ahmed body.pdf

8/17/2019 ahmed body.pdf

1/12

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

295

NUMERICAL SIMULATION OF THE FLOW AROUND THE AHMED

VEHICLE MODEL

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

†

Universidad Tecnológica Nacional (UTN), Facultad Regional Santa Fe (FRSF ) Lavaise 610, 3000-Santa Fe, Argentina, [email protected] , http://www.utn.edu.ar

‡ Centro Internacional de Métodos Computacionales en Ingeniería (CIMEC) Instituto de Desarrollo Tecnológico para la Industria Química (INTEC )

Universidad Nacional del Litoral -CONICET , Güemes 3450, 3000-Sant a Fe, ARGENTINA ph.: (54342)-4511 594/5, fx: (54342)-4511 169

(nnigro, mstorti, jdelia)@intec.unl.edu.ar http://www.cimec.org.ar

Abstract— The unsteady flow around the Ahmed

vehicle model is numerically solved for a Reynolds

number of 4.25 million based on the model length. A

viscous and incompressible fluid flow of Newtonian

type governed by the Navier-Stokes equations is as-sumed. A Large Eddy Simulation (LES) technique is

applied together with the Smagorinsky model as

Subgrid Scale Modeling (SGM) and a slightly modi-

fied van Driest near-wall damping. A monolithic

computational code based on the finite element me-

thod is used, with linear basis functions for both

pressure and velocity fields, stabilized by means of

the Streamline Upwind Petrov-Galerkin (SUPG)

scheme combined with the Pressure Stabilizing Pet-

rov-Galerkin (PSPG) one. Parallel computing on a

Beowulf cluster with a domain decomposition tech-

nique for solving the algebraic system is used. The

flow analysis is focused on the near-wake region,where the coherent macro structures are estimated

through the second invariant of the velocity gradient

(or Q-criterion) applied on the time-average flow. It

is verified that the topological features of the time-

average flow are independent of the averaging time

T and grid-size

Keywords— Ahmed vehicle, bluff aerodynamics,

incompressible viscous fluid, large eddy simulation

(LES), time-average flow, finite element method,

fluid mechanics.

I. INTRODUCTION

A. The Ahmed vehicle modelGround vehicles can be classified as bluff-bodies thatmove close to the road surface and are fully submergedin the fluid. In general, for the usual velocities of com-mercial passenger cars, buses and trucks, compressiblee ects can be neglected and an incompressible viscousfluid model can be assumed. As the Reynolds numbersbased on the body length are usually too high, the flowregimes are fully turbulent. In addition, a key feature ofthe flow field around a ground vehicle is the presence ofseveral separated flow regions, while the net aerody-namic force is the result of complicated interactionsamong them. Even simple basic vehicle configurations

with smooth surfaces, free from appendages and wheels,generate a variety of quasi two dimensional and fully

three dimensional regions of separated flows where thelargest one is the wake. In a time-averaged sense, theseparated flow regions exhibit complicated kinematicmacro structures and those present in the wake deter-mine mostly the body drag. Nowadays, numerical simu-lations, wind tunnel and road tests are jointly used in theautomotive industry for the aerodynamic study fromseveral perspectives.

The Ahmed vehicle model is a very simplified bluff-body which is frequently employed as a benchmark invehicle aerodynamics. It has been used in several ex-periments (Ahmed et al., 1984; Sims-Williams, 1998;Bayraktar et al., 2001; Spohn and Gilliéron, 2002;Lienhart and Becker, 2003) and numerical studies (Han,1989; Basara et al., 2001; Basara, 1999; Basara andAlajbegovic, 1998; Gilliéron and Chometon, 1999; Ka-padia et al., 2003). A slightly modified version was also

studied by Duell and George (1999); Barlow et al. (1999); Krajnović and Davidson (2003).The shape of this body is free from all accessories

and wheels but it still retains the primary behavior of thevehicle aerodynamics, as seen in Fig. 1. Special atten-tion is focused on the time-averaged flow in the near-wake and the dependence of drag on the slant angle α atthe top rear end. The rear end is a simplification of a so-called fastback one such as on a Volkswagen Golf I.This model was tested in open jet wind tunnels and novelocity distributions were reported in the inlet and out-let boundaries (Ahmed et al., 1984). Previous numericalsimulations had assumed the incoming flow to be lami-

nar and steady (Han, 1989; Krajnović

and Davidson,2004; Basara and Alajbegovic, 1998). Wind tunnel ex-periments on this model show two critical slant angles:αm≈12.5º and α M ≈30.0

º, called first and second criticalangles, respectively, where the main topological struc-ture of the time-averaged flow in the near-wake changessignificantly as summarized in Table 1 (Ahmed et al.,1984; Sims-Williams, 1998; Bayraktar et al., 2001;Spohn and Gilliéron, 2002; Lienhart and Becker, 2003).

Among the experimental tests, Janssen and Hucho(1975) showed the dependence of the flow on the slantangle at the rear end for an industrial vehicle model,while Morel (1978) performed tests on the Morel body

which is previous to the Ahmed one. Both models rep-


2/12


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 angle α, where αm≈12.5º andα M ≈

30.0º.slant angle α time-averaged flow in the near-wakeα < αm almost 2D attachedαm < α < α M massively separated 3D

α > 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; Basara et 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 approach

The 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 ofthe 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 angle α, 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 approach

In 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


3/12

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

297

from an engineering perspective. The smallest scales offluid 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 ofboundary 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 all

the 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 PARAMETERS

The 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 edges

were tested, where the slant length is kept fixed to ls =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.


4/12


298

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

III. FINITE ELEMENT DISCRETIZATION

A. Mesh generation

The 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 Taylor ηT = 265.68×

10-5 L and Kol-mogorov ηK =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 are hm ≈ {63.50, 31.75}×10-5 Lmesh # 1 mesh # 2

η T /hm 4.18 8.37

η K /hm 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 through

the quality factor eee d h minmin / =γ , for e =1, 2, ..., E ,

where ehmin

and ed min

are 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 spacing

The Kolmogorov length scale ηK is the length scale forthe smallest turbulent motions, while the Taylor one ηT gives more emphasis to intermediate turbulent motionswhose length scales are near some integral length scaleof the mean flow. Empirical correlations for both are

given 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),


5/12


299

where A is an O(1) constant, L~

is some integral scale

and 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 aredefined 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 conditions

A Newtonian viscous fluid model is adopted, with non-perturbed speed

∞U = 60m / s and kinematic viscosity

ν=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 is

imposed, 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 withvelocity u=

∞U ; (iii) null pressure at outlet; (iv) no-slip

at 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 ofbasis functions needs to be stabilized to achieve stablesolutions. This is performed with a SUPGPSPG schemewhose implementation details are given in Garibaldi etal. (2008).

A. Large Eddy Simulation

In LES techniques, the momentum balance equations

are solved with an “effective” kinematic viscosity νe = ν

+ νt , which is the sum of the molecular part calculatedas ν = µ/ ρ, plus a “turbulent” one νt . The last one is es-timated in the PETSc-FEM (Sonzogni et 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

veS t ε ε

(1)

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

flows in ducts) and )(:)( uu ε ε is the trace of the strain

rate )(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 /uτ, where uτ=(τ w / ρ)

1/2 is the

local friction speed and τ 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 viscosity νt 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 speed uτ 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 the

threshold distance d , giving the modified onev f

~.

B. Parallel computing

The 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 (Battaglia et al., 2006; D’Elía et al.,2000), added mass computations (Storti and D’Elía,2004) or inertial waves in closed flow 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 FLOW

Even 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 collecting

several wind tunnel tests on this model (Ahmed et al.,1984; Sims-Williams, 1998; Bayraktar et al., 2001;


6/12


300

Spohn and Gilliéron, 2002; Lienhart and Becker, 2003)show two critical slant angles: αm≈12.5º and α 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


7/12


301

VII. NUMERICAL RESULTS

A. Instantaneous flow in the near-wake

A 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 the Q-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-wake

The 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 ofBeaudoin et 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 as T =6T L where T 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 ofthe time-averaged flow on the body surface are ex-tracted using the Q-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.


8/12


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 (Ahmed et 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, where S 10 ,S 11 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 center C , a saddle S s, 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 flow

section

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 (Chong et al., 1990), a singular pointP in a cut plane ξ , η in a three dimensional flow is apoint where velocity vanishes. For its classification,Chong et al. (1990) scheme is adopted here, which isbased on the eigenvalues z1 = a1 +ib1 and z2 = a2 +ib2 ofthe Jacobian matrix H = ∂(u,v) / ∂(ξ ,η) evaluated at thesingular point P, 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 ofthe time-averaged flow that were obtained in the sym-metrical section ( x=0), where S 0 to S 13 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 (Chong et al., 1990).

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

repulsionattractioncentersaddle Shalf-saddlefocusnode

a1, a2 > 0a1, a2 < 0a1 = a2 > 0a1·a2 < 0when S 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.


9/12


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 points S 1 ,S 2, a focus one N 1 on the topside and the corresponding S 3, S 4 and N 4 on the bottomside. Figure 15 shows the two toroidal vortices F 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 the F 2 vortex a stagnation point P isidentified. A bubble formed behind the upper edge ofthe slanted surface is shown in Fig. 9. On the otherhand, on the symmetrical vertical section, the topologi-cal relation T =1-n must be verified (Hunt et 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

1

2

1

S S N N

T ; (2)

while ∑ N is the sum of nodes and focus points, ∑ * N is the sum of the half-node points, ∑S is the sum ofsaddle points, ∑ *S is 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 is n=1 (e.g. a flow without a body)while for the flow around a single body, as in the pre-sent case, is n=2. From the present visualizations, thefollowing values are obtained: ∑ = N 7 , ∑ =* 0 N , ∑ =S 1 and 41

*∑ =S . Therefore T ≡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 averaging

time T and grid-size of both meshes.


10/12


304

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

Figure 19: Pressure coefficient C D: time-evolutions, meanvalues and experimental measured (Ahmet et al., 1984), for aslant angle α =12.5º.

D. Drag and pressure coefficients

The drag coefficient is computed as C 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 and U ∞ is thenon-perturbed speed. The pressure coefficient is definedas ( ) ( )2 / 2

∞∞−= U p pC

p ρ , where p∞ is the undisturbed

pressure at the inlet boundary. The pressure coefficientfor a slant angle α = 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 = D

C while the numerical are C D=0.246 for the

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

+6% and εr %≈-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 bluffmodel 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-efficient C 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 ofturbulence and some average profiles, as velocities andturbulent stresses, which could be useful for RANSstudies.

ACKNOWLEDGMENTS

This 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.

REFERENCES

Ahmed, S., G. Ramm and G. Faltin, “Some salient fea-tures of the time-averaged ground vehicle wake,”SAE Report, 840300, 1–31 (1984).

Barlow, J., R. Guterres, R. Ranzenbach and J. Williams,“Wake structures of rectangular bodies with radi-used edges near a plane surface,” SAE Congress,1999-01-0648 (1999).

Basara, B., “Numerical simulation of turbulent wakesaround a vehicle,” ASM E Fluid Engineering Divi-sion Summer Meeting, FEDSM99-7324, San Fran-cisco, USA (1999).

Basara, B. and A. Alajbegovic, “Steady state calcula-tions of turbulent flow around Morel body,” 7t h Int .Symp. of Flow Modelling and Turbulence Measure-ments, Taiwan, 1–8 (1998).

Basara, B., V. Przulj and P. Tibaut, “On the calculation

of external aerodynamics industrial benchmarks,”SAE Conf., 01–0701 (2001).


11/12


305

Battaglia, L., J. D’Elía, M. Storti and N.M. Nigro,“Numerical simulation of transient free surfaceflows,” ASME-Journal of Applied Mechanics, 73,1017– 1025 (2006).

Bayraktar, D., D. Landman and O. Baysal, “Experimen-tal and computational investigation of Ahmed body

for ground vehicle aerodynamics,” SAE Report , 01-2742 (2001).Beaudoin, J., K. Gosse, O. Cadot, P. Paranthoën, B.

Hamelin, M. Tissier, J. Aider, D. Allano, I. Muta-bazi, M. Gonzales and J. Wesfreid, “Using cavita-tion technique to characterize the longitudinal vor-tices of a simplified road vehicle,” Technical report,PSA Peugeot Citroën Directions de la Researche,France (2003).

Beowulf Geronimo cluster, http://www.cimec.org.ar/geronimo (2005).

Calo, V., Residual-based Multiscale Turbulence Model-ing: finit e volume simulations of bypass transition.

Ph.D. thesis, Stanford University (2005).Calvo N.A., Meshsuite: 3d mesh generation,http://wwww.cimec. org.ar (1997).

Chong, M., A. Perry and B. Cantwell, “A general classi-fication of three-dimensional flow field,” Phys. Flu-id s A, 2, 765–777 (1990).

Comte, P., J. Silvestrini and P. Bégou, “Streamwisevortices in large eddy simulation of mixing layers,”

Eur . J . Mech, 17, 615–637 (1998).D’Elía, J., M. Storti and S. Idelsohn, “A panel-Fourier

method for free surface methods,” ASME-Journal ofFluid s Engineering, 122, 309–317 (2000).

D’Elía, J., N. Nigro and M. Storti, “Numerical simula-

tions of axisymmetric inertial waves in a rotatingsphere by finite elements,” Int. Journal of Computa-tional Fluid Dynamics, 20, 673–685 (2006).

Dubief, Y. and F. Delcayre, “On coherent-vortex identi-fication in turbulence,” J . Turbulence, 1, 1–22(2000).

Duell, E. and A. George, “Experimental study of aground vehicle body unsteady near wake,” SAE Re-

port 1999-01-0812, 197–208 (1999).ERCOFTAC, European Research Community on Flow,

Turbulence and Combustion. http://www.ercoftac.org, (2007).

Garibaldi, J., M. Storti, L. Battaglia and J. D’Elía,

“Numerical simulations of the flow around a spin-ning projectil in subsonic regime,” Latin American

Applied Research, 38, 241-248 (2008).Gilliéron, P. and F. Chometon, “Modeling of stationary

three-dimensional separated air flows around anAhmed reference model,” ESAIM , 7, 173–182(1999).

Gilliéron, P. and A. Spohn, “Flow separations generatedby a simplified geometry of an automotive vehicle,”Technical report, Technocentre Renault - CNRS,Lab. d’ Etudes A´erodynamiques de Poitiers (2002).

Han, T., “Computational analysis of three-dimensionalturbulent flow around a bluff body in ground prox-

imity,” AIIA Journal, 27, 1213-1219 (1989).

Howard, R. and M. Pourquie, “Large eddy simulation ofan Ahmed reference model,” Journal of Turbulence,3, 12 (2002).

Hucho, W., Aerodynamics of road vehicles. Society ofAutomative Engineers (1998).

Hunt, J., A. Wray and P. Moin, “Eddies, stream, and

convergence zones in turbulent flows,” Repor tCTRS88 , Center For Turbulent Research (1988).Janssen, L. and W. Hucho, “Aerodynamische Formop-

tienierung der Typen VW -Golf und VW –Scirocco,” Volkswagen Golf I ATZ , 77, 1–5 (1975).

Kapadia, S., S. Roy and K. Wurtzler, “Detached eddysimulation over a reference Ahmed model car,” 41st

Aerospace Sciences Meeting and Exhibit , AIAA-2003-0857, 1–10 (2003).

Kenwright, D., “Automatic detection of open and closedseparation and attachment lines,” Proceedings of

IEEE Visualization (1998).Krajnović, S., Large-Edd y Simulation for computing the

flow around vehicles. Ph.D. thesis, Chalmers Uni-versity of Technology, Sweden (2002).Krajnović, S. and L. Davidson, “Numerical study of the

flow around the bus-shaped body,” ASME-J . of Flu-id s Engineering, 125, 500–509 (2003).

Krajnović, S. and L. Davidson, “Large-eddy simulationof the flow around simplified car model,” SAE World Congress, Detroit, USA, 2004–01–0227(2004).

Krajnović, S. and L. Davidson, “Flow around a simpli-fied car, part 1: Large eddy simulation,” Journal ofFluid s Engineering, 127, 907–918 (2005a).

Krajnović, S. and L. Davidson, “Flow around a simpli-

fied car, part 2: Understanding the flow,” Journal ofFluid s Engineering, 127, 919–928 (2005b).

Lienhart, H. and S. Becker, “Flow and turbulence struc-tures in the wake of a simplified car model (Ahmedmodel),” SAE Report 2003-01(0656) (2003).

McDonough, J., “On intrinsic errors in turbulence mod-els based on Reynolds-Averaged Navier-Stokes equ-ations,” Int . J . Fluid Mech. Res., 22, 27–55 (1995).

Message Passing Interface MPI. http://www.mpiforum.org/docs/docs.html (2007).

Morel, T., “Aerodynamics drag of bluff body shapes.characteristics of hatch-back cars,” SAE Report ,780267 (1978).

Paz, R. and M. Storti, “An interface strip preconditionfor domain decomposition methods. application tohidrology,” Int . J . for Num. Meth. in Engng., 62,1873–1894 (2005).

PETSc-2.3.3. http://www.mcs.anl.gov/petsc (2007).PETSc-FEM. A general purpose, parallel, multi-physics

FEM program, GNU General Public License,http://www.cimec.ceride.gov.ar/petscfem (2008).

Sagaut, P., Large edd y simulation for incompressible flows, an introduction. Springer, Berlin (2001).

Sims-Williams, D., “Experimental investigation intounsteadinless and instability in passanger car aero-dynamics,” SAE Report , 980391 (1998).

Sonzogni, V., A. Yommi, N. Nigro and M. Storti, “Aparallel finite element program on a Beowulf Clus-


12/12


306

ter,” Advances in Engineering Software, 33, 427–443 (2002).

Spohn, A. and P. Gilliéron, “Flow separations generatedby a simplified geometry of an automotive vehicle,”

IUTAM Symposium: unstead y separated flows. Tou-louse, France (2002).

Storti, M.A. and J. D’Elía, “Added mass of an oscillat-

ing hemisphere at very-low and very-high frequen-cies,” ASME Journal of Fluid s Engineering, 126,1048–1053 (2004).

Wilcox, D., Turbulence Modeling for CFD. DCW, 2ndedition (1998).

Received: February 3, 2008

Accepted: August 15, 2008

Recommended by Subject Editor: Eduardo Dvorkin

ahmed body.pdf

Documents

Transcript of ahmed body.pdf