Solution to industry benchmark problems with the lattice-Boltzmann code XFlow David M. Holman1 , Ruddy M. Brionnaud1 , and Zaki Abiza1 1
Next Limit Technologies, Angel Cavero 2, 28043, Madrid (Spain)
[email protected]
Abstract
This contribution presents some of the capabilities of the Computational Fluid Dynamics (CFD) code XFlow, which uses a proprietary particle-based kinetic kinetic solver based based on the Lattice-Bolt Lattice-Boltzmann zmann Method. Method. Using traditional traditional CFD software, industrial problems require time consuming meshing process which which often leads to errors errors or even even divergence divergence of the simulation simulation.. Due to its particle-based and fully Lagrangian approach, the complexity of the geometry surfaces is not a limiting factor in XFlow even in the presence of moving parts, allowing allowing to solve solve real industrial industrial problems. The performance performance of XFlow XFlow will be demonstrated for different industry benchmarks. The first example is the Ahmed body which is a classical benchmark in the automotive industry. The second benchmark presented will be the NASA trapezoidal wing. XFlow results will be described and show good agreement with experimental data. Ahmed body, Keywords: Lattice-Boltzmann, Lagrangian, particle-based, Ahmed NASA trapezoidal wing 1. Introductio Introduction n
For the past 20 years, the field of Computational Fluid Dynamics (CFD) has reached reached a high high level level of maturi maturity ty,, but it has only been recent recently ly that that CFD has been broadly applied to the improvement of several processes at differe different nt stages: stages: resear research ch,, design design,, manufa manufact cturi uring, ng, optimi optimizat zation ion,, etc. etc. The need for robust and reliable analysis tools is therefore growing rapidly, in proportion proportion to the increasing increasing complexit complexity y of simulations. simulations. To provide provide quick, accurate feedback to realistic engineering problems is consequently essential for companies to be competitive. Preprint submitted to ECCOMAS 2012
The traditional numerical methodologies employed so far are based on methods methods invo involvi lving ng finite finite volum volumes es and finite finite elemen elements, ts, app applie lied d to Navie NavierrStoke Stokess equat equation ions. s. Howe Howeve ver, r, even even though though such such methods methods have have been widely widely inve investi stigat gated, ed, they they still still hold hold major drawba drawback cks, s, limiti limiting ng their their capaci capacity ty to solve solve real industrial industrial problems: problems: uncertain uncertainties ties induced induced by the meshing process; highly empirical approaches to the turbulence modeling (RANS); the treatment of the nonlinear convective term; artificial stabilization parameters; and so on. Because Because of this, in most cases engineers engineers are not able to model real systems; they are forced to fall back on simplified models and approximations. These methods require a time-consuming meshing process, are not tolerant to moving parts, and are usually limited to steady-state analysis, ignoring transient dynamics. Particle-based Particle-based methods have been in development development for several decades, and are now starting starting to come to the fore. Among them, the promising promising LatticeLatticeBoltzmann Method (LBM) surmounts many of the drawbacks of traditional CFD methods. XFlow XFlow CFD uses a particle-based particle-based and fully Lagrangian Lagrangian approach based on LBM. With this method, classic fluid-domain meshing is not required and surface complexity is not a limiting factor. XFlow has been validated in several benchmarks, demonstrating the validity of the method to solve industrial problems. The first example presented in this paper is the Ahmed body, a classic benchmark for the automotive industry dustry. The car’s geometry geometry has a variable variable slant angle and is a challenging challenging test case in terms of turbulence modelling and drag estimation. The NASA trapezoidal wing is the second benchmark presented in this paper, a three elem elemen entt airf airfoi oill compo compose sed d of a slat slat,, a main main blade blade and and a flap. flap. Th Thee goal goal is to assess the aerodynamic coefficients on a large range of incidence angles, including including the post-stall post-stall region. region. 2. Numerical Numerical methodology methodology
Over the last few years, schemes based on minimal kinetic models for the Boltzmann Boltzmann equation are becoming becoming increasingly increasingly popular as a reliable reliable alternative to conventional CFD approaches. The Lattice Boltzmann method (LBM) was originally developed as an improved modification of the Lattice Gas Automata to remove statistical noise and achieve achieve better Galilean Galilean invarianc invariancee [1, 2]. Due to the flexibility flexibility afforded by its close connection to kinetic theory, the LBM can be adapted to model several physical phenomena. Recent research has led to major improvements, 2
including physically consistent models for multiphase and multicomponent flow and fully compressible flow [3, 4, 5]. 2.1. Lattice Gas Automata
The Lattice Gas Automata (LGA) is a simple scheme for modeling the behavior of gases. The basic idea behind the LGA is that particles with specific velocities ( ei , i = 1,...,b) propagate through a d-dimensional lattice, at discrete times t = 0, 1, 2,... and collide according to specific rules designed to preserve the mass and the linear momentum when different particles reach the 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 square lattice and in four directions (d = 2, b = 4). The state of an element of the lattice at instant t is given by the occupation number ni (r, t), with ni = 1 being presence and n i = 0 absence of particles with velocity ei . The stream-and-collide equation that governs the evolution of the system is (1) ni (r + ei , t + dt) = ni (r, t) + Ω i (n1 ,...,nb ), i = 1,...,b, where Ωi is the collision operator that computes a post-collision state conserving mass and linear momentum. If one were to assume Ωi = 0, only an streaming operation would be performed. From a statistical point of view, the system is made up of a large number of elements which are macroscopically equivalent to the problem investigated. The macroscopic density and linear momentum can be computed as: 1 b ρ = ni b i=1
1 n
(2)
b
ρv =
b
i ei
(3)
i=1
2.2. Lattice Boltzmann method
While the LGA schemes use boolean logic to represent the occupation stage, the LBM method makes use of statistical distribution functions f i with real variables, preserving by construction the conservation of mass and linear momentum. The Boltzmann transport equation is defined as follows: ∂f i + ei · ∇f i = Ωi , i = 1,...,b, ∂t 3
(4)
where f i is the particle distribution function in the direction i, ei the corresponding discrete velocity and Ω i the collision operator. The stream-and-collide scheme of the LBM can be interpreted as a discrete approximation of the continuous Boltzmann equation. The streaming or propagation step models the advection of the particle distribution functions along discrete directions, while most of the physical phenomena are modeled by the collision operator which also has a strong impact on the numerical stability of the scheme. In the most common approach, a single-relaxation time (SRT) based on the Bhatnagar-Gross-Krook (BGK) approximation is used 1 ΩBGK = (f ieq − f i ), i τ
(5)
where τ is the relaxation time parameter, related to the macroscopic viscosity as follows 1 (6) ν = c 2s (τ − ). 2 f ieq is the local equilibrium function usually defined as eq
f i = ρwi
e iα uα u α uβ 1+ 2 + 2c2s cs
e e
iα iβ
c2s
− δ αβ
.
(7)
Here cs is the speed of sound, u the macroscopic velocity, δ the Kronecker delta and the wi are weighting constants built to preserve the isotropy. The α and β subindexes denote the different spatial components of the vectors appearing in the equation and Einstein’s summation convention over repeated indices has been used. By means of the Chapman-Enskog expansion the resulting scheme can be shown to reproduce the hydrodynamic regime for low Mach numbers [5, 6, 7]. The single-relaxation time approach is commonly used because of its simplicity. However it is not well-posed for high Mach number applications and it is prone to numerical instabilities. Some of the limitations of the BGK are addressed with multiple-relaxation-time (MRT) collision operators where the collision process is carried out in moment space instead of the usual velocity space ˆij (meq ΩMRT = M ij 1 S (8) i − mi ), i −
ˆij is diagonal, m eq where the collision matrix S i is the equilibrium value of the moment m i and M ij is the transformation matrix [8, 9]. 4
An alternative method that aims to overcome the limitations of the BGK approach is the entropic lattice Boltzmann (ELBM) scheme, which may rely on a single-relaxation-time where the attractors of the particle distribution functions are based on the minimization of a Lyapunov-type functional enforcing the H-theorem locally in the collision step. However, this method is expensive from the computational point of view [10] and thus not used in practical engineering applications. The collision operator in XFlow is based on a multiple relaxation time scheme. However, as opposed to standard MRT, the scattering operator is implemented in central moment space. The relaxation process is performed in a moving reference frame by shifting the discrete particle velocities with the local macroscopic velocity, naturally improving the Galilean invariance and the numerical stability for a given velocity set [11]. Raw moments can be defined as k l m
µx y z
N
k l m i ix iy iz
f e e e =
(9)
i
and the central moments as k l m
˜x y z µ
N
k
l
m
f (e − u ) (e − u ) (e − u ) = i
ix
x
iy
y
iz
z
(10)
i
2.3. Turbulence modeling
The approach used for turbulence modeling is the Large Eddy Simulation (LES). This scheme introduces an additional viscosity, called turbulent eddy viscosity ν t , in order to model the subgrid turbulence. The LES scheme we have used is the Wall-Adapting Local Eddy viscosity model, that provides a consistent local eddy-viscosity and near wall behavior [12]. The actual implementation is formulated as follows: ν t = S αβ = Gdαβ = gαβ =
(Gdαβ Gdαβ )3/2 ∆ (S αβ S αβ )5/2 + (Gdαβ Gdαβ )5/4 gαβ + gβα 2 1 2 1 2 2 (gαβ + gβα ) − δ αβ gγγ 2 3 ∂u α ∂x β 2 f
5
(11) (12)
(13) (14)
where ∆f = C w ∆x is the filter scale, S is the strain rate tensor of the resolved scales and the constant C w is typically 0.325. A generalized law of the wall that takes into account for the effect of adverse and favorable pressure gradients is used to model the boundary layer [13]: U U 1 + U 2 uτ U 1 u p U 2 = = + uc uc uc uτ uc u p d pw /dx u p τ w uτ + uτ + u p = + f y f y 1 2 ρu2τ uc uc |d pw /dx| uc uc uc y y+ = ν uc = uτ + u p
uτ =
|τ | /ρ ν d p ρ dx
w
(15)
(16) (17) (18) (19)
1/3
u p =
w
.
(20)
Here, y is the normal distance from the wall, u τ is the skin friction velocity, τ w is the turbulent wall shear stress, d pw /dx is the wall pressure gradient, u p is a characteristic velocity of the adverse wall pressure gradient and U is the mean velocity at a given distance from the wall. The interpolating functions f 1 and f 2 given by Shih et al. [13] are depicted in figure 1.
Figure 1: Unified laws of the wall
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3. Ahmed body b enchmark
The Ahmed Body is a classic benchmark for the automotive industry. It was first defined and its characteristics described in the experimental work of Ahmed [14]. The car geometry was studied at various slant angles from 0 to 40 degrees. The experimental measurements were conducted by Ahmed in the DFVLR subsonic wind tunnels at Braunschweig and G¨ottingen which have 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 coefficient against the slant angle obtained by Ahmed in [14], and the second one is to analyze the mean recirculation structures on the slant surface of the Ahmed body and in the downstream region. 3.1. Simulation setup
A strictly identical geometry to the one used by Ahmed was imported into the virtual wind tunnel featured in XFlow. This virtual wind tunnel consists 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 the top boundaries, and zero gauge pressure was imposed at the outlet. Periodic boundary conditions were set on the side walls, and a free-slip wall with no velocity was imposed at the bottom boundary. The geometry of the Ahmed body was separated into two parts in order to simplify the setup modification for variable slant angles. The first part is the fore body that has an invariable geometry. The second part is the rear body which is replaced when the slant angle changes. These two parts are shown on figure 2.
Figure 2: Fore body geometry and rear geometry
The simulation settings are gathered in table 1, and correspond to a Reynolds number based on the car length equal to 4.29 million. The sim7
Table 1: Simulation specifications of the Ahmed body benchmark
Inlet velocity Density Dynamic viscosity Car length Reynolds number Slant angles Turbulence intensity
60 m/s 1 kg/m3 1.46014 × 10 5 Pa.s 1044 mm 4.29 × 106 0 ; 5 ; 10 ; 12.5 ; 15 ; 20 ; 25 ; 30 ; 40 degrees 0.5% −
ulation time was two seconds and the time step ∆ t = 7.69231 × 10 automatically estimated by XFlow to ensure the numerical stability.
5
−
s is
3.2. Spatial discretization
Since XFlow is a particle based technology it does not require a timeconsuming meshing process. The preprocessor generates the initial octree lattice structure based on the input geometries and the user-specified resolution for each geometry. The lattice may have several levels of detail which are hierarchically arranged. Each level solves spatial and temporal scales two times smaller than the previous level, thus forming the aforementioned octree structure. The lattice structure may be modified later by the solver if the computational domain changes (due to the presence of moving parts) or if the resolution changes dynamically in order to adapt to the flow patterns (adaptive wake refinement). The adaptive wake refinement feature in XFlow is based on the module of the vorticity field: in the lattice elements where the vorticity reaches a threshold value the lattice is automatically refined. Similarly, when the vorticity is lower than another threshold, eight adjacent lattice elements are merged to form a coarser lattice element. This saves computational resources and removes the need to refine your solution in advance. Consequently, as in illustrated figure 3, three resolutions are required by 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 wake that allows us to get good results in an acceptable time, a resolution dependency study is conducted before starting the validation of the Ahmed body. This preliminary study consists in refining the resolutions and seeing how this affects the accuracy of the results, but also checking if the code is converging 8
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 predicted by XFlow for a slant angle of 35 degrees which is a reference angle for this benchmark. The far field is taken constant as 0.08 m, and four resolutions are considered for the walls and the wake as described table 2. The drag coefficient is computed for the four cases and compared with the experimental value measured by Ahmed [14]. The drag points from the simulations are plotted in figure 4 in function of the number of elements at t = 0.3 s. The point corresponding to the resolution h/22 = 0.01 m gives good results and in an acceptable time for a slant angle φ = 35 , and will therefore 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. As the wake refinement algorithm creates a significant number of elements as it develops, its importance in the drag contribution must be assessed accurately to get a good compromise between solution quality and computational time. Hence, a second study is conducted on the wake resolution starting from the elected near wall resolution (0.01 m) and then increasing by multiples of two, due to the lattice structure. The figure 5 demonstrates the importance of solving the wake accurately: using the same resolution near the walls and within the wake the drag coefficient history shows a nice prediction, but ◦
9
Figure 4: Drag coefficient against the number of lattice nodes for different resolutions at φ = 35 ◦
as soon as the wake resolution is the double or quadruple of the near wall resolution affects the results quite dramatically. Hence, for all our runs, the spatial discretization chosen for all the different slant angles is done with an automatic wake refinement with a resolution of 0.08 m for the far field, and 0.01 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 and mainly consists in geometry importation, the flow and boundary specifications, and the resolution setup. The calculation time is almost the same for all the slant angles and varies between 6 and 8 hours with the previously selected resolutions on two Intel Xeon E5620 (2.4GHz). The first result given by Ahmed is the curve representing the drag coefficient against the slant angle φ, and gives the drag contributions of every part of the Ahmed body: the front C k , the rear vertical surface C b , the rear slant surface C s and the friction drag C r . The total drag Ahmed found was C w and was the sum of the different contributions. Hence, the total drag obtained from XFlow for the different slant angles is superimposed with the C w 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 10
Figure 5: Drag coefficient history for different wake refinement resolution at φ = 35
◦
is the critical angle 12.5 degrees, as measured by Ahmed. The absolute drag values predicted by XFlow are accurate and the relative error varies from only 0.4% to 3.2% for most of the angles, except around the drag breakdown and at 0 degree angle where it reaches a maximum of only 7.1%. These small discrepancies can be explained, on the one hand, by the complexity around the flow around 30 degrees of slant angle which is switching from a massive 3D separation in the near-wake region to an almost 2D attached structure at higher angles [15], and, on the other hand, by stronger gradients produced by 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 main recirculation structures resulting from the flow around the Ahmed body. For this study, the averaging of the flow fields is required in order to filter the temporal fluctuations and to identify the main structures of the turbulent wake. The averaging of the fields started from t = 0.3 s when the flow was established, as indicated for example by figure 5, to cut off the transient period. Ahmed provides pictures of the oil flow on the slanted surface for φ = 12.5, 25 and 30 degrees. It can be compared with XFlow which features Line Integral Convolution (LIC) that approximates the surface streamlines on a body. The figure 7 shows similar structure for the three angles: a quite 11
smooth and attached flow at 12.5 degrees, smooth flow patterns with two small and symmetric fringes on the sides at 25 degrees, and two large and symmetric separation bubbles at 30 degrees. Ahmed also provides different velocity vectors plots in the symmetry plane of the car, showing the near-wake region. This allows the study of the separation bubble on the rear slant and within the wake for different slant angles. Figure 8 compares the near-wake region for a slant angle of 5 degrees between the experimental results measured by Ahmed and results obtained by XFlow at the same scale. This allows us to check the length of the bubble separation located around the non-dimensional coordinate x/Lref = 0.375, predicted in an extremely similar way in the two pictures. Two main eddy structures are detected - highlighted in red boxes on figure 8 - which are symmetrical from the top and bottom of the separation bubble. The code tends to locate them slightly further downstream, though with reasonable overall flow patterns. The near-wake structure for a slant angle of 25 degrees also show good similarities. This figure 8 shows an equivalent triangular separation bubble, ending around the non-dimensional coordinate x/L ref = 0.2 for both cases. 4. NASA trapezoidal wing benchmark
The NASA trapezoidal wing benchmark comes from the 1st AIAA CFD High Lift Prediction Workshop (HiLiftPW-1), sponsored by the Applied Aerodynamics Technical Committee, which took place in June 2010 in Chicago, IL. The challenge was to simulate a half aircraft configuration composed of a body and a 3-element airfoil with a plane of symmetry as shown in figure 9 for a wide range of angles of attack. The trapezoidal wing is composed of slat, main element and flap. The latter can be in two different configurations: Configuration 1 at 25 degrees and Configuration 8 at 20 degrees of angle-of-attack. The objectives of the benchmark are multiple [16]: • Assess the prediction capability of CFD codes in landing/taking-off configuration, • Develop practical modeling guidelines for the analysis of high-lift configurations, 12
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 0.005 Wake (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 existing CFD 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. The Mach number was 0.2, the Reynolds number based on the mean aerodynamic chord (MAC) was 4.3 million. The angles of attack run for this benchmark were: -4, 1, 6, 13, 21, 25, 28, 32, 34 and 37 degrees. The hardware used in all 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 benchmark using the four resolutions described in Table 3 and a constant far field resolution of 1.28 m. An incidence angle of 13 degrees which is one of the reference angles of the first workshop was employed. The drag coefficient obtained with each of the four simulations is plotted in figure 10 as a function of the number of elements at t = 0.3 s. The point corresponding to resolution h/23 gives the best estimation of the drag compared to the experimental data, with only 1% of relative error. This value will therefore be used as the reference near wall resolution for the rest of the study. However, two different wake resolutions have been used depending on the incidence of the NASA trapezoidal wing. Indeed, for large angles of attack, a significant wake develops and the number of lattice elements introduced by the adaptive wake refinement increases. At 32 degrees, the simulation reaches 25 million lattice elements, which is the maximum number of elements that can fit in the 12 GB of RAM available on the workstation. Special care is thus required in order to keep this number within the memory constraints for higher angles. The wake resolution has been limited to double the normal value for those cases (Resolution 2 in table 4). 13
Table 4: Resolutions used for the 1 st High Lift Prediction Workshop
Walls (m) Wake (m) Far Field (m) Max. # of Particles Angles Resolution 1 0.005 0.01 1.28 25 × 106 [-4 ; 32 ] 6 Resolution 2 0.005 0.02 1.28 10 × 10 [34 ; 37 ]
4.2. Numerical results
The experimental data were produced at the 14x22 wind-tunnel at the well-known NASA Langley. Forces, moments, and Cp distribution were provided with free transition [17]. Data were provided as lower and upper values which are assumed to be the range of uncertainty in the wind tunnel measurements. On figure 11, the drag coefficient against the angle of attack α is shown. XFlow results show very good agreement with the experimental data along the whole range of angles. The drag slope is accurate and still behaves correctly 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 and absolute lift coefficient values. Starting from 32 degrees, the critical angle is reached and the code also succeeds in predicting this: the wind tunnel data indicates the maximum lift point at around 33 degrees, and it happens between 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 bubble of 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 polar curve on Figure 11 is hence matching the experimental results, especially in the pre-stall region. The pitching moment coefficients also lie between the upper and lower limits of the experimental results within almost the whole range. 5. Conclusions
The CFD code XFlow features a kinetic particle-based solver that differs from the traditional approaches, which are usually mesh-based. The latticeBoltzmann method employed is able to solve advanced industrial problems even in the presence of complex geometries or moving parts. 14
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◦
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◦
The methodology has demonstrated it can solve industrial benchmarks efficiently. For instance the Ahmed body is a classic benchmark for the automotive industry that XFlow solved with a high degree of accuracy. XFlow did not face convergence issues even for extreme slant angles, and changing the rear of the car did not add additional workload. The code has been demonstrated to be robust and accurate in terms of drag and flow pattern prediction, and closely matches the data measured by Ahmed in the DFVLR subsonic wind tunnel of Braunschweig including the drag breakdown around 30 degrees and the low slant angles where gradients are stronger. The High Lift Prediction Workshop benchmark has also been successfully validated by XFlow. The NASA trap wing geometry was tested within a range of incidence between -4 and 37 degrees, which includes the post-stall region. The drag, lift and pitching moment coefficients predicted by the code are in good agreement with the experimental tests conducted in the NASA Langley 14x22 wind tunnel. The stall angle is also accurately predicted around 33 degrees. XFlow has therefore demonstrated its robustness and accuracy in different benchmarks. The method is well-suited for external aerodynamics and shows strong potential for more advanced topics, such as analysis involving complex geometries, the presence of moving parts and fluid-structure interaction. References
[1] U. Frisch, B. Hasslacher, Y. Pomeau, Lattice-gas automata for the navier-stokes equation, Physical review letters 56 (14) (1986) 1505–1508. [2] G. R. McNamara, G. Zanetti, Use of the Boltzmann equation to simulate lattice-gas automata, Physical Review Letters 61 (1988) 2332–2335. doi:10.1103/PhysRevLett.61.2332. [3] S. Chen, G. Doolen, Lattice boltzmann method for fluid flows, Annual review of fluid mechanics 30 (1) (1998) 329–364. [4] S. Succi, The lattice boltzmann equation, For Fluid Dynamics and Beyond. [5] Z. Ran, Y. Xu, Entropy and weak solutions in the thermal model for the compressible euler equations, Arxiv preprint arXiv:0810.3477.
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[6] Y. H. Qian, D. D’Humi`eres, P. Lallemand, Lattice BGK models for Navier-Stokes equation, EPL (Europhysics Letters) 17 (1992) 479. doi:10.1209/0295-5075/17/6/001. [7] F. J. Higuera, J. Jim´enez, Boltzmann approach to lattice gas simulations, EPL (Europhysics Letters) 9 (1989) 663. doi:10.1209/02955075/9/7/009. [8] X. Shan, H. Chen, A general multiple-relaxation-time boltzmann collision model, International Journal of Modern Physics C 18 (4) (2007) 635–643. [9] D. d’Humi`eres, Multiple–relaxation–time lattice boltzmann models in three dimensions, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 360 (1792) (2002) 437–451. [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 moment lattice boltzmann method, Journal of Statistical Physics (2011) 1–48. [12] F. Ducros, F. Nicoud, T. Poinsot, Wall-adapting local eddy-viscosity models for simulations in complex geometries, in: Proceedings of 6th ICFD Conference on Numerical Methods for Fluid Dynamics, 1998, pp. 293–299. [13] T. Shih, L. Povinelli, N. Liu, M. Potapczuk, J. Lumley, A generalized wall function, NASA Technical Report. [14] S. Ahmed, G. Ramm, G. Faitin, Some salient features of the timeaveraged ground vehicle wake, Tech. rep., Society of Automotive Engineers, Inc., Warrendale, PA (1984). [15] G. Franck, N. Nigro, M. Storti, J. D’El´ıa, Numerical simulation of the flow around the ahmed vehicle model, Latin American applied research 39 (4) (2009) 295–306. [16] C. Rumsey, The 1st aiaa cfd high lift prediction workshop (Jun. 2010). URL http://hiliftpw.larc.nasa.gov/index-workshop1.html 16
[17] C. McGinley, L. Jenkins, R. Watson, A. Bertelrud, 3-d high-lift flowphysics experiment–transition measurements, AIAA Paper 5148 (2005) 2005.
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Figure 6: Drag coefficient against the slant angle φ
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Figure 7: Averaged Line Integral Convolution (LIC) on the slanted surface from Ahmed (left) and XFlow (right)
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Figure 8: Near-wake structure at scale for: a) φ = 5 , b) φ = 25 ◦
Figure 9: NASA trapezoidal wing geometry
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◦
Figure 10: Drag coefficient against the number of lattice nodes for different resolutions at α = 13 ◦
Figure 11: Drag (a) and lift (b) coefficients against the angle of attack, the polar curve (c), and the pitching moment coefficient (d)
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Figure 12: Averaged Line Integral Convolution (LIC) at 37 degrees incidence
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