F~s = P
i
F~i
−
→Ts = P
i
−−→OMi∧F~i
Connaissant sa masse M et son moment d'inertie I, la trajectoire de l'objet solide se déduit des lois de
Newton :
M.d ~dtU = F~s
I.d~dtΩ = T~s
oùM =ρs.π.R2et I=12.M.R2 pour un disque de rayonRen dimension 2.
7.4 Coder en python
On peut aussi regarder le code Pyhton de Mora [21] qui tourne sur de gros clusters aussi vite qu'un code C.
On peut aussi regarder la librairie 'pylbm' de Paris-Sud à l'adresse suivante :https://github.com/pylbm/
pylbm_gallery/blob/master/2D/Rayleigh_Benard/Rayleigh-Benard.pyLe chier .py fait 175 lignes.
7.4. Coder en python
Figure 7.3 : Source : polycopié de Wohlmut, TU Münich.
Bibliographie
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[2] Cheng Chang, Chih-Hao Liu, and Chao-An Lin. Boundary conditions for lattice boltzmann simulations with complex geometry ows. Computers & Mathematics with Applications, 58(5) :940949, 2009. 59 [3] Qing Chen, Xiaobing Zhang, and Junfeng Zhang. Improved treatments for general boundary conditions
in the lattice boltzmann method for convection-diusion and heat transfer processes. Physical Review E, 88(3) :033304, 2013. 64
[4] James DeBonis. Solutions of the taylor-green vortex problem using high-resolution explicit nite dif-ference methods. In 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, page 382, 2013. 55,56
[5] François Dubois, Pierre Lallemand, and Mohamed Mahdi Tekitek. On anti bounce back boundary condition for lattice boltzmann schemes. Computers & Mathematics with Applications, 79(3) :555575, 2020. 65
[6] I Ginzbourg and PM Adler. Boundary ow condition analysis for the three-dimensional lattice boltz-mann model. Journal de Physique II, 4(2) :191214, 1994. 30
[7] Zhaoli Guo, Chuguang Zheng, and Baochang Shi. Discrete lattice eects on the forcing term in the lattice boltzmann method. Physical Review E, 65(4) :046308, 2002. 29,30,57
[8] Xiaoyi He, Qisu Zou, Li-Shi Luo, and Micah Dembo. Analytic solutions of simple ows and analysis of nonslip boundary conditions for the lattice boltzmann bgk model. Journal of Statistical Physics, 87(1-2) :115136, 1997. 30
[9] Chih-Fung Ho, Cheng Chang, Kuen-Hau Lin, and Chao-An Lin. Consistent boundary conditions for 2d and 3d lattice boltzmann simulations. Computer Modeling in Engineering and Sciences (CMES), 44(2) :137, 2009. 59,61
[10] Timm Krüger, Halim Kusumaatmaja, Alexandr Kuzmin, Orest Shardt, Goncalo Silva, and Erlend Ma-gnus Viggen. The lattice boltzmann method. Springer International Publishing, 10 :9783, 2017. 64 [11] Anthony JC Ladd. Numerical simulations of particulate suspensions via a discretized boltzmann
equa-tion. part 1. theoretical foundaequa-tion. Journal of uid mechanics, 271 :285309, 1994. 18, 33,64
[12] Anthony JC Ladd. Numerical simulations of particulate suspensions via a discretized boltzmann equa-tion. part 2. numerical results. Journal of uid mechanics, 271 :311339, 1994. 33, 65
[13] Jonas Latt, Bastien Chopard, Orestis Malaspinas, Michel Deville, and Andreas Michler. Straight velocity boundaries in the lattice boltzmann method. Physical Review E, 77(5) :056703, 2008. 61
[14] Qing Li, Kai Hong Luo, QJ Kang, YL He, Q Chen, and Q Liu. Lattice boltzmann methods for multiphase ow and phase-change heat transfer. Progress in Energy and Combustion Science, 52 :62105, 2016.28, 30
[15] Tian-Fu Li, Kang Luo, and Hong-Liang Yi. Eect of unipolar charge injection direction on the onset of rayleigh-bénard convection : a lattice boltzmann study. International Communications in Heat and Mass Transfer, 112 :104496, 2020. 57
[16] Chih-Hao Liu, Kuen-Hau Lin, Hao-Chueh Mai, and Chao-An Lin. Thermal boundary conditions for thermal lattice boltzmann simulations. Computers & Mathematics with Applications, 59(7) :21782193, 2010. 59
[17] Kang Luo, Jian Wu, Hong-Liang Yi, and He-Ping Tan. Lattice boltzmann modelling of electro-thermo-convection in a planar layer of dielectric liquid subjected to unipolar injection and thermal gradient.
International Journal of Heat and Mass Transfer, 103 :832846, 2016. 57
[18] Li-Shi Luo. Lattice-gas automata and lattice Boltzmann equations for two-dimensional hydrodynamics.
PhD thesis, School of Physics, Georgia Institute of Technology, 1993. 30
[19] Nicos S Martys and Hudong Chen. Simulation of multicomponent uids in complex three-dimensional geometries by the lattice boltzmann method. Physical review E, 53(1) :743, 1996. 29, 30
[20] AA Mohamad and A Kuzmin. A critical evaluation of force term in lattice boltzmann method, natural convection problem. International Journal of Heat and Mass Transfer, 53(5-6) :990996, 2010. 28 [21] Peter Mora, Gabriele Morra, and David A Yuen. A concise python implementation of the lattice
boltzmann method on hpc for geo-uid ow. Geophysical Journal International, 220(1) :682702, 2020.
66
[22] Xiaowen Shan. Simulation of rayleigh-bénard convection using a lattice boltzmann method. Physical Review E, 55(3) :2780, 1997. 57
[23] Xiaowen Shan and Gary Doolen. Multicomponent lattice-boltzmann model with interparticle interac-tion. Journal of Statistical Physics, 81(1-2) :379393, 1995. 29,30, 57
[24] Sauro Succi and S Succi. The lattice Boltzmann equation : for complex states of owing matter. Oxford University Press, 2018. 65
[25] Shi Tao, Ao Xu, Qing He, Baiman Chen, and Frank GF Qin. A curved lattice boltzmann boundary scheme for thermal convective ows with neumann boundary condition. International Journal of Heat and Mass Transfer, 150 :119345, 2020. 65
[26] Wim M Van Rees, Anthony Leonard, DI Pullin, and Petros Koumoutsakos. A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical ows at high reynolds numbers. Journal of Computational Physics, 230(8) :27942805, 2011. 55,56
[27] AJ Wagner. Thermodynamic consistency of liquid-gas lattice boltzmann simulations. Physical Review E, 74(5) :056703, 2006. 30
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[29] Lina Xu, Parthib Rao, and Laura Schaefer. A novel scheme for curved moving boundaries in the lattice boltzmann method. International Journal of Modern Physics C, 27(12) :1650144, 2016. 62,64
[30] Ting Zhang, Baochang Shi, Zhaoli Guo, Zhenhua Chai, and Jianhua Lu. General bounce-back scheme for concentration boundary condition in the lattice-boltzmann method. Physical Review E, 85(1) :016701, 2012. 18,64
[31] Qisu Zou and Xiaoyi He. On pressure and velocity boundary conditions for the lattice boltzmann bgk model. Physics of uids, 9(6) :15911598, 1997. 59
Tableau des symboles
Symbole Dénition Equation Page
B Opérateur de collision 1.20 4
BBGK Opérateur de collision BGK 1.27 5
~c Vitesse (continue) des particules 2
~ci Vitesse prédénie 9
cr Vitesse de réseau 2.6 10
cs Vitesse du son isotherme 1
cp Capacité calorique massique 2.26 17
D Dimension spatiale (1,2,3) 2
DT Diusivité thermique 2.26 17
˜
ecin Energie cinétique volumique 1.10 2
f Fonction de distribution en masse à 1 particule 2
f(eq) Fonction de distribution à l'équilibre 5
fi Fonction de distribution discrète (masse) 2.2 9 fipost Fonction de distribution post-collision 4.3 25 f(M) Fonction de distribution de Maxwell-Boltzmann 6 F(~x,~c,t) Fonction de distribution en nombre à 1 particule 1 gi Fonction de distribution discrète (énergie) 2.27 18
Gr Nombre de Grashof 3.5 22
h Coecient d'échange thermique 3.9 23
H Entropie thermodynamique 1.29 5
k Ratio dec2rsurc2s 2.16 13
Ma Nombre de Mach 1.4 1
M(n) Moment continu d'ordren(tenseur) 1.11 3
M˜(n) Moment discret d'ordren(tenseur) 2.3 9
mp Masse d'une particule 2
M˜ Masse molaire 1
n(~x,t) Densité de particules [m−3] 1.5 2
N Nombre de particules dans le système 1
Nu Nombre de Nusselt 3.7 22
opp(i) Direction opposée à la vitesse~ci 34
Pr Nmmbre de Prandtl 3.3 21
Q Nombre de vitesses prédénies 9
R Constante des gaz parfait 1
R˜ Constante des gaz parfaits modiée 1.1 1
Ra Nombre de Rayleigh 3.4 22
Re Nombre de Reynolds 3.1 21
T Température 2.25 17
~v Vitesse macroscopique (mesurable exp.) 2
vs Vitesse du son adiabatique 1
wi Coecient de pondération 2.1 9
Wi Coecient de pondération Wi 12
Bibliographie
Symbole Dénition Equation Page
βV Coe. de dilatation thermique volumique 2.32 18
δij Symbole de Kronecker 11
∆t Pas de temps 10
∆tcrit Pas de temps critique (optimal) 2.19 16
∆x Pas d'espace 10
γ Coecient adiabatique 5.3 37
λ Conductivité thermique 2.26 17
ν Viscosité cinématique 2.20 16
ν∗ Viscosité cinématique adimensionnée 2.22 17
ρ Masse volumique [kg.m−3] 1.8 2
τ Temps de relaxation (pour la fonction f) 1.27 5 τT Temps de relaxation (pour la fonction g) 2.27 18 τ∗ Temps de relaxation adimensionné (pourf) 2.7 11