numerical simulation of the flow around the ahmed vehicle model

8/7/2019 Numerical Simulation of the Flow Around the Ahmed Vehicle Model

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Latin American Applied Research 39:295-306(2009)

296

Figure 1: Ahmed model, dimensions in mm.Table 1: Time-averaged flow in the near-wake of the Ahmedvehicle as a function of the slant angleα, whereαm≈ 12.5º andα M ≈

30.0º.

slant angleα time-averaged flow in the near-wakeα < αm almost 2D attachedαm < α < α M massively separated 3Dα > α M almost 2D attached

resent two benchmarks widely used in the automotiveindustry (Basara and Alajbegovic, 1998).

On one hand, among numerical simulations, a LESaround a bus-like vehicle is developed by Krajnović (2002), Krajnović and Davidson (2003), while Gilliéronand Spohn (2002) showed a Detached Eddy Simulations(DES) over another simplified vehicle model. Othersources of related work can be found in Hucho (1998)and ERCOFTAC (2007).

Regarding numerical benchmarks with the presentmodel, several computations were performed using theReynolds Averaged Navier-Stokes (RANS) equations(Han, 1989; Basaraet al. , 2001; Basara, 1999; Basaraand Alajbegovic, 1998). On the other hand, LES wereemployed by Gilliéron and Chometon (1999) for a slantangle of 35◦ with an unstructured mesh, while Kapadiaet al. (2003) and Krajnović and Davidson (2004) haveconsidered a slant angle of 25◦. As a more recent reviewof the subject, see Krajnović and Davidson (2005a,b).B. The RANS approachThe RANS equations determine mean flow quantitiesbut they require turbulence models to close the system,i.e. an equal number of equations and unknowns. Thereare many closure models proposed (Wilcox, 1998) butunfortunately it is very difficult to find one that can ac-curately represent the Reynolds stresses in the detachedflow regions of bluff bodies, where the flow regime isstrongly dependent on the geometry, specially at therear part, and they often lead to complex flow patterns.A deficiency of the most used closure models for RANSschemes is their inherent inability to deal with mas-

sively separated flows containing many coherent struc-tures. Thus, the mean pressure could not be accurately

predicted for these cases leading to a poor estimation of the mean aerodynamic forces. Physical turbulence con-sists of an intricate interplay between random and co-herent structures, whose intermittency (an increasingsparseness of the small-scale eddies) is a typical prop-erty. Moreover, it was shown that solutions of RANSequations agree with the time-averaged solutions of theunsteady N-S equations only if the Reynolds stress ten-sor used in the RANS equations is well approximated(McDonough, 1995). However, this requirement is notgenerally reached at present, which leads to large differ-ences between the numerical results obtained with bothstrategies. Krajnović and Davidson (2004) have notedthat previous RANS numerical simulations of the flowaround the Ahmed model performed either relativelywell or poorly according to the slant angleα, which in-fluences the main topology of the time-averaged flow inthe near wake, as it was summarized in Table 1. Theyargued that possible reasons of the failures could be dueto (i) higher level of flow unsteadiness; (ii) the fact thatmost RANS turbulent models miss the flow separationat the rear part for such slant angles; and (iii) the factthat optimized RANS turbulent models for predictingthe separation onset fail in the massively separated flowregions. For instance, even though the k -ω simulationof Han (1989) used good quality grids, an accurate nu-merical scheme and proved grid convergence, a dragcoefficient 30 % higher than the experimental one waspredicted (Wilcox, 1998). From the numerical results,Krajnović and Davidson (2004) conclude that the widerange of the turbulent scales around the rear end which,together with very unsteady reattachments in this re-gion, are possible reasons for failures.C. The LES approachIn LES only the larger unsteady turbulent motions arecomputed while the effect of the small (unresolved)scales on the large (resolved) ones is modeled. The lar-ger scales contain most of the kinetic energy of the flowand they are affected by the geometry and forcing.

Thus, they are in general highly inhomogeneous andirregular, making their flow description most relevant



G. FRANCK, N. NIGRO, M. STORTI, J. D’ELÍA

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from an engineering perspective. The smallest scales of fluid motion are considered “universal”, in the sensethat the dynamics of the small scales is independentfrom the flow geometry and forcing, and solely con-trolled by energy transfers due to scale interactions (Ca-lo, 2005). Most of the computational effort in Direct

Numerical Simulation (DNS) is spent on solving thesmallest dissipative scale, while LES models focus ontheir global effect. The computational efficiency of LESversus DNS arises as trade-off with accuracy since sucha modeling reduces the accuracy of the method. In con-trast to RANS, it can be shown that LES proceduresgenerally converge to DNS (McDonough, 1995) andthus their computed solutions can be expected to con-verge to Navier-Stokes solutions, as space and time stepsizes are refined.

On the other hand, many theoretical problems in deal-ing with the boundary conditions for LES can be identi-fied (Sagaut, 2001). From a mathematical point of view,

two problems dealing with boundary conditions for LESare: (i) incidental interaction between spatial filter andexact boundary conditions; and (ii) the specification of boundary conditions leading to a well-posed problem. Acommon approach consists in considering that the filterwidth decreases nearby the boundaries so that the inter-action terms cancels out. Then, the basic filtered equa-tions are left unchanged and the remaining problem is todefine classical boundary conditions for the filteredfield. Another option is the use of “embedded” bound-ary conditions which consists of filtering through theboundary leading to a layer along the boundary whosewidth is order of the filter cutoff length scale, but some

additional source terms must be computed or modeled.From a physical point of view, the boundary conditionsrepresent the whole flow domain beyond the computa-tional one and they must be applied to all scales (to allspace-time modes).

There are some problems with the boundary condi-tions at solid walls and inflow sections which are moreclosely related to LES while the corresponding ones atthe outflow section are not specific to this method (Sa-gaut, 2001). In the case of inflow conditions, severalstrategies are proposed (Sagaut, 2001), for instance: (i)“stochastic reconstructions” from statistical one-pointdescriptions; (ii) “deterministic computations”, as the“precursor simulation”, where a first computation isperformed for the attached boundary layer flow withoutthe body and the data from some extraction plane areused as inlet boundary condition; and (iii) “semi-deterministic reconstructions”: they propose an inter-mediate approach between the previous ones to recovertwo-point correlations of the inflow with no preliminarycomputations. Also, mixed strategies have been de-signed as the hybrid RANS-LES approaches (e.g. “zon-al decompositions”, “nonlinear disturbance” equationsand “universal modeling” (Sagaut, 2001)), which areemployed to decrease the rather large computationalcost of LES due to (i) the goal of directly capturing allthe scales motions of turbulence production and (ii) anobserved inability of most subgrid models to correctly

account for anisotropy.From a practical point of view, most of the LES

schemes use structured grids but in these the unknownnumbers easily become too large in wall-bounded flows(Krajnović and Davidson, 2004). As the mesh pointsshould be concentrated where they are needed, e.g. on

boundary layers and separated flow regions, some strat-egies are proposed. On structured meshes, for instance,this was achieved combining structured multi-blockusing O and C grids (Krajnović and Davidson, 2004),but other options can be proposed with the same aim. Arecent work is the Krajnović and Davidson (2004) one,where a LES-Smagorinsky model for the SGS stresseswas used with a three dimensional Finite Volume Me-thod (FVM) on a fairly structured mesh and a slant an-gle of 25º.

The main objectives of the present work are: (i) toperform large eddy simulations with unstructured mesh-es and finite elements of the unsteady flow around the

same model but for a slant angle of 12.5◦ (a rear endvariant) using mixed meshes, i.e. unstructured meshesplus several layers of prismatic elements close to theboundary layers; (ii) an assessment of the LES turbu-lence model combined with a SUPG-PSPG stabilizationmethod to predict the main flow dynamic structures thatare involved in vehicle aerodynamics without a deepanalysis of sensitivity of the results with the turbulence;(iii) a visualization of the time-averaged flow in thenear-wake and determination of coherent macro struc-tures (i.e. high space localized and time-persisting vor-tex flows) by means of the second invariant of the gra-dient velocity or Q-criterion (Dubief and Delcayre,

2000); and (iv) a topological validation of the numberand type of the singular points (where velocity vanishes)on the symmetrical vertical section.

II. GEOMETRICAL PARAMETERSThe Ahmed model has a length L = 1044 mm, which isapproximately a quarter of a typical passenger carlength. The height H and the width B are defined ac-cording to the ratio ( L: B: H ) = (3.36:1.37:1). It has threemain geometrical sectors: the front one, with boundariesrounded by elliptical arcs to induce an attached flow, amiddle sector which is a box shaped sharp body with arectangular cross section and, finally, a rear end sector.A set of interchangeable rear ends with sharp edgeswere tested, where the slant length is kept fixed tols =222 mm. The 12.5º slant angle was analyzed here.

Figure 2 shows the dimensions of the computationalflow domain relative to L. A Cartesian coordinate sys-tem O( x, y, z) is employed whose origin is placed on theground at the front sector. The body is suspended to zbase =0.048 L from the wind tunnel ground. The parallelepi-pedic domain has 10 L×

2 L×

1.5 L in the streamwise y,

spanwise x and stream-normal (vertical) z Cartesiandirections respectively. The inlet flow section is placed2.45 L upstream of the model front while the outlet flowsection is placed 6.6 L downstream from the model rear

end.




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Figure 2: Dimensions of the computational flow domain as function of the body length L.

III. FINITE ELEMENT DISCRETIZATIONA. Mesh generationThe meshing process is performed with the Meshsuitemesh generator (Calvo, 1997). It involves a basic tetra-

hedral generation and the addition of layers of wedgeelements for a better resolution close to the body sur-face. The meshing is done following four stages:1. Lines meshing: a subdivision of the lines defining

the boundaries (wire model) using CAD resources;2. Surfaces meshing: they are generated from the

geometrical definition of the surface boundaries anda refinement function implicitly defined by the spac-ing of the one-dimensional mesh and, if necessary,some interior surface points. The bluff body and itswake are surrounded by an auxiliary surface whichis also meshed;

3. Volume meshing: the auxiliary surface defines a

sub-domain which is used for better user-control onthe volumetric meshing, where refinement in zonesclose to the body surface and wake is chosen.

4. Wedge elements addition : wedge elements (pris-matic elements of triangular basis) are added to thetetrahedral mesh in order to obtain better simulationsof boundary layers without excessive refinements inthe remaining flow domain. Since the final mesh hasboth wedge and tetrahedral elements, special consid-eration is needed for parallel implementation.

A general view of one of the meshes for the wholecomputational flow domain is given in Fig. 3.B. Some parameters of the combined wedge-

tetrahedral meshesFor a better comparison, it is assumed that each wedgeis nearly equivalent to 2.5 tetrahedrons. Two combinedwedge-tetrahedral meshes are employed and they havethe following features:a) mesh # 1: it has 151 k-wedges close to the body sur-

face, 450 k-tetrahedrons, and 170 k-nodes (83 k-nodes on the wedge part plus 87 k-nodes on the tet-rahedral part), which is equivalent to a pure tetrahe-dral mesh with 828 k-elements and 170 k-nodes;

b) mesh # 2: it has 202 k-wedges close to the body sur-face, 450 k-tetrahedrons, and 198 k-nodes (111 k-nodes on the wedge part plus 87 k-nodes on the tet-rahedral part), which is equivalent to a pure tet-

Figure 3: Unstructured finite element mesh for the computa-tional flow domain around the Ahmed model.

Table 2: Length scales of TaylorηT = 265.68× 10-5 L and Kol-

mogorovηK =1.27×10-5 L, relative to the averaged mesh sizeh

m. The normal element spacing of the first wedge-layer on the

body surface arehm ≈ {63.50, 31.75}×10-5 Lmesh # 1 mesh # 2

η T /hm 4.18 8.37η K /h m 0.02 0.04

rahedral mesh with 955 k-elements and 198 k-nodes.The mesh generator eliminates major degeneracies

with exception of slivers and cups and then, as a meshquality assessment, is sufficient to measure how near isa node from the opposite face. This is computed throughthe quality factor eee d h minmin / =γ , for e =1, 2, ..., E ,where eh

minand ed

minare the minimum height and edge

length, respectively, on each element. In the presentcase, the quality factors, without smoothing, are

minγ =0.02 and maxγ =0.95 which are taken as accept-able.

C. Turbulent scales and mesh spacingThe Kolmogorov length scaleηK is the length scale forthe smallest turbulent motions, while the Taylor oneηT gives more emphasis to intermediate turbulent motionswhose length scales are near some integral length scaleof the mean flow. Empirical correlations for both aregiven by L A LK

~Re 4 / 3~

4 / 1 −−=η and L A LT

~Re15 2 / 1~

2 / 12 / 1 −−=η ,

respectively, e.g. see Howard and Pourquie (2002),




299

where A is an O(1) constant, L~ is some integral scaleand v LU L

/ ~Re~ ∞= is the corresponding Reynolds num-

ber, where ν is the kinematic viscosity of the fluid and∞U the non-perturbed speed. In the present case, the

following values are adopted: A =0.5, =≡ L L~ 1.044 m

(i.e., the body length), while the remaining values aredened in Sec. D. Then, 6

~ 1025.4Re ×= L

,

LT 31066.2 −×≈η and LK

51027.1 −×≈η . On the otherhand, the normal element spacing of the first wedge-layer (on the body surface) are

{ } Lhm410175.3,350.6 −×≈ . Therefore, close to the body

surface, the mesh spacing normal to the body surfaceare smaller than the Taylor length scale but bigger thanthe Kolmogorov one (see Table 2). It can be remarkedthat the turbulent eddies down to the Taylor length scalesize should be well modeled.

D. Fluid flow parameters and boundary conditionsA Newtonian viscous fluid model is adopted, with non-perturbed speed ∞U = 60m / s and kinematic viscosityν=14.75×10-6

m2 / s (air at 15 C). The Reynolds number

(based on the model length L) is Re = 4.25×106. At theinflow section a uniform axial velocity profile∞U isimposed, while a non-slip boundary condition at theground is prescribed which, in turn, is moving with thesame stationary velocity ∞U . These boundary condi-tions emulate road tests or wind tunnel tests with aboundary layer control and they have been employed inmany related studies (Han, 1989; Krajnović and David-son, 2004; Basara and Alajbegovic, 1998) (wind tunnelshave several mechanisms to ensure that the boundarylayers are well attached to the tunnel walls, e.g. by suc-tion techniques). Under these conditions the equilibriumflow is uniform flow, and no boundary layers are devel-oped. Thus, the vehicle model is assumed at rest and theboundary conditions are: (i) uniform flow at inlet∞U ;(ii) no-slip at the ground which in turn is moving withvelocityu= ∞U ; (iii) null pressure at outlet; (iv) no-slipat the body surface (u=o ) and (v) slip at the lateral andupper sides.

IV. STABILIZED FINITE ELEMENTSThe finite element method is used to solve the momen-tum and continuity equations for the velocity and pres-sure at each node and at each time step. For simplicity,equal order spatial discretization for pressure and veloc-ity is highly attractive, e.g. linear tetrahedral or pris-matic elements, but it is well known that this kind of basis functions needs to be stabilized to achieve stablesolutions. This is performed with a SUPGPSPG schemewhose implementation details are given in Garibaldiet al. (2008).

A. Large Eddy SimulationIn LES techniques, the momentum balance equationsare solved with an “effective” kinematic viscosityνe = ν

+ νt , which is the sum of the molecular part calculatedas ν = µ/ ρ, plus a “turbulent” oneνt . The last one is es-timated in the PETSc-FEM (Sonzogniet al. , 2002) codeby means of the Smagorinsky sub-grid model coupledwith a modified van Driest near-wall damping factor

v f ~ , and given by

);(~); / exp(1

;)(:)(22

+

++

−=

−−=

=

yd H f f

A y f

uu f hC v

vv

v

veSt ε ε

(1)

in whichC S is the Smagorinsky constant (C S ≈ 0.10 for

flows in ducts) and )(:)( uu ε ε is the trace of the strainrate )(uε , while )( +− yd H is the Heavside function andd is a certain threshold distance from the body surface.A constant value A+ =25 is adopted, while y+ = y/yw is thenon-dimensional distance from the nearest wall ex-pressed in wall units yw= ν /u τ, whereu τ=(τ w / ρ)1/2 is thelocal friction speed andτ w is the local wall shear stress.The van Driest near-wall damping factor f ν reduces the“turbulent” kinematic viscosity close to the solid walls,but it introduces a non-local effect in the sense that the“turbulent” kinematic viscosityνt inside an element alsodepends on the state of the fluid at the closest wall. It isa near-wall modification of the Prandtl’s mixing lengthturbulence model and was found to be useful in attachedflows. It is written in terms of y+ so that in regions withsignificant wall friction the factor is relevant only in athin layer near the body skin. On the other hand, in re-gions where the local friction speeduτ takes very low ornull values (usually in the separation and reattachmentregions) the influence of the damping factor f ν is signifi-cant at large distance from the body. In order to preventthis drawback, it is further restricted to act within thethreshold distanced , giving the modied onev f

~ .

B. Parallel computingThe numerical simulations were performed using a do-main decomposition technique (Paz and Storti, 2005) inthe PETSc-FEM (2008) code, which is a parallel mul-tiphysics finite element library based on the MessagePassing Interface MPI (2007) and PETSc-2.3.3 (2007),and used in several contexts, for instance, for modelingfree surface flows (Battagliaet al. , 2006; D’Elíaet al. ,2000), added mass computations (Storti and D’Elía,2004) or inertial waves in closed ow domains (D’Elíaet al ., 2006). The present problem was solved using theBeowulf Geronimo cluster (2005), with 11 P4 nodesand 2 GBytes of RAM memory each.

V. EXPERIMENTAL NEAR-WAKE FLOWEven though the wake flow of any bluff body is basi-cally unsteady, the time-averaged flow exhibits somecoherent macro structures (i.e. high space localized andtime-persisting vortex flows) that appear to govern thedrag generated at the rear end by massive flow separa-tions. The experimental information obtained collectingseveral wind tunnel tests on this model (Ahmedet al. ,1984; Sims-Williams, 1998; Bayraktaret al. , 2001;




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Spohn and Gilliéron, 2002; Lienhart and Becker, 2003)show two critical slant angles:αm≈ 12.5º andα M ≈ 30.0º,where the main topological structure of the time-averaged flow is summarized in Table 1. These experi-mental tests show the following overall behavior of thetime-averaged flow as a function of the slant angle:

1. When the slant angle is lower than the first criticalone (α< αm), see Fig. 4, I and II, the time-averagedflow in the near-wake is almost two-dimensional andit remains attached on the top and slanted surfaces. Itonly separates past the vertical rear surface. In thenear-wake there are two re-circulatory time-averaged sub-flows A and B located one over theother (see sketch in Fig. 5). The shear layer comingoff the slant side edge rolls up into a longitudinalvortex. At the top and bottom edges of the verticalsurface, the shear layers roll up into the two re-circulatory time-averaged sub-flows A and B. Oil-flow pictures on the vertical surface suggest that

these re-circulatory time-averaged sub-flows are notbounded but they extend downstream. Therefore,they can be thought as generated through two horse-shoe vortices placed one over the other, inside theseparation bubble D . The bound legs of these vor-tices are almost parallel to the vertical surface whilethe trailing legs of the upper vortex A align them-selves in the direction of the onset flow and mergewith the vortexC coming off the slant side edges.The streamlines of the toroidal vortices on the verti-cal surface produce the singular point N (see sketchin Fig. 5) and when the slant angle increases, thesingular point N moves downward;

2. When the slant angle is nearly equal to the first criti-cal one (α = αm), another bubble appears on the up-per slanted surface, while the streamlines over theslanted surface are more aligned with the slant sideedges enforcing the vortexC sketched in Fig. 5;

3. When the slant angle is between both critical ones(αm < α< α M ), see Fig. 4 III, the time-averaged flowin the near-wake quickly shows a strong three di-mensional behavior, where the streamlines inside thefirst bubble are fed by a centered rotation motionand they result in a helical motion, while the bubbleboundary increases when the slant angle increases,and the side vortexC is stronger and the flow de-taches faster, while a separation point is formed be-tween the flows close to the slanted and vertical sur-faces;

4. When the slant angle is greater than the second criti-cal one (α> α M ), see Fig. 4 IV, there is a flow sepa-ration on the slanted surface and the singular point N moves upwards, while the flow is again almost two-dimensional.

VI. THE Q-CRITERIONCoherent structures in a turbulent flow are sub-flowswith some “cohesive” character, where the turbulence is“self-reorganized” in such a way that the resulting mean

motion is rather more regular than chaotic. They have a

Figure 4: Sketch of the time-averaged flow in the near-wakeas a function of the slant angleα, being I y II forα< αm, III forαm < α< α M , and IV forα> α M .

Figure 5: Sketch of vortices and bubble of the time-averagedflow in the near-wake of the Ahmed model.

relatively high concentration of vorticity intensityω , sothey typically show a spiraled motion and they persistfor a time scale sizeτ c longer than the typical eddy timeturnoverω -1. Previous studies of coherent structures bymeans of the pressure isosurfaces suggest that a pressurecriteria can be better than a vorticity one (Comteet al. ,1998). Nevertheless, the threshold value for the pressureisosurfaces shows a strong dependence on the meanpressure around the coherent structures and it can fail ina flow subregion with a high vortex concentration. The

so-called Q-criterion (Huntet al. , 1988) is another re-source for its study. It is based on both pressure andvorticity intensities and it can be defined asQ=( ij ij -SijSij) / 2, with ij=(u i,j-

u i,j) / 2 andSij =(u i,j+u i,j) / 2, are the

symmetrical and skew-symmetrical components of thevelocity gradient∇ u respectively. It can be shown thatthe scalarQ is the second invariant of the velocity gra-dient∇ u

and that it measures the ratio between rotation

and deformation rates. Thus, isosurfaces withQ>0 showregions where rotation rates are bigger than the strainones, so they can develop coherent structures. Sinceω 2=2 2, another equivalent definition isQ=(ω 2-2SijSij) / 4, showing that the coherent structures have a

relatively high concentration of vorticity intensityω .




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VII. NUMERICAL RESULTSA. Instantaneous flow in the near-wakeA view of the instantaneous flow on the front, top, onelateral side, and partially over the slant surface is shownin the upper plot of Fig. 6, where the flow structure ishighlighted through the isosurfaces of theQ-criterion

using a threshold of 40 Hz2

, while at the bottom an ex-perimental observation in a tunnel is included (Beau-doin et al. , 2003). The flow pattern is very sensitive tothe curvature radii and to the Reynolds number. Thereare detachments on the curved surfaces at the front partas well as on the intersection between front and topmiddle surfaces. The transversal Kelvin-Helmholtz vor-tices are convected downstream from the front side andconverted into the hairpin vortices λ, see Fig. 6 (top).The flow separates at the two tilted edges, between theslanted surface and the lateral ones producing two largecounter-rotating cone-like trailing vortices. The samefigure shows another flow separation on the sharp edge

between the body roof and the slant surface, where vor-tex parallel to the separation line is formed. As it maybe observed there is good agreement between numericaland experimental results.

B. Time-averaged flow in the near-wakeThe time used for the averaging of the instantaneousflow must be long enough to produce a steady flow.This situation is reached after several hundred of timesteps because of stability and accuracy requirements

Figure 6: Transversal Kelvin-Helmholtz vortices are down-stream convected from the front side and converted into hair-

pinλ

vortices: present computation (top) and experimental of Beaudoinet al. (bottom).

(Krajnović and Davidson, 2004). The symmetry of thetime-averaged flow around the vertical symmetricalplane was used as another indicator of the convergencein time of the average flow. The visualizations shown inFigs. 7-11 are all computed through a time-averagedprocedure of the large eddy simulations of the unsteady

flow, where the averaged-time must be big enough toprevent spurious effects. In this case, the averaged-timeis chosen asT =6T L whereT L= L/U ∞ is the elapsed timerequired by a fluid particle to travel one model length.The visualizations on the body surface are performed bymeans of trace lines (or friction ones) of the time-averaged flow. The detachments and re-attachments of the time-averaged flow on the body surface are ex-tracted using theQ-criterion. From these numerical re-sults the following features can be observed:

Figure 7: Stream-ribbons of the toroidal horseshoes vortexgenerated in the separation bubble of the time-averaged flow.

Figure 8: Trace lines of the time-averaged flow showing the

singular point N .

Figure 9: A bubble on the slanted surface of the time-averagedflow and some singular points.




302

1. In the near-wake there are two large re-circulatorysub-flows A and B situated one over the other whichagree with the experimental ones (Ahmedet al. ,1984; Gilliéron and Chometon, 1999), see stream-ribbons in Fig. 7. The vortex cores are obtainedthrough a phase plane technique (Kenwright, 1998);

2. The streamlines of the toroidal vortices over the ver-tical surface produce the singular point N shown inFig. 8;

3. There is a bubble on the slanted surface shown inFig. 9, whereS10 ,S11 are half-saddle points while N 6 is a nucleus point;

4. Flow detachments occur on the sharp edges of themodel, see Fig. 10. There are three flow detachmentson the frontal part: one on the upper side (T ) and twomore on both lateral sides ( L). On the same figure,on both lateral sides, there is a separation zone Z onthe top side which encloses the triple border pointformed by the front, upper and lateral sides, with a

small flow channel due to a high adverse pressuregradient. Also, there is a separation zone L on themiddle lateral side;

5. Fig. 11 shows trace lines (or friction ones) of thetime-averaged flow showing (i) on the lateral surface(top-left): a centerC , a saddleSs, an unstable focus

D and two separation lines; (ii) on the slant surface(top-right): a saddle, a stable focus and a bubble; and(iii) on the top surface (bottom): a saddle, a stablefocus, an unstable focus and two separation lines.

C. Topological validation at the symmetrical flowsection

A topological validation of the time-averaged flow vi-sualizations is made on the symmetrical section ( x=0) inorder to assess if all the main flow structures in thesymmetrical section have been identified.

As it is known (Chonget al. , 1990), a singular pointP in a cut planeξ , η in a three dimensional flow is apoint where velocity vanishes. For its classification,Chonget al. (1990) scheme is adopted here, which isbased on the eigenvalues z1 = a 1 +ib1 and z2 = a 2 +ib2 of the Jacobian matrix H = ∂(u,v) / ∂(ξ ,η) evaluated at thesingular pointP , where (u,v) are the flow velocities onthe cut plane. The resulting taxonomy is summarized inTable 3. It can be noted that in a focus point there is aflow rotation around it as opposed to a node case. Fig-ure 12 shows streamlines as well as singular points of the time-averaged flow that were obtained in the sym-metrical section ( x=0), whereS0 to S13 are half saddlepoints, N 1 to N 7 are focus points (in the vortex nuclei)and D is another saddle point. Figures 13 and 14 show

Figure 10: Trace lines of the time-averaged flow on the bodysurface. The flow run left to right.

Figure 11: Trace lines of the time-averaged flow showingsingular points on the lateral surface (top-left) and on the slantsurface (top-right).

Table 3: A taxonomy for singular points P on a cut plane in athree dimensional flow (Chonget al. , 1990).

point type z=eig(∇ u)|P={ z1, z2} zi= a i=+i b i, for i=1,2

repulsionattractioncentersaddleShalf-saddlefocusnode

a1, a2 > 0a1, a2 < 0a1 = a 2 > 0a1·a 2 < 0whenS is at a boundaryb1, b2 ≠ 0 (rotation)b1, b2 = 0 (no-rotation)

Figure 12: Streamline sketch in a time-averaged sense ob-tained from the present numerical simulations on the (vertical)symmetry plane x = 0 showing singular points.




303

Figure 13: Trace lines of the time-averaged flow on the frontpart at the upper side showing some singular points.

Figure 14: Trace lines of the time-averaged flow on the frontpart at the bottom side showing some singular points.

the streamlines projected on the ZY -plane at x=0. Thereare half-saddle pointsS1 ,S2, a focus one N 1 on the topside and the correspondingS3, S4 and N 4 on the bottomside. Figure 15 shows the two toroidal vorticesF 1 (upper) and F2 (bottom), whose nucleus are N4 and N5, re-

spectively. Figures 16 and 17 show four tiny and smallvortices near the vertical surface at the rear end. Twovortices are on the upper region and the remaining twoones on the inferior one. There are vortices whose nu-cleus B, N 2 and B, N 3 rotate in a clockwise and counterclockwise fashion respectively, and they also have half-saddle points. Near theF 2 vortex a stagnation pointP isidentified. A bubble formed behind the upper edge of the slanted surface is shown in Fig. 9. On the otherhand, on the symmetrical vertical section, the topologi-cal relationT =1-n must be verified (Huntet al. , 1988),where

Figure 15: Streamlines of the time-averaged flow projectedonto the vertical symmetry plane x = 0 showing some singularpoints.

Figure 16: Tiny vortices of the time-averaged flow on thevertical base surface of the rear end at the upper side B, N 3.

Figure 17: Tiny vortices of the time-averaged flow on thevertical base surface of the rear end at the bottom side B, N 2.

+−+= ∑∑∑∑** 2

121

SS N N

T ; (2)

while ∑ N is the sum of nodes and focus points,∑ * N

is the sum of the half-node points,∑Sis the sum of

saddle points,∑ *Sis the sum of half saddle points, and

n is the connectivity of the two-dimensional flow sec-tion. For instance, the connectivity of any two-dimensional section in a simply connected three dimen-sional flow region isn=1 (e.g. a flow without a body)while for the flow around a single body, as in the pre-sent case, isn=2. From the present visualizations, thefollowing values are obtained:∑ =

N 7, ∑ =

* 0 N

, ∑ =S

1

and 41*∑ =S

. ThereforeT ≡ 1, i.e. the topological restric-tion is satisfied. It can be concluded that all the mainflow structures in the symmetrical section have beenidentified. These topological features of the time-average flow are found independently of the averagingtimeT and grid-size of both meshes.




304

Figure 18: Pressure coefficientC p as a function of the y-streamwise coordinate at the top and bottom body surface onthe symmetry plane for a slant angleα =12.5º.

Figure 19: Pressure coefficientC D: time-evolutions, meanvalues and experimental measured (Ahmetet al. , 1984), for aslant angleα =12.5º.

D. Drag and pressure coefficientsThe drag coefficient is computed asC D = D p / γ, with

2 / 2 proy AU ∞= ρ γ , where D p is the drag force, A proy is the

frontal surface area, ρ is the fluid density andU ∞ is thenon-perturbed speed. The pressure coefficient is definedas ( )( )2 / 2 ∞∞

−= U p pC p ρ , where p∞ is the undisturbedpressure at the inlet boundary. The pressure coefficientfor a slant angleα = 12.5º is plotted in Fig. 18 as a func-tion of the streamwise coordinate at the top and bottombody surface on the symmetry plane. In Fig. 19 the un-steady nature of the drag coefficient )(t C num

D is shown

for the coarser mesh (#1) and the finer one (#2). In eachcase, oscillations in its values are observed during star-tup but, after some time-steps, these oscillations becomesmall and )(t C num

D approaches a constant value. Themean drag coefficient measured in the wind tunnel tests(Ahmed et al. , 1984) for a slant angle of 12.5º was

230.0exp= DC while the numerical areC D=0.246 for the

coarser mesh, andC D=0.228 for the finer one, with apercentage relative error of εr %≈

+6% andεr %≈ -1%,

respectively.

VIII. CONCLUSIONS

Large Eddy Simulations (LES) with unstructured mesh-es and finite elements of the unsteady flow around the

Ahmed vehicle model were performed for a slant angleof 12.5º. Two meshes were employed, showing meshconvergence through the drag coefficient (see Fig. 19).The symmetry of the time-averaged flow was used asanother indicator of the solution convergence. Flowvisualizations of the time-averaged flow in the near-

wake qualitatively agree with the experimental observa-tions. Most of the features of the flow around this bluff model in ground proximity were predicted, such as theformation of trailing vortices, massive separation andre-circulatory flows. The turbulent eddies up to the Tay-lor length scale were well resolved. The time-averagedflow was dominated by large coherent macro structuresformed by massive flow separation. The mean drag co-efficientC D agrees quite well the experimental one, witha percentual relative error of +6 % for the coarser meshand -1 % for the finer one. It is found that the topologi-cal features of the time-averaged flow are independentof the averaging time T and grid-size. It is concluded

that according to the current computational resourcesLES may be used as a feasible turbulence model for realvehicles aerodynamics. Future modeling efforts wouldbe focused on the representations of the structure of turbulence and some average profiles, as velocities andturbulent stresses, which could be useful for RANSstudies.

ACKNOWLEDGMENTSThis work was performed with the Free Software Foun-dation GNU-Project resources as Linux OS and Octave,as well as other Open Source resources as PETSc,MPICH and OpenDX, and supported through ConsejoNacional de Investigaciones Científicas y Técnicas(CONICET, Argentina, grants PIP-02552/2000, PIP5271/05) Universidad Nacional del Litoral (UNL, Ar-gentina, grant CAI+D 2005-10-64) and Agencia Na-cional de Promoción Científica y Tecnológica (ANP-CyT, Argentina, grants PICT 12-14573/2003, PME209/2003, PICT 1506/2006). N. Calvo has participatedin fruitful discussions on mesh quality and mesh genera-tion.

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Received: February 3, 2008Accepted: August 15, 2008Recommended by Subject Editor: Eduardo Dvorkin

numerical simulation of the flow around the ahmed vehicle model

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