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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)
[email protected]
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
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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].
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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)
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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.
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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
16
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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.
17
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Figure 6: Drag coefficient against the slant angle
18
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Figure 7: Averaged Line Integral Convolution (LIC) on the
slanted surface from Ahmed(left) and XFlow (right)
19
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Figure 8: Near-wake structure at scale for: a) = 5, b) = 25
Figure 9: NASA trapezoidal wing geometry
20
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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
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Figure 12: Averaged Line Integral Convolution (LIC) at 37
degrees incidence
22