slip condition on the solidsurface. It is thus important that y be small enough thatthese approximations are accurate.

The thickness of the region near the wall surface thatneeds to be covered by triangular prisms is an importantconsideration in the present methodology. It must belarge enough to fully encompass the high vorticity and itsgradients that are produced next to solid walls via

viscosity. This region generally constitutes the viscoussublayer and an immediately adjacent part of theturbulent boundary layer. Here there is a large viscousflux of vorticity out of the solid surface leading to theappearance of vorticity in the flow. The vorticity field inthe mesh is determined as a solution to (1) via a finitevolume scheme. The result of the computation is toestablish the amount of vorticity that convects anddiffuses to the top sheet layer, to subsequently be turnedinto new vortex tubes.

For a given Reynolds number the thickness of the sheetregion is targeted to be within y + = 25 50, where y += U y/ denotes wall coordinates and U is the frictionvelocity. To utilize approximately 11 sheet levels, it isgenerally required that y + be taken to be approximately3. This places the sheet resolution in the normal directionat the high end of what may be appropriate for DNS andallows for a reasonable determination of the wall vorticityvia a finite difference approximation.

The sizes, dimensions, and overall comportment of thesurface triangles can have a significant effect on theaccuracy and performance of the finite volume scheme.For example, long narrow triangles need to be avoided.An especially important aspect of the mesh is the

average aspect ratio defined as the average edge lengthof the triangles divided by y . The larger this ratio is thecoarser the spanwise resolution is and the more likelythat there will be significant discretization error. On theother hand, the smaller the aspect ratio the moretriangles are needed to cover a surface and the moreexpensive the computation becomes. There is also alower limit for the aspect ratio since the triangular prismsmust be sheet-like" in order to use velocity formulas thathave been developed with this condition in mind. It isfound empirically that an aspect ratio of 10 is perhapsideal. If y + 3 and the triangular prisms have anaverage aspect ratio of 10, then the typical spanwiseextent of the triangles is about 30 wall units, which isadequate for resolving streamwise vortical wallstructures directly in terms of the vorticity field.

FINITE VOLUME ALGORITHM

The vorticity field is determined in the mesh region via afinite volume solution to the 3D vorticity equation. Foreach triangular prism the vorticity is that at its center,while the velocities are computed at the centers of the

top and bottom triangles. The velocities at the six nodalcorners of the prism are determined via area-weightedaveraging of the velocities at the centers of the top andbottom triangles. The finite volume scheme uses asimple explicit time discretization. The convection term isapproximated by applying the divergence theorem to itsequivalent conservative form yielding an expression forthe net flux of vorticity from a prism as a sum ofcontributions from each of its five sides. For thiscalculation the vorticity on the triangular faces is takenfrom the upwind prism. On the quadrilateral sides thevorticity is taken from a linear least square fit of thevorticities in prisms that are contiguous to the prism on

the upwind side of the surface on the same and theimmediately neighboring levels above and below, plusthe one prism on the downwind side.

The evaluation of the stretching term is done at thecenter of the prisms so the vorticity appearing in theexpression is directly available. The velocity gradient iscomputed using the scalar divergence theorem followinga similar approach as done in computing the convectionterm.

The evaluation of the diffusion term in (1) distinguishesbetween diffusion normal and parallel to the surface. TheLaplacian is expressed in a local rectangular Cartesiancoordinate system and the term with normal derivativesis evaluated using a standard second order finitedifference formula using the vorticity values at thecenters of the prisms. The two tangential terms arecomputed by differentiation of a polynomial determinedby a second order least-square fit of the vorticities ofprisms located within a given radius of the prism centerand in the layers immediately above and below theprism.

Several aspects of the time integration are noteworthy.First of all, the time step t is limited by CFL conditions in


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the normal and tangential directions as well as adiffusive stability requirement. In practice t satisfyingthese conditions is smaller than it would have to be inorder to compute accurate motion of the vortex tubes.Consequently, a number of iterations of the finite volumescheme, referred to as subcycles, are performed beforeupdating the positions of the vortex tubes. Thecontribution of the vortex tubes to the velocities in themesh is held fixed during the sub-cycling.

To solve the finite volume equations a boundarycondition is needed for the vorticity at the top sheet level.An effective approach in this regard is to solve the finitevolume equations only up to the second to last sheetlevel assuming the viscous flux in the wall normaldirection to be constant over this sheet. A calculation ofthe total vorticity flux due to convection and diffusion intothe top sheet level supplies the vorticity that is put intonew tubes. In this, when the convective flux is away fromthe surface the vorticity at the top sheet is not needed (inview of the upwind model), while if the flux is toward thesurface it is taken to be zero since all incoming vorticity isexpected to be in the form of vortex tubes.

The finite volume algorithm described here is found tooffer stable solutions to the 3D vorticity equation over awide range of flows that have been treated thus far.Moreover, tests have shown each separate part of thenumerical discretization to be consistent and convergentunder mesh refinement.

NEW VORTICES

Going hand in hand with the computation of the evolvingvorticity field in the wall mesh is the necessity ofcollecting information on the amount of vorticity that

fluxes outward into the top level of prisms. The fluxderives from both viscous and convective components.The vorticity that accumulates during the time steps inthe sub-cycling is turned into a vortex tube if itsmagnitude surpasses a threshold. The orientation of thenew vortex is clear from the relative magnitudes of thevorticity components. Its center is at the center of theprism whose vorticity it will take. The length of the newtube is set by the condition that its ends just intersect thesides of the prism. Its strength is determined via therelation

|s | = | | V (3)

in which V is the volume of the prism. This conditionforces the far field velocity from the new tube to matchthat of the prism.

VELOCITY EVALUATION

According to the Helmholtz decomposition theorem thevelocity in incompressible flow can be written as the sumof a divergence free part and a potential flow thatenforces the non-penetration boundary condition. For agiven vorticity field the solenoidal part takes the formof the Biot-Savart law

This forms the basis for (2) in the case of tubes, and forthe prisms it is integrated in closed form over the twolateral directions assuming constant vorticity. Since theprisms have high aspect ratio a simple midpointapproximation is used for the integration in the normaldirection. In view of the expense of evaluating thecomplex formulas that result from integration of (4), theyare used only in the near field. Elsewhere, the velocitydue to the prisms is computed assuming an equivalentvortex tube with strength given by (3). The potential flowis determined via a boundary element method thatmakes use of the surface triangularization. Thus thecomputation of the complete velocity field is done withoutimposition of a grid covering the flow domain.

The locations where the velocity field is to be evaluatedare known as field points." These may be considered tobe of three types: 1. the end points of vortex tubes notlocated within the mesh region; 2. the end point of tubes

that are in the mesh region; and, 3. the centers of thetriangles in the wall mesh. The contribution of the vortextubes to all categories of field points is the same: theyare evaluated using a parallel implementation of the FastMultipole Method (FMM) [8,14]. The contribution of thesheets to the first and third category of mesh points is viathe local or far field velocity sheet formulas as may berelevant. For the second category of points, however, thevelocity contributed by the sheets is computed via acombination of interpolation and least square fitting. Inparticular, the velocities from the third category are usedto estimate the velocities at the node points using areaweighted averaging. To then get the velocity for the endpoint of a tube that lies in a particular prism, a least-square fit is done using the six nodes of the prism. Amotivation for this special approach is to enhanceefficiency since the evaluation of the local sheet formulasfor large numbers of vortices can be expensive. It is alsothe case that this approach reduces some discretizationerror, particularly in the normal velocity component thatarises from the assumption of piecewise constantvorticity over coarsely sized triangles.

There are three separate calls to the FMM solver in thevelocity computation. The first is for the purpose ofevaluating the contributions of the tubes at all fieldpoints. In this the flow domain is covered by an oct-treewhose boxes, in principle, should be subdividedadaptively until there is an optimal number of tubes ineach box (approximately 100). Regions containinggreater numbers of tubes undergo more subdivisionsthan others. A tube is placed into a box according to thelocation of its center point. A similar oct-tree is developedfor the field points. The two trees are similar but mightnot be identical. The best performance of the FMM canbe expected to occur when the oct-trees have beengrown to their optimal levels.

The FMM works by combining the influence of tubes in aparticular box into a single multi-pole expansion, and


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then gathering such expansions together for parentboxes that hold smaller child boxes. Similarly,expansions used in computing field points are broughtfrom parent to child boxes down through the tree.Adjacent boxes at the smallest size affect each otheronly through the exact formulas, that is, such interactionscannot be included in the multipole expansions. For thisreason, having more than the optimal number of tubes ina box can hamper efficiency of the FMM solver.

In its application to computing the velocity due to tubes, itis generally not the case that the oct-trees can be grownto their optimal levels. Preventing this is the need tosmooth the Biot-Savart kernel at small distances: theFMM depends on the unsmoothed kernel for itsderivation. To maintain the effectiveness of the FMM incases where many particles accumulate in the smallestboxes, a ``middleman" scheme is used in which theFMM is used to evaluate the velocities at certain nodalpoints within the boxes and then 3D linear interpolation isused to get the velocities at the field points in theseboxes. Typically, the nodal points are the corners ofboxes constructed by two further subdivisions of the

smallest boxes in the FMM oct-tree.

The second call to the FMM is for contributions ofsheets. In this case the positions of tubes lying within themesh region are excluded from the field points. For thiscalculation the standard Biot-Savart kernel is used fornon-near neighbor boxes while the integral formulas fortriangles are used to compute the local contributions.Finally, the last call to the FMM is for the purpose ofcomputing the contributions of the wall sources to thevelocity field.

The use of the FMM reduces the nominal O(N 2 ) cost of

evaluating the velocities at the locations of N vortices toa much more affordable O (N log(N )) or even O(N)calculation. With parallelization, problems containingseveral million vortices can be treated in a reasonabletime frame. For example, with 4 million vortices, theVorCat code maintains excellent parallel scalabilitythrough at least 16 processors, with scalability increasingas the number of vortices increases. In a typical result, acalculation with 4 million vortices on 64 processors takesunder 1 minute of CPU time. This is sufficient to enableuseful calculations for many complex flows in areasonable time.

RESULTS

The previously described algorithm has been applied to anumber of ground vehicle flows with a view towardestablishing its accuracy and physical correctness. Thiswork is ongoing in the sense that most of thecomputations have yet to be performed at Reynoldsnumbers of O(10 6) that are typical of experimentalstudies of the Ahmed and Morel body flows. In particular,triangularizations appropriate to high Reynolds numberflow entail O(10 5) surface triangles and several millionprisms to achieve a fine coverage of turbulent structuresnext to boundaries. Such computations can be expected

to produce O(10 8) tubes and thus require a significantinvestment in computational resources (e.g., a largernumber of parallel processors than are typically availablefor this developmental work).

In the interest of more rapidly exploring the performanceof the scheme over a wide range of cases the presentresults have been obtained from simulations withapproximately 500,000 prisms and several million vortextubes. These more modest computations can becompleted within a day or two using 16 or 32 processorparallel computers. The results given here are forReynolds numbers no greater than 500,000 since this isthe upper limit for which a surface mesh containingapproximately 30,000 triangles can provide resolutionthat is at least adequate for representing the physics ofthe viscous sublayer.

Figure 1. U + at Re=360,000. , DNS from [13]; gridfree scheme with fine mesh; _ _ , gridfree schemewith coarse mesh.

BOUNDARY LAYER

The particular issue of surface resolution has beenstudied via analysis of the velocity statistics of theturbulent boundary layers that form on the top andbottom surfaces of the Morel and Ahmed bodies. Forexample, Fig. 1 is a comparison of the mean velocityscaled in wall coordinates as computed from a Morelbody at Re = 360,000 (based on streamwise bodylength) compared to a DNS calculation [13] and the

classic log law relation U = 1/0.41 log(y +

) + 5. In thisdiscussion (x,y,z) respectively denote the streamwise,normal and transverse coordinates. The DNS solutionhas Reynolds number based on momentum thickness R = 500, and the computed solution has R = 622. Both acoarse mesh and a fine mesh solution are shown, withthe latter used just on the top surface of the body to limitthe overall number of triangles. It is seen that in thisinstance the better resolution makes a significantdifference in so far as the accuracy of the prediction isconcerned. The different mesh characteristics arereflected in their spanwise resolution. For the fine meshon average z+ = 107.2 with aspect ratio 16.3, while for


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the coarse mesh z+ = 240.4 with aspect ratio 36.9.Even though the fine mesh is less than ideal, it is stillseen to produce a mean flow that is consistent with thatof a turbulent boundary layer and, in fact, has a clear loglaw. In contrast, the coarser mesh is prone to significanterror. For both computations, y+ 6.5. The results forthe Ahmed body flow presented below are obtainedusing meshes (depending on Reynolds number) thatvary between the coarse and fine resolutions seen here.

The normal Reynolds stresses corresponding to themean velocity in Fig. 1 are somewhat over predicted asshown in Fig. 2. With better resolution these statisticsimprove. For example Figs. 3 and 4 show results for themean velocity, Reynolds shear stress and turbulentkinetic energy for a better resolved simulation at Re =64,800. The first two statistics are excellent. Thedisplacement in peak kinetic energy away from the wallmay reflect the need for yet an additional enhancementto the wall resolution or may be due to differencesbetween the Morel body and boundary layer flows.

Figure 2. Normal Reynolds stresses for Re=360,000boundary layer. , DNS from [13]; , gridfree scheme.

Figure 3. Mean U at Re=64,800. , DNS from [13]; ,gridfree scheme.

Figure 4. Reynolds strear stress and turbulent kineticenergy for boundary layer at Re=64,800. , DNS from[13]; , gridfree scheme.

AHMED BODY FLOW STATISTICS

Several calculations have been made of the Ahmed bodyflow including an inviscid rather than viscous groundplane. Though this prevents the simulation from includingthe effect of vorticity generated on the floor of the windtunnel, it has the advantage of reducing the scale of thecomputations while still allowing for meaningfulcomparisons with experiments. A viscous ground planewill be included in future computations of the Ahmedbody flow. Some idea of the quantitative accuracy of theapproach is obtained by comparing computed andmeasured mean velocities and Reynolds stresses for thecase with 25 o base slant angle. The computations are forRe = 500,000, a value that is at the upper limit of whatcan be tolerated with available computer resources, butstill less than the value R e = 2,784,000 used inexperiments. In the computations the Ahmed body isrepresented using a surface mesh containing 32,432triangles with an additional 3920 triangles on the groundplane. With 10 layers of prisms above the surface prismsthere are thus 324,320 total. A posteriori it is determinedthat the average prism in the simulation has thickness12.8 in wall coordinates and average spanwise extent of280. While this surface mesh is clearly less than optimalas suggested by the boundary layer results, nonethelessthe predicted statistics show much agreement with thephysical experiments. This is now illustrated in a number

of comparisons for streamwise, normal and transversemean velocities, U, V and W, respectively, turbulent

kinetic energy, K, and Reynolds shear stress, uv , atvarious locations situated along the centerline of thebody and above the rear window. The numericalcalculation is run from an impulsive start untilapproximately t = 1.5 with averaging performed overessentially the last 0.5 time units. The front surface of thebody is at x = 0, the rear is at x = 1, the ground plane isat y = 0, and the bottom and top surfaces of the Ahmedbody are at y=0.04675 and 0.3253, respectively.Moreover, the body extends in the spanwise direction


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from z = -0.1874 to 0.1874 and the rear slanted windowbegins at approximately x = 0.837 and ends at x = 1.

Figures 5 - 8 pertain to the mean streamwise velocity, U,while Figs. 9 12 are for V. Figures 5 and 9 are forlocations on the centerline in front of the body and on thetop surface. Ahead of the body the close agreementbetween experiment and computation reflects the moreor less potential nature of the flow in this region. Near y =0 the effect of using an inviscid boundary is visible. Theplots for x > 0 show that the acceleration of the fluid overthe top surface is captured as well as the boundary layerprofiles over the rear window. In these results it can beexpected that the modeling of the turbulent physics playsa role. It may also be noticed that the qualitativeagreement with experiment is quite good, in addition tothe quantitative accuracy.

The predicted wake characteristics on the centerline areshown in Figs. 6 and 10. Just downstream of the body, atx = 1.0364 the wake profile is fairly well capturedincluding the recirculating zone. The comparisons furtherdownstream suggest that the extent of the wake deficit is

smaller than experiment. It may be expected that anumber of factors contribute to this disparity, with themost obvious being the smaller Reynolds number of thesimulation in comparison to experiment. This issupported by a calculation at R e = 100,000 that wasobserved to have a smaller wake deficit than thecalculation at higher Reynolds number. The flow in thewake is highly non-steady and it is clear that theaveraging interval that is used here to obtain statistics isnot sufficiently long to produce entirely smooth results. Infact, this limitation is felt most strongly in the last fewmeasurement stations and may also have someresponsibility for the smaller than measured wake deficit.

For example, it may be inferred from the last plot in Fig.10 that the vortices produced by the body have barelyreached this location by the end of the simulation.

Figures 7 and 8 for U and Figs. 11 and 12 for V illustratethe quality of the computed solution at different spanwiselocations at two different x positions centered over therear window. Many details of the boundary layer in thisregion appear to be well captured. It is interesting to notethat the profiles at z = 0.21073 are just off to the side ofthe body. These are embedded in the turbulent,separated boundary layer that is generated on the side ofthe vehicle. These curves are clearly quite noisy andrequire a longer averaging interval to produce smoothstatistics. Figure 13 shows W at three z locations oneither side of the body. This plot demonstrates that theaverage flow over the window is toward the centerline, asis expected in the case of the 25 o base slant angle.

Figure 5 Computed streamwise velocity on centerline, ; vs. experiment [10], .

Figure 6. Computed streamwise velocity on wakecenterline, ; vs. experiment [10], .

Figure 7. Computed streamwise velocity at spanwiselocations at x = 0.8678, ; vs. experiment [10], .


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Figure 8. Computed streamwise velocity at spanwiselocations at x = 0.9157, ; vs. experiment [10], .

Figure 9 Computed normal (V) velocity on centerline, ;vs. experiment [10], .

Figure 10. Computed normal (V) velocity on wakecenterline, ; vs. experiment [10], .

Figure 11. Computed normal (V) velocity at spanwiselocations at x = 0.8678, ; vs. experiment [10], .

Figure 12. Computed normal (V) velocity at spanwiselocations at x = 0.9157, ; vs. experiment [10], .

Figure 13. Computed transverse (W) velocity atspanwise locations at x = 0.8678, ; vs. experiment[10], .


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Figure 14 compares the predicted K vs. experiment atpoints along the centerline. The first two locations areover the window and the remaining four are in the wake.The accuracy of the individual curves varies somewhatfrom location to location with some trends quiteaccurately captured. There is a clear need for longeraveraging to produce smoother statistics and this mighthave some effect on the uniformity of the predictions. Infact, it is not hard to imagine that higher order statisticssuch as K are particularly vulnerable to the influence ofindividual large scale flow events if the averaging is overtoo short a time interval. In any case, it is interesting tonote that the double-peaked K distribution in the nearwake is well reproduced, particularly at x = 1.0364.Figures 15 and 16 show K for locations over the rearwindow. Here, more averaging is clearly required if theaccuracy of the predictions is to be carefully evaluated. Aclear trend toward over prediction is visible in theseresults. This is perhaps related to the use of an overlycoarse surface mesh, very much like the situationpreviously encountered in regards to Fig. 2.

Figure 14. Computed turbulent kinetic energy oncenterline, ; vs. experiment [10], .

Figure 15. Computed K at spanwise locations at x =0.8678, ; vs. experiment [10], .

Figure 16. Computed turbulent kinetic energy atspanwise locations at x = 0.9157, ; vs. experiment[10], .

Figure 17. Computed Reynolds shear stress at spanwiselocations at x = 0.9157, ; vs. experiment [10], .

The Reynolds shear stress predictions in the middle ofthe rear window are shown in Fig. 17. The numericalmodel clearly is able to reproduce the preferential

correlation that underlies uv as was previously seen inFig. 4. This generally requires that the vortical structuresof turbulence that cause the Reynolds shear stresses arecorrectly accounted for in the simulation. More insight

into this aspect of the solution can be achieved byvisualizing the gridfree vortical elements that make upthe flow, as will now be considered.

VISUALIZATIONS

Visualization results are shown primarily for a relativelywell resolved computation in which the base slant angleis 30 o and the Reynolds number R e = 40,000. In thiscalculation there are 28,538 triangles on the vehiclesurface. Three-dimensional renderings of the vorticesgive an indication of separation regions and the locationsand degree of turbulence production. Such views often


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show where many small vortices have combined to formlarger scale structures that might be of interest. It is alsopossible to obtain information about the origination onthe body of vorticity appearing in the wake.

Shown here in Figs. 18 - 20 are images at time t = 1.91of the approximately 2.5 million vortex tubes that havebeen generated in the course of a calculation of theAhmed body flow impulsively moved from rest at t=0.The figures respectively show views from the side, thefront and the top. Both Figs. 18 and 20 show boundarylayers forming from the leading edge of the body, theirdownstream transition to turbulence, and the formationof turbulent structures. The latter are most visible in thelarge mainly streamwise agglomerations of vortices aswell as the corrugated outer edges of the vortex layers.Such structures are consistent with the knowncharacteristics of turbulent boundary layers and helpexplain the good agreement in predicting velocitystatistics shown previously. Another perspective on theflow is given in Fig. 21 showing contours of normalvelocity just above the top surface of the body. There isclear evidence here for the existence of the streaky

structure that is the footprint of coherent streamwisevortices in boundary layer flow [5].

Figure 18. Side view of vortices in Ahmed body flow.

Figure 19. Front view of vortices in Ahmed body flow.

Figure 20. Top view of vortices in Ahmed body flow.

Figure 21. Wall normal velocity above top surface of theAhmed body.

The flow over the rear window is an important feature ofthe Ahmed body flow. Instantaneous quiver plots ofvelocity vectors in planes parallel to the symmetry planeare shown in Figs. 22 and 23 for base slant angles of12.5 o and 30 o, respectively. In the former case there isno separation over the window, though a largerecirculating vortex forms against the rear. These resultsare consistent with the known behavior of the flow at thisangle. For the larger angle there is clearly a considerabledegree of separation over both the rear window andback. Animations of the velocity field show the separatedflow regions to vary rapidly in time and spatial position.Once again the behavior is consistent with the high dragconfiguration of the Ahmed body that occurs at baseslant angles in the vicinity of 30 o.


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Figure 22. Instantaneous velocity vectors over the rear ofthe Ahmed body at 12.5 o base slant angle.

Figure 23. Instantaneous velocity vectors over the rear ofthe Ahmed body at 30 o base slant angle.

Some insights into the vortical nature of the flow over thewindow may be obtained from Figs. 24 and 25. Theformer shows contours of streamwise vorticity close tothe body surface in which there is a clear pattern ofalternating sign in the vorticity just as the flow movesover the window. This hints at the presence of somespanwise structure to the separated flow at this location,not unlike similar phenomena in mixing layers [4]. Thevisualization of vortices in Fig. 25 gives some evidence

of the vortical flow over the C-pillars. This turbulent statehas evolved from the situation shown in Fig. 26 that isnot long after the start up of the calculation and beforethe flow has transitioned to turbulence. Here the vorticesare clearly seen to curl over the C-pillars. It is alsointeresting to note the rollup of the vorticity shedding offthe bottom rear edge of the body.

Figure 24. Streamwise vorticity contours on the topsurface of the Ahmed body flow.

Figure 25. Detail of vortices near the rear window at anearly time showing wrapping of vortices over the C-pillars.

Figure 26. Vortices shed off rear shortly after startup ofthe simulation.


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FORCES

The pressure in a vortex method is determined at anyfixed time step as a post-processing step using theinstantaneous vorticity distribution that is contained in thetubes and prisms. A particularly convenient way to getpressure on the solid viscous surfaces appearing in acomputation is by an application of Greens third identity[15]. Due to the appearance of a singularity in theGreens function, however, some care must be given tohow the numerical quadrature is carried out. Forexample, the computed pressure can be a sensitivefunction of the evolving vorticity field, and if so, relativelylong time averages are needed to get mean pressures.Integrated quantities depending on pressure such as thedrag coefficient are less affected by transientfluctuations.

In the present case a higher order pressure solver that isuseful for obtaining local pressure fields is in the finalstate of development and results from this will bereported in the future. A less accurate, preliminarypressure solver has been developed in order to verify the

concept. This is useful for obtaining integrated quantitiessuch as drag but requires long time averaging to getconverged pressure statistics. Fig. 27 shows thecomputed drag force for the 25 o Ahmed body as afunction of time. To the extent that a comparison can bemade between flows at quite different Reynoldsnumbers, it may be noted that the value in the figure atthe last time is in the vicinity of 0.42 and this may becontrasted with the experimental value of 0.36.

Figure 27. Time dependence of computed dragcoefficient for 25 o Ahmed body simulation at R e =500,000.

CONCLUSION

The computational results presented here demonstratesome of the capabilities of a gridfree approach to largeeddy simulation in modeling ground vehicle flows. Whilemany of the difficulties of grid generation that typicallyaffect LES schemes are avoided in this work, the study

does show the importance of having a reasonably finenear-wall mesh to resolve the important vorticityproducing motions upon which the gridfree elementsdepend. The wall resolution issue reflects the fact thatwith the absence of a turbulence model per se, theReynolds number is a real parameter in this method andits effect on the physics has to be taken into accountwhere necessary.

The present computations demonstrate thatquantitatively accurate predictions of velocity statisticscan be obtained as long as appropriate wall resolution ismaintained. The simulations are seen to provide a wealthof visual information about the nature of the turbulentand rotational motions produced by ground vehicles.Future computations will be focused on scaling up thesimulations to exactly match the conditions ofexperiments. This is primarily a question of securingcomputational resources commensurate with the largestorage requirements for simulations containing tenmillion or more vortices and several million wall prisms.

ACKNOWLEDGMENTS

This research has been supported through an ATP/NISTaward to VorCat, Inc. Computer time is provided in partby NCSA.

REFERENCES

1. Ahmed, S. R., et al. (1984) Some salient features ofthe time-averaged ground vehicle wake, SAE Paper No.840300.

2. Basara, B., et al. (2001) On the calculation ofexternal aerodynamics: industrial benchmarks, SAEPaper No. 2001-01-0701.

3. Bernard, P.S. and Wallace, J.M. (2002) Turbulent Flow: Analysis, Measurement and Prediction , John Wiley& Sons.

4. Bernard, P. S., et al. (2003) Studies of turbulentmixing using the VorCat implementation of the 3D vortexmethod, AIAA Paper No. AIAA-2003-3599.

5. Bernard, P. S., et al. (2004) Gridfree Simulation ofTurbulent Boundary Layers Using VorCat, AIAA PaperAIAA-2003-3424.

6. Chorin, A. J. (1993) Hairpin removal in vortexinteractions II, J. Comput. Phys . 107 , 1 - 9.

7. Duncan, B. D., et al. (2002) Numerical simulation andspectral analysis of pressure fluctuations in vehicleaerodynamic noise generation, SAE Paper No. 2002-01-0597.


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8. Greengard, L. and Rokhlin, V. (1987) A fast algorithmfor particle simulations, J. Comput. Phys . 73 , 325-348.

9. Krajnovic, S. and Davidson, L. (2001) Large-eddysimulation of the flow around a ground vehicle body, SAEPaper No. 2001-01-0702.

10. Lienhart, H. Stoots, C. and Becker, S. (2000) Flowand turbulence structures in the wake of a simplified carmodel (Ahmed model), DGLR Fach. Symp. Der AG STAB, Stuttgart Univ.

11. Morel, T. (1978) Aerodynamics drag of bluff bodyshapes characteristic of hatch-back cars, SAE Paper780267.

12. Puckett, E. G. (1993) Vortex methods: anintroduction and survey of selected research topics, inIncompressible computational fluid dynamics: trends and advances , edited by M. D. Gunzburger and R. A.Nicolaides, Cambridge University Press, Cambridge, pp.335-407.

13. Spalart, P.R. (1988) Direct simulation of a turbulentboundary layer up to R = 1410, J. Fluid Mech . 187 , 61-98.

14. Strickland, J. H. and Baty, R. S. (1994) A three-dimensional fast solver for arbitrary vorton distributions,Report SAND93-1641, Sandia National Laboratory.

15. Uhlman, J. S. (1992) An integral equation formulationof the equations of motion of an incompressible fluid,Tech. Report 10086, NUWC-NPT.

16. Verzicco, R., et al (2002) Large eddy simulation of aroad vehicle with drag-reduction devices, AIAA J. 402447-2455.

17. Zhang, Y. J., et al. (2003) A hybrid unstructuredfinite-volume algorithm for road vehicle flowcomputations with ground effects, J. Vehicle Des . 33365 - 380.

CONTACT

Please address inquiries to Peter Bernard [email protected].

peter s. bernard, pat collins and mark potts- vortex method simulation of ground vehicle...

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