Solution to industry benchmark problems with the

lattice-Boltzmann code XFlow

David M. Holman1, Ruddy M. Brionnaud1, and Zaki Abiza1

1Next Limit Technologies, Angel Cavero 2, 28043, Madrid (Spain)

david.holman@nextlimit.com

Abstract

This contribution presents some of the capabilities of the ComputationalFluid Dynamics (CFD) code XFlow, which uses a proprietary particle-basedkinetic solver based on the Lattice-Boltzmann Method. Using traditionalCFD software, industrial problems require time consuming meshing processwhich often leads to errors or even divergence of the simulation. Due to itsparticle-based and fully Lagrangian approach, the complexity of the geometrysurfaces is not a limiting factor in XFlow even in the presence of movingparts, allowing to solve real industrial problems. The performance of XFlowwill be demonstrated for different industry benchmarks. The first example isthe Ahmed body which is a classical benchmark in the automotive industry.The second benchmark presented will be the NASA trapezoidal wing. XFlowresults will be described and show good agreement with experimental data.

Keywords: Lattice-Boltzmann, Lagrangian, particle-based, Ahmed body,NASA trapezoidal wing

1. Introduction

For the past 20 years, the field of Computational Fluid Dynamics (CFD)has reached a high level of maturity, but it has only been recently thatCFD has been broadly applied to the improvement of several processes atdifferent stages: research, design, manufacturing, optimization, etc. Theneed for robust and reliable analysis tools is therefore growing rapidly, inproportion to the increasing complexity of simulations. To provide quick,accurate feedback to realistic engineering problems is consequently essentialfor companies to be competitive.

Preprint submitted to ECCOMAS 2012

The traditional numerical methodologies employed so far are based onmethods involving finite volumes and finite elements, applied to Navier-Stokes equations. However, even though such methods have been widelyinvestigated, they still hold major drawbacks, limiting their capacity tosolve real industrial problems: uncertainties induced by the meshing pro-cess; highly empirical approaches to the turbulence modeling (RANS); thetreatment of the nonlinear convective term; artificial stabilization parame-ters; and so on. Because of this, in most cases engineers are not able to modelreal systems; they are forced to fall back on simplified models and approxi-mations. These methods require a time-consuming meshing process, are nottolerant to moving parts, and are usually limited to steady-state analysis,ignoring transient dynamics.

Particle-based methods have been in development for several decades, andare now starting to come to the fore. Among them, the promising Lattice-Boltzmann Method (LBM) surmounts many of the drawbacks of traditionalCFD methods. XFlow CFD uses a particle-based and fully Lagrangian ap-proach based on LBM. With this method, classic fluid-domain meshing isnot required and surface complexity is not a limiting factor.

XFlow has been validated in several benchmarks, demonstrating the va-lidity of the method to solve industrial problems. The first example presentedin this paper is the Ahmed body, a classic benchmark for the automotive in-dustry. The cars geometry has a variable slant angle and is a challengingtest case in terms of turbulence modelling and drag estimation. The NASAtrapezoidal wing is the second benchmark presented in this paper, a threeelement airfoil composed of a slat, a main blade and a flap. The goal isto assess the aerodynamic coefficients on a large range of incidence angles,including the post-stall region.

2. Numerical methodology

Over the last few years, schemes based on minimal kinetic models for theBoltzmann equation are becoming increasingly popular as a reliable alterna-tive to conventional CFD approaches.

The Lattice Boltzmann method (LBM) was originally developed as an im-proved modification of the Lattice Gas Automata to remove statistical noiseand achieve better Galilean invariance [1, 2]. Due to the flexibility affordedby its close connection to kinetic theory, the LBM can be adapted to modelseveral physical phenomena. Recent research has led to major improvements,

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including physically consistent models for multiphase and multicomponentflow and fully compressible flow [3, 4, 5].

2.1. Lattice Gas Automata

The Lattice Gas Automata (LGA) is a simple scheme for modeling thebehavior of gases. The basic idea behind the LGA is that particles withspecific velocities (ei, i = 1, ..., b) propagate through a d-dimensional lattice,at discrete times t = 0, 1, 2, ... and collide according to specific rules designedto preserve the mass and the linear momentum when different particles reachthe same lattice position.

The simplest LGA model is the HPP approach, introduced by Hardy,Pomeau and de Pazzis, in which particles move in a two-dimensional squarelattice and in four directions (d = 2, b = 4). The state of an element of thelattice at instant t is given by the occupation number ni(r, t), with ni = 1being presence and ni = 0 absence of particles with velocity ei.

The stream-and-collide equation that governs the evolution of the systemis

ni(r+ ei, t+ dt) = ni(r, t) + i(n1, ..., nb), i = 1, ..., b, (1)

where i is the collision operator that computes a post-collision state con-serving mass and linear momentum. If one were to assume i = 0, only anstreaming operation would be performed.

From a statistical point of view, the system is made up of a large numberof elements which are macroscopically equivalent to the problem investigated.The macroscopic density and linear momentum can be computed as:

=1

b

bi=1

ni (2)

v =1

b

bi=1

niei (3)

2.2. Lattice Boltzmann method

While the LGA schemes use boolean logic to represent the occupationstage, the LBM method makes use of statistical distribution functions fiwith real variables, preserving by construction the conservation of mass andlinear momentum.

The Boltzmann transport equation is defined as follows:

fit

+ ei fi = i, i = 1, ..., b, (4)

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where fi is the particle distribution function in the direction i, ei the corre-sponding discrete velocity and i the collision operator.

The stream-and-collide scheme of the LBM can be interpreted as a dis-crete approximation of the continuous Boltzmann equation. The streamingor propagation step models the advection of the particle distribution func-tions along discrete directions, while most of the physical phenomena aremodeled by the collision operator which also has a strong impact on thenumerical stability of the scheme.

In the most common approach, a single-relaxation time (SRT) based onthe Bhatnagar-Gross-Krook (BGK) approximation is used

BGKi =1

(f eqi fi), (5)

where is the relaxation time parameter, related to the macroscopic viscosityas follows

= c2s( 1

2). (6)

f eqi is the local equilibrium function usually defined as

f eqi = wi

(1 +

eiuc2s

+uu2c2s

(eieic2s

)). (7)

Here cs is the speed of sound, u the macroscopic velocity, the Kroneckerdelta and the wi are weighting constants built to preserve the isotropy. The and subindexes denote the different spatial components of the vectors ap-pearing in the equation and Einsteins summation convention over repeatedindices has been used.

By means of the Chapman-Enskog expansion the resulting scheme can beshown to reproduce the hydrodynamic regime for low Mach numbers [5, 6, 7].

The single-relaxation time approach is commonly used because of its sim-plicity. However it is not well-posed for high Mach number applications andit is prone to numerical instabilities. Some of the limitations of the BGK areaddressed with multiple-relaxation-time (MRT) collision operators where thecollision process is carried out in moment space instead of the usual velocityspace

MRTi = M1ij Sij(m

eqi mi), (8)

where the collision matrix Sij is diagonal, meqi is the equilibrium value of the

moment mi and Mij is the transformation matrix [8, 9].

4

An alternative method that aims to overcome the limitations of the BGKapproach is the entropic lattice Boltzmann (ELBM) scheme, which may relyon a single-relaxation-time where the attractors of the particle distributionfunctions are based on the minimization of a Lyapunov-type functional en-forcing the H-theorem locally in the collision step. However, this method isexpensive from the computational point of view [10] and thus not used inpractical engineering applications.

The collision operator in XFlow is based on a multiple relaxation timescheme. However, as opposed to standard MRT, the scattering operator isimplemented in central moment space. The relaxation process is performedin a moving reference frame by shifting the discrete particle velocities withthe local macroscopic velocity, naturally improving the Galilean invarianceand the numerical stability for a given velocity set [11].

Raw moments can be defined as

xkylzm =Ni

fiekixe

liye

miz (9)

and the central moments as

xkylzm =Ni

fi(eix ux)k(eiy uy)l(eiz uz)m (10)

2.3. Turbulence modeling

The approach used for turbulence modeling is the Large Eddy Simulation(LES). This scheme introduces an additional viscosity, called turbulent eddyviscosity t, in order to model the subgrid turbulence. The LES scheme wehave used is the Wall-Adapting Local Eddy viscosity model, that provides aconsistent local eddy-viscosity and near wall behavior [12].

The actual implementation is formulated as follows:

t = 2f

(GdGd)

3/2

(SS)5/2 + (GdGd)

5/4(11)

S =g + g

2(12)

Gd =1

2(g2 + g

2)

1

3g

2 (13)

g =ux

(14)

5

where f = Cwx is the filter scale, S is the strain rate tensor of the resolvedscales and the constant Cw is typically 0.325.

A generalized law of the wall that takes into account for the effect ofadverse and favorable pressure gradients is used to model the boundary layer[13]:

U

uc=

U1 + U2uc

=uuc

U1u

+upuc

U2up

(15)

=wu2

uucf1

(y+uuc

)+

dpw/dx

|dpw/dx|upucf2

(y+upuc

)(16)

y+ =ucy

(17)

uc = u + up (18)

u =|w| / (19)

up =

(

dpwdx)1/3

. (20)

Here, y is the normal distance from the wall, u is the skin friction velocity,w is the turbulent wall shear stress, dpw/dx is the wall pressure gradient, upis a characteristic velocity of the adverse wall pressure gradient and U is themean velocity at a given distance from the wall. The interpolating functionsf1 and f2 given by Shih et al. [13] are depicted in figure 1.

100 101 102

y+ u/uc

0

5

10

15

20

25

f 1

f1 (y+ u/uc )

100 101 102

y+ up /uc

0

5

10

15

20

25

30

35

40

45

f 2

f2 (y+ up /uc )

Figure 1: Unified laws of the wall

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3. Ahmed body benchmark

The Ahmed Body is a classic benchmark for the automotive industry. Itwas first defined and its characteristics described in the experimental workof Ahmed [14]. The car geometry was studied at various slant angles from0 to 40 degrees. The experimental measurements were conducted by Ahmedin the DFVLR subsonic wind tunnels at Braunschweig and Gottingen whichhave a square nozzle of (3 x 3) m and a length of 5.8 m.

The first goal of this study is to validate the curve of the drag coefficientagainst the slant angle obtained by Ahmed in [14], and the second one is toanalyze the mean recirculation structures on the slant surface of the Ahmedbody and in the downstream region.

3.1. Simulation setup

A strictly identical geometry to the one used by Ahmed was importedinto the virtual wind tunnel featured in XFlow. This virtual wind tunnelconsists of a rectangular domain and was set to dimensions of (8 x 2 x 2)m. A far-field velocity boundary condition was used at the inlet and thetop boundaries, and zero gauge pressure was imposed at the outlet. Periodicboundary conditions were set on the side walls, and a free-slip wall with novelocity was imposed at the bottom boundary.

The geometry of the Ahmed body was separated into two parts in orderto simplify the setup modification for variable slant angles. The first part isthe fore body that has an invariable geometry. The second part is the rearbody which is replaced when the slant angle changes. These two parts areshown on figure 2.

Figure 2: Fore body geometry and rear geometry

The simulation settings are gathered in table 1, and correspond to aReynolds number based on the car length equal to 4.29 million. The sim-

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Table 1: Simulation specifications of the Ahmed body benchmark

Inlet velocity 60 m/sDensity 1 kg/m3

Dynamic viscosity 1.46014 105 Pa.sCar length 1044 mm

Reynolds number 4.29 106Slant angles 0 ; 5 ; 10 ; 12.5 ; 15 ; 20 ; 25 ; 30 ; 40 degrees

Turbulence intensity 0.5%

ulation time was two seconds and the time step t = 7.69231 105 s isautomatically estimated by XFlow to ensure the numerical stability.

3.2. Spatial discretization

Since XFlow is a particle based technology it does not require a time-consuming meshing process. The preprocessor generates the initial octreelattice structure based on the input geometries and the user-specified reso-lution for each geometry. The lattice may have several levels of detail whichare hierarchically arranged. Each level solves spatial and temporal scalestwo times smaller than the previous level, thus forming the aforementionedoctree structure.

The lattice structure may be modified later by the solver if the com-putational domain changes (due to the presence of moving parts) or if theresolution changes dynamically in order to adapt to the flow patterns (adap-tive wake refinement). The adaptive wake refinement feature in XFlow isbased on the module of the vorticity field: in the lattice elements wherethe vorticity reaches a threshold value the lattice is automatically refined.Similarly, when the vorticity is lower than another threshold, eight adjacentlattice elements are merged to form a coarser lattice element. This savescomputational resources and removes the need to refine your solution in ad-vance. Consequently, as in illustrated figure 3, three resolutions are requiredby the user: the far field, the wake and the near wall resolutions.

In order to select the best resolution near the walls and within the wakethat allows us to get good results in an acceptable time, a resolution depen-dency study is conducted before starting the validation of the Ahmed body.This preliminary study consists in refining the resolutions and seeing how thisaffects the accuracy of the results, but also checking if the code is converging

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Figure 3: Example of lattice structure using the near wall and adaptive wake refinement

Table 2: Near walls and wake resolutions used in the resolution dependency study

h h/2 h/22 h/23

Resolution (m) 0.04 0.02 0.01 0.005# of Elements at t = 0.3 s 88,316 222,337 1,132,292 8,316,626

to the right solution. It is done by measuring the drag coefficient predictedby XFlow for a slant angle of 35 degrees which is a reference angle for thisbenchmark. The far field is taken constant as 0.08 m, and four resolutionsare considered for the walls and the wake as described table 2.

The drag coefficient is computed for the four cases and compared withthe experimental value measured by Ahmed [14]. The drag points from thesimulations are plotted in figure 4 in function of the number of elements att = 0.3 s. The point corresponding to the resolution h/22 = 0.01 m givesgood results and in an acceptable time for a slant angle = 35, and willtherefore become the reference near wall resolution for the rest of the study.The figure 4 also confirms the convergence of the code to the correct solution.

A second question arises regarding the value of the wake resolution. Asthe wake refinement algorithm creates a significant number of elements as itdevelops, its importance in the drag contribution must be assessed accuratelyto get a good compromise between solution quality and computational time.Hence, a second study is conducted on the wake resolution starting from theelected near wall resolution (0.01 m) and then increasing by multiples of two,due to the lattice structure. The figure 5 demonstrates the importance ofsolving the wake accurately: using the same resolution near the walls andwithin the wake the drag coefficient history shows a nice prediction, but

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0 1000 2000 3000 4000 5000 6000 7000 8000 9000N (103 nodes)

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

Drag Coefficient, Cx

h/23h/22

h/2

h

Figure 4: Drag coefficient against the number of lattice nodes for different resolutions at = 35

as soon as the wake resolution is the double or quadruple of the near wallresolution affects the results quite dramatically. Hence, for all our runs, thespatial discretization chosen for all the different slant angles is done with anautomatic wake refinement with a resolution of 0.08 m for the far field, and0.01 m around the Ahmed body and within the wake.

3.3. Numerical results

The time required in XFlow to set up the case is about 10 minutes andmainly consists in geometry importation, the flow and boundary specifica-tions, and the resolution setup. The calculation time is almost the same forall the slant angles and varies between 6 and 8 hours with the previouslyselected resolutions on two Intel Xeon E5620 (2.4GHz).

The first result given by Ahmed is the curve representing the drag coef-ficient against the slant angle , and gives the drag contributions of everypart of the Ahmed body: the front Ck, the rear vertical surface Cb, the rearslant surface Cs and the friction drag Cr. The total drag Ahmed found wasCw and was the sum of the different contributions. Hence, the total dragobtained from XFlow for the different slant angles is superimposed with theCw from Ahmed, as shown in figure 6.

From the figure 6 we observe a good overall drag prediction by the code:the drag breakdown occurs right after 30 degrees and the minimum drag point

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0.0 0.1 0.2 0.3 0.4 0.5Time (s)

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Drag Coefficient, Cx

Wake 0.01mWake 0.02mWake 0.04mExperimental

Figure 5: Drag coefficient history for different wake refinement resolution at = 35

is the critical angle 12.5 degrees, as measured by Ahmed. The absolute dragvalues predicted by XFlow are accurate and the relative error varies fromonly 0.4% to 3.2% for most of the angles, except around the drag breakdownand at 0 degree angle where it reaches a maximum of only 7.1%. These smalldiscrepancies can be explained, on the one hand, by the complexity aroundthe flow around 30 degrees of slant angle which is switching from a massive3D separation in the near-wake region to an almost 2D attached structure athigher angles [15], and, on the other hand, by stronger gradients producedby the rear of the car at 0 degree angle.

3.4. Flow field results

The second part of the results analysis is done by analyzing the mainrecirculation structures resulting from the flow around the Ahmed body. Forthis study, the averaging of the flow fields is required in order to filter thetemporal fluctuations and to identify the main structures of the turbulentwake. The averaging of the fields started from t = 0.3 s when the flow wasestablished, as indicated for example by figure 5, to cut off the transientperiod.

Ahmed provides pictures of the oil flow on the slanted surface for =12.5, 25 and 30 degrees. It can be compared with XFlow which featuresLine Integral Convolution (LIC) that approximates the surface streamlineson a body. The figure 7 shows similar structure for the three angles: a quite

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smooth and attached flow at 12.5 degrees, smooth flow patterns with twosmall and symmetric fringes on the sides at 25 degrees, and two large andsymmetric separation bubbles at 30 degrees.

Ahmed also provides different velocity vectors plots in the symmetryplane of the car, showing the near-wake region. This allows the study ofthe separation bubble on the rear slant and within the wake for differentslant angles.

Figure 8 compares the near-wake region for a slant angle of 5 degreesbetween the experimental results measured by Ahmed and results obtainedby XFlow at the same scale. This allows us to check the length of the bubbleseparation located around the non-dimensional coordinate x/Lref = 0.375,predicted in an extremely similar way in the two pictures. Two main eddystructures are detected - highlighted in red boxes on figure 8 - which aresymmetrical from the top and bottom of the separation bubble. The codetends to locate them slightly further downstream, though with reasonableoverall flow patterns.

The near-wake structure for a slant angle of 25 degrees also show goodsimilarities. This figure 8 shows an equivalent triangular separation bubble,ending around the non-dimensional coordinate x/Lref = 0.2 for both cases.

4. NASA trapezoidal wing benchmark

The NASA trapezoidal wing benchmark comes from the 1st AIAA CFDHigh Lift Prediction Workshop (HiLiftPW-1), sponsored by the AppliedAerodynamics Technical Committee, which took place in June 2010 in Chicago,IL. The challenge was to simulate a half aircraft configuration composed ofa body and a 3-element airfoil with a plane of symmetry as shown in figure9 for a wide range of angles of attack. The trapezoidal wing is composedof slat, main element and flap. The latter can be in two different configu-rations: Configuration 1 at 25 degrees and Configuration 8 at 20 degrees ofangle-of-attack.

The objectives of the benchmark are multiple [16]:

Assess the prediction capability of CFD codes in landing/taking-offconfiguration,

Develop practical modeling guidelines for the analysis of high-lift con-figurations,

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Table 3: Resolutions used for the resolution-dependency at 13 degrees incidence

h h/2 h/22 h/23

Near wall (m) 0.04 0.02 0.01 0.005Wake (m) 0.08 0.04 0.02 0.01

# of Elements at t = 0.3 s 201,513 653,211 2,893,687 21,880,186

Provide an impartial forum for evaluating the effectiveness of existingCFD codes and modeling techniques,

Identify areas that require additional research and development.4.1. Simulation setup

XFlow simulations were run for the Configuration 1 with no brackets. TheMach number was 0.2, the Reynolds number based on the mean aerodynamicchord (MAC) was 4.3 million. The angles of attack run for this benchmarkwere: -4, 1, 6, 13, 21, 25, 28, 32, 34 and 37 degrees. The hardware used inall the computations was a single workstation with two Intel Xeon E5620 @2.4 GHz processors (8 cores) and 12GB of RAM.

A resolution dependency study has also been performed for this bench-mark using the four resolutions described in Table 3 and a constant far fieldresolution of 1.28 m. An incidence angle of 13 degrees which is one of thereference angles of the first workshop was employed.

The drag coefficient obtained with each of the four simulations is plottedin figure 10 as a function of the number of elements at t = 0.3 s. Thepoint corresponding to resolution h/23 gives the best estimation of the dragcompared to the experimental data, with only 1% of relative error. Thisvalue will therefore be used as the reference near wall resolution for the restof the study.

However, two different wake resolutions have been used depending on theincidence of the NASA trapezoidal wing. Indeed, for large angles of attack,a significant wake develops and the number of lattice elements introduced bythe adaptive wake refinement increases. At 32 degrees, the simulation reaches25 million lattice elements, which is the maximum number of elements thatcan fit in the 12 GB of RAM available on the workstation. Special care isthus required in order to keep this number within the memory constraintsfor higher angles. The wake resolution has been limited to double the normalvalue for those cases (Resolution 2 in table 4).

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Table 4: Resolutions used for the 1st High Lift Prediction Workshop

Walls (m) Wake (m) Far Field (m) Max. # of Particles AnglesResolution 1 0.005 0.01 1.28 25 106 [-4; 32]Resolution 2 0.005 0.02 1.28 10 106 [34; 37]

4.2. Numerical results

The experimental data were produced at the 14x22 wind-tunnel at thewell-known NASA Langley. Forces, moments, and Cp distribution were pro-vided with free transition [17]. Data were provided as lower and upper valueswhich are assumed to be the range of uncertainty in the wind tunnel mea-surements.

On figure 11, the drag coefficient against the angle of attack is shown.XFlow results show very good agreement with the experimental data alongthe whole range of angles. The drag slope is accurate and still behavescorrectly at both low and high incidences, with a slight slope decrease.

The lift coefficient is also very well predicted for the whole range of angles.Within the range [1, 28] degrees, XFlow predicts accurately both slope andabsolute lift coefficient values. Starting from 32 degrees, the critical angleis reached and the code also succeeds in predicting this: the wind tunneldata indicates the maximum lift point at around 33 degrees, and it happensbetween the point of 32 degrees and 34 degrees. Starting from that point,the lift drops, due to a large bubble of separation on the wing. The bubbleof separation grows on the tip of the wing, as shown in the Figure 12.

Since both drag and lift coefficients are quite well predicted, the polarcurve on Figure 11 is hence matching the experimental results, especially inthe pre-stall region.

The pitching moment coefficients also lie between the upper and lowerlimits of the experimental results within almost the whole range.

5. Conclusions

The CFD code XFlow features a kinetic particle-based solver that differsfrom the traditional approaches, which are usually mesh-based. The lattice-Boltzmann method employed is able to solve advanced industrial problemseven in the presence of complex geometries or moving parts.

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The methodology has demonstrated it can solve industrial benchmarksefficiently. For instance the Ahmed body is a classic benchmark for the au-tomotive industry that XFlow solved with a high degree of accuracy. XFlowdid not face convergence issues even for extreme slant angles, and changingthe rear of the car did not add additional workload. The code has beendemonstrated to be robust and accurate in terms of drag and flow patternprediction, and closely matches the data measured by Ahmed in the DFVLRsubsonic wind tunnel of Braunschweig including the drag breakdown around30 degrees and the low slant angles where gradients are stronger.

The High Lift Prediction Workshop benchmark has also been successfullyvalidated by XFlow. The NASA trap wing geometry was tested within arange of incidence between -4 and 37 degrees, which includes the post-stallregion. The drag, lift and pitching moment coefficients predicted by the codeare in good agreement with the experimental tests conducted in the NASALangley 14x22 wind tunnel. The stall angle is also accurately predictedaround 33 degrees.

XFlow has therefore demonstrated its robustness and accuracy in differentbenchmarks. The method is well-suited for external aerodynamics and showsstrong potential for more advanced topics, such as analysis involving complexgeometries, the presence of moving parts and fluid-structure interaction.

References

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[6] Y. H. Qian, D. DHumie`res, P. Lallemand, Lattice BGK models forNavier-Stokes equation, EPL (Europhysics Letters) 17 (1992) 479.doi:10.1209/0295-5075/17/6/001.

[7] F. J. Higuera, J. Jimenez, Boltzmann approach to lattice gas simu-lations, EPL (Europhysics Letters) 9 (1989) 663. doi:10.1209/0295-5075/9/7/009.

[8] X. Shan, H. Chen, A general multiple-relaxation-time boltzmann col-lision model, International Journal of Modern Physics C 18 (4) (2007)635643.

[9] D. dHumie`res, Multiplerelaxationtime lattice boltzmann models inthree dimensions, Philosophical Transactions of the Royal Society ofLondon. Series A: Mathematical, Physical and Engineering Sciences360 (1792) (2002) 437451.

[10] P. Asinari, Entropic multiple-relaxation-time lattice boltzmann models,Tech. rep., Politecnico di Torino, Torino, Italy (2008).

[11] K. Premnath, S. Banerjee, On the three-dimensional central momentlattice boltzmann method, Journal of Statistical Physics (2011) 148.

[12] F. Ducros, F. Nicoud, T. Poinsot, Wall-adapting local eddy-viscositymodels for simulations in complex geometries, in: Proceedings of 6thICFD Conference on Numerical Methods for Fluid Dynamics, 1998, pp.293299.

[13] T. Shih, L. Povinelli, N. Liu, M. Potapczuk, J. Lumley, A generalizedwall function, NASA Technical Report.

[14] S. Ahmed, G. Ramm, G. Faitin, Some salient features of the time-averaged ground vehicle wake, Tech. rep., Society of Automotive En-gineers, Inc., Warrendale, PA (1984).

[15] G. Franck, N. Nigro, M. Storti, J. DEla, Numerical simulation of theflow around the ahmed vehicle model, Latin American applied research39 (4) (2009) 295306.

[16] C. Rumsey, The 1st aiaa cfd high lift prediction workshop (Jun. 2010).URL http://hiliftpw.larc.nasa.gov/index-workshop1.html

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[17] C. McGinley, L. Jenkins, R. Watson, A. Bertelrud, 3-d high-lift flow-physics experimenttransition measurements, AIAA Paper 5148 (2005)2005.

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Figure 6: Drag coefficient against the slant angle

18

Figure 7: Averaged Line Integral Convolution (LIC) on the slanted surface from Ahmed(left) and XFlow (right)

19

Figure 8: Near-wake structure at scale for: a) = 5, b) = 25

Figure 9: NASA trapezoidal wing geometry

20

0 5 10 15 20 25Number of Lattice Nodes (106 )

0.30

0.32

0.34

0.36

0.38

0.40

Drag Coefficient, CD

h/23

h/22

h/2

h XFlowExperiment

Figure 10: Drag coefficient against the number of lattice nodes for different resolutions at = 13

10 0 10 20 30 40 (deg)

0.0

0.2

0.4

0.6

0.8

1.0

Drag Coefficient, CD

(a)

ExperimentalExperimental LowerExperimental UpperXFlow

10 0 10 20 30 40 (deg)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Lift Coefficient, CL

(b)

0.0 0.2 0.4 0.6 0.8 1.0Drag Coefficient, CD

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Lift Coefficient, CL

(c)

10 0 10 20 30 40 (deg)

0.5

0.4

0.3

0.2

0.1

0.0

Pitching Moment, C m

(d)

Figure 11: Drag (a) and lift (b) coefficients against the angle of attack, the polar curve(c), and the pitching moment coefficient (d)

21

Figure 12: Averaged Line Integral Convolution (LIC) at 37 degrees incidence

22