CHAPITRE 5 CONCLUSION ET RECOMMANDATIONS
5.1 Recommandations
Cette thèse se termine par des recommandations découlant d’observations et constatations faites lors de la réalisation des travaux, lesquelles pourraient être utiles pour des recherches futures.
1. Il est mentionné plus haut qu’un avantage de la LBM consiste à ne pas devoir résoudre une équation différentielle non-linéaire. Or, l’implémentation du modèle de turbulence Spalart-Allmaras est basée sur une telle approche et déroge donc de la simplicité inhérente de la LBM. Comme il existe des modèles LBM développés pour simuler le phénomène d’advection-diffusion [120], il serait intéressant d’étudier la possibilité d’utiliser une telle approche pour concevoir un modèle de turbulence entièrement basé sur la collision et propagation de distributions. Une idée similaire avait d’ailleurs été proposée par Succi et al. [121] pour créer un pendant LBM du modèle de turbulence k-ε. Cependant, aucune suite ne semble avoir été donnée à cette idée dans la littérature.
2. Une limitation des programmes conçus pour les deuxième et troisième parties du travail découle du très grand temps de calcul requis pour obtenir des solutions stationnaires produisant une convergence à la quatrième décimale des coefficients aérodynamiques. Hormis l’amélioration de la construction des programmes et le passage éventuel vers un autre langage de programmation que Matlab, l’implémentation de fonctions de paroi pourrait permettre de limiter le raffinement nécessaire près des profils d’ailes. Le temps de calcul serait réduit parce qu’il y aurait moins de nœuds sur le réseau et que le pas de temps utilisé serait plus grand.
3. Le programme de la deuxième partie du travail utilise un réseau multi-domaines pour lequel le réseau le plus raffiné couvre tout le périmètre du profil d’aile. Cette configuration est utilisée parce que des essais effectués avec des raffinements différents près de la paroi ont produit des discontinuités des champs macroscopiques aux interfaces. Comme la méthode choisie pour le transfert d’information en est une d’interpolation, il serait intéressant d’étudier le problème plus en détail et de voir si des améliorations pourraient être apportées à l’algorithme d’interpolation.
4. L’opérateur de collision en cascade est utilisé dans le programme multi-domaines parce qu’il est très stable. Or, Geier et al. [33] affirment aussi que cet opérateur «ne vise pas à
éliminer les hautes fréquences de l’écoulement mais de les traiter de façon précise sans affecter la précision des basses fréquences», et que cette méthode est correcte pour «accomplir une séparation des échelles résolues et non-résolues dans la simulation d’écoulements turbulents». Geier et al. affirment aussi que son objectif est de «capturer correctement le comportement physique des ondes les plus courtes résolues par le raffinement du réseau». Ces affirmations font en sorte que cette méthode s’apparente à la méthode de simulation des grandes échelles (LES) pour les écoulements turbulents. Il serait donc très intéressant, si les ressources computationnelles le permettent, de concevoir un programme de simulation 3D pour écoulements turbulents sur ailes d’avion comparant l’utilisation stricte de l’opérateur de collision en cascade avec un autre opérateur de collision couplé à un modèle LES, tel le modèle de Smagorinsky. Ce type de simulation a déjà été effectué pour l’écoulement autour d’une sphère [58] et il serait intéressant d’étendre la méthode à une aile.
RÉFÉRENCES
[1] Chapman S, Cowling TG. The mathematical theory of non-uniform gases. Third. Cambridge University Press; 1970.
[2] The Editors of Encyclopædia Britannica. Kinetic theory of gases. Encycl Br 2016. https://www.britannica.com/science/kinetic-theory-of-gases (accessed November 11, 2016).
[3] Mason EA. Gas. Encycl Br 2016. https://www.britannica.com/science/gas-state-of- matter/Kinetic-theory-of-gases#ref507018 (accessed November 11, 2016).
[4] The Editors of Encyclopædia Britannica. Perfect gas. Encycl Br 2016. https://www.britannica.com/science/perfect-gas (accessed November 11, 2016).
[5] Müller-Kirsten HJW. Statistical Mechanics of an Ideal Gas (Maxwell). Basics Stat. Phys. Second, World Scientific; 2013, p. 11–25. doi:10.1142/9789814449540_0002.
[6] Viggen EM. The lattice Boltzmann method: fundamentals and acoustics. Norwegian University of Science and Technology, 2014.
[7] Succi S. The lattice Boltzmann equation for fluid dynamics and beyond. Oxford University Press; 2001.
[8] Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys Rev 1954;94:511–25. doi:https://doi.org/10.1103/PhysRev.94.511.
[9] Cédric Villani. A review of Mathematical Topics in Collisional Kinetic Theory. In: Friedlander S, Serre D, editors. Handb. Math. Fluid Dyn. Vol. 1, Elsevier Science; 2002, p. 71–305.
[10] Wolf-Gladrow DA. Lattice-Gas Cellular Automata and Lattice Boltzmann Models - An Introduction. Berlin: Springer; 2005.
[11] Gardner M. Mathematical Games. The fantastic combinations of John Conway’s new solitaire game “life”. Sci Am 1970;223:120–3.
[12] Wolfram S. A new kind of science. Wolfram Media Inc; 2002.
[13] Hardy J, Pomeau Y, De Pazzis O. Time evolution of a two-dimensional classical lattice system. Phys Rev Lett 1973;31:276–9. doi:10.1103/PhysRevLett.31.276.
[14] Hardy J, De Pazzis O, Pomeau Y. Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions. Phys Rev A 1976;13:1949–61. doi:10.1103/PhysRevA.13.1949.
[15] Viggen EM. The lattice Boltzmann method with applications in acoustics. Norwegian University of Science and Technology, 2009.
[16] Frisch U, Hasslacher B, Pomeau Y. Lattice as Automata for the Navier-Stokes Equation. Phys Rev Lett 1986;56:1505–8. doi:10.1103/PhysRevLett.56.1505.
[17] Chopard B, Dupuis A, Masselot A, Luthi P. Cellular automata and lattice Boltzmann techniques: an approach to model and simulate complex systems. Adv Complex Syst 2002;5:103–246. doi:10.1142/S0219525902000602.
[18] D’Humières D, Lallemand P. Numerical simulations of hydrodynamics with lattice gas automata in two dimensions. Complex Syst 1987;1:599–632.
[19] McNamara GR, Zanetti G. Use of the Boltzmann equation to simulate lattice-gas automata. Phys Rev Lett 1988;61:2332–5. doi:10.1103/PhysRevLett.61.2332.
[20] Higuera FJ, Jiménez J. Boltzmann approach to lattice gas simulations. EPL (Europhysics Lett 1989;9:663–8.
[21] Koelman JMVA. A Simple Lattice Boltzmann Scheme for Navier-Stokes Fluid Flow. Europhys Lett 1991;15:603–7. doi:10.1209/0295-5075/15/6/007.
[22] Qian YH, D’Humières D, Lallemand P. Lattice BGK models for Navier-Stokes equation. Europhys Lett 1992;17:479–84.
[23] He X, Luo L-S. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Phys Rev E 1997;56:6811–7. doi:10.1103/PhysRevE.56.6811.
[25] Chen S, Doolen GD. Lattice Boltzmann method for fluid flows. Annu Rev Fluid Mech 1998;30:329–64. doi:10.1146/annurev.fluid.30.1.329.
[26] Zou Q, He X. On pressure and velocity flow boundary conditions and bounceback for the lattice Boltzmann BGK model. Phys Fluids 1997;9:1–19.
[27] Bouzidi M, Firdaouss M, Lallemand P. Momentum transfer of a Boltzmann-lattice fluid with boundaries. Phys Fluids 2001;13:3452–9. doi:10.1063/1.1399290.
[28] Guo Z, Zheng C, Shi B. An extrapolation method for boundary conditions in lattice Boltzmann method. Phys Fluids 2002;14:2007. doi:10.1063/1.1471914.
[29] Feng Z-G, Michaelides EE. The immersed boundary-lattice Boltzmann method for solving fluid–particles interaction problems. J Comput Phys 2004;195:602–28. doi:10.1016/j.jcp.2003.10.013.
[30] D’Humières D. Generalized lattice-Boltzmann equations. Rarefied gas Dyn. theory simulations, American Institute of Aeronautics and Astronautics, Inc.; 1994, p. 450–8. [31] Ansumali S, Karlin I V. Entropy function approach to the lattice Boltzmann method. J Stat
Phys 2002;107:291–308.
[32] Geier MC. Ab initio derivation of the cascaded lattice boltzmann automaton. University of Freiburg, 2006.
[33] Geier M, Greiner A, Korvink JG. Cascaded digital lattice Boltzmann automata for high Reynolds number flow. Phys Rev E 2006;73:66705.
[34] Lagrava D, Malaspinas O, Latt J, Chopard B. Advances in multi-domain lattice Boltzmann grid refinement. J Comput Phys 2012;231:4808–22. doi:10.1016/j.jcp.2012.03.015.
[35] Imamura T, Suzuki K, Nakamura T, Yoshida M. Flow simulation around an airfoil by lattice Boltzmann method on generalized coordinates. AIAA J 2005;43:1968–73. doi:10.2514/1.7554.
[36] He X, Doolen G. Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder. J Comput Phys 1997;134:306–15. doi:10.1006/jcph.1997.5709.
[37] Stiebler M, Tölke J, Krafczyk M. An upwind discretization scheme for the finite volume
doi:10.1016/j.compfluid.2005.09.002.
[38] Moreau S, Sanjos M, Kim M, P F. Direct simulation of trailing-edge noise generated by a controlled diffusion airfoil using a lattice-Boltzmann method. Seventh Int. Symp. Turbul. Shear Flow Phenom., Ottawa: Turbulence and Shear Flow Phenomena; 2011, p. 1–6. [39] Sanjosé M, Moreau S. Direct numerical simulation of self-noise of an installed control-
diffusion airfoil. Can Acoust 2011;39:2–3.
[40] Holman DM, Brionnaud RM, Modena MC. Aerodynamic analysis of the 2nd high lift prediction workshop by a lattice-Boltzmann method solver. 32nd AIAA Appl. Aerodyn. Conf., Atlanta, GA: AIAA; 2014. doi:10.2514/6.2014-2568.
[41] König B, Fares E, Nölting S. PowerFLOW Analysis HiLiftPW-2 Configuration. 52nd AIAA Aerosp Sci Meet 2014.
[42] Satti R, Li Y, Shock R, Duncan B. Computational analysis of a wingtip vortex in the near- field using LBM-VLES approach. 49th AIAA Aerosp. Sci. Meet. Incl. New Horizons Forum Aerosp. Expo., Orlando, FL: AIAA; 2011, p. 1–12.
[43] Satti R, Li Y, Shock R, Noelting S. Unsteady flow analysis of a multi-element airfoil using lattice Botlzmann method. AIAA J 2012;50. doi:10.2514/1.J050906.
[44] Koenig B, Fares E, Jammalamadaka A, Li Y. Investigation of the NACA 4412 trailing Edge separation using a lattice-Boltzmann approach. 44th AIAA Fluid Dyn. Conf., American Institute of Aeronautics and Astronautics; 2014. doi:doi:10.2514/6.2014-3324.
[45] König B, Fares E, Broeren A. Lattice-Boltzmann analysis of three-dimensional ice shapes on a NACA 23012 airfoil. SAE Tech Pap 2015. doi:10.4271/2015-01-2084.
[46] Li Y, Jammalamadaka A, Gopalakrishnan P, Zhang R, Chen H. Computation of high- subsonic and transonic flows by a lattice Boltzmann method. 54th AIAA Aerosp. Sci. Meet., San Diego, CA: American Institute of Aeronautics and Astronautics, AIAA SciTech Forum; 2016, p. 1–15. doi:10.2514/6.2016-0043.
[47] Filippova O, Succi S, Mazzocco F, Arrighetti C, Bella G, Hänel D, et al. Multiscale lattice Boltzmann schemes with turbulence modeling. J Comput Phys 2001;170:812–29. doi:10.1006/jcph.2001.6764.
[48] Succi S, Filippova O, Smith G, Kaxiras E. Applying the lattice Boltzmann equation to multiscale fluid problems. Comput Sci … 2001;3:26–37.
[49] Teixeira CM. Incorporating turbulence models into the lattice-Boltzmann method. Int J Mod Phys C 1998;9:1159–75. doi:10.1142/S0129183198001060.
[50] Premnath KN, Pattison MJ, Banerjee S. Dynamic subgrid scale modeling of turbulent flows using lattice-Boltzmann method. Phys A Stat Mech Its Appl 2009;388:2640–58. doi:http://dx.doi.org/10.1016/j.physa.2009.02.041.
[51] Li Y, Shock R, Zhang R, Chen H, Shih T. Simulation of flow over an iced airfoil by using a lattice-Boltzmann method. 43rd AIAA Aerosp. Sci. Meet. Exhib., Reno, NV: American Institute of Aeronautics and Astronautics; 2005, p. 1–8. doi:http://dx.doi.org/10.2514/6.2005-1103.
[52] Li Y, Shock R, Zhang R, Chen H. Numerical study of flow past an impulsively started cylinder by the lattice-Boltzmann method. J Fluid Mech 2004;519:273–300. doi:10.1017/S0022112004001272.
[53] Chen H, Filippova O, Hoch J, Molvig K, Shock R, Teixeira C, et al. Grid refinement in lattice Boltzmann methods based on volumetric formulation. Phys A Stat Mech Its Appl 2006;362:158–67. doi:10.1016/j.physa.2005.09.036.
[54] Chi X, Li Y, Chen H, Addy H, Choo Y, Shih T. A comparative study using CFD to predict iced airfoil aerodynamics. AIAA Pap 2005;1371:1–12. doi:10.2514/6.2005-1371.
[55] Chen X-P. Applications of lattice Boltzmann method to turbulent flow around two- dimensional airfoil. Eng Appl Comput Fluid Mech 2012;6:572–80.
[56] Li K, Zhong C, Zhuo C, Cao J. Non-body-fitted Cartesian-mesh simulation of highly turbulent flows using multi-relaxation-time lattice Boltzmann method. Comput Math with Appl 2012;63:1481–96. doi:10.1016/j.camwa.2012.03.080.
[57] Si H, Shi Y. Study on lattice Boltzmann method/large eddy simulation and its application at high Reynolds number flow. Adv Mech Eng 2015;7:1–8. doi:10.1177/1687814015573829. [58] Ishida T. Lattice Boltzmann method for aeroacoustic Simulations with block-structured Cartesian grid. 54th AIAA Aerosp. Sci. Meet., American Institute of Aeronautics and
Astronautics; 2016, p. 1–8. doi:doi:10.2514/6.2016-0259.
[59] Imamura T, Suzuki K, Nakamura T, Yoshida M. Acceleration of steady-state lattice Boltzmann simulations on non-uniform mesh using local time step method. J Comput Phys 2005;202:645–63. doi:10.1016/j.jcp.2004.08.001.
[60] Filippova O, Hänel D. Lattice-Boltzmann simulation of gas-particle flow in filters. Comput Fluids 1997;26:697–712.
[61] Mei R, Luo L-S, Shyy W. An accurate curved boundary treatment in the lattice Boltzmann method. J Comput Phys 1999;155:307–30. doi:10.1006/jcph.1999.6334.
[62] Lallemand P, Luo L-S. Lattice Boltzmann method for moving boundaries. J Comput Phys 2003;184:406–21. doi:10.1016/S0021-9991(02)00022-0.
[63] Rohde M, Kandhai D, Derksen J, Van den Akker H. Improved bounce-back methods for no- slip walls in lattice-Boltzmann schemes: Theory and simulations. Phys Rev E 2003;67:66703. doi:10.1103/PhysRevE.67.066703.
[64] Ginzburg I, d’Humières D. Multireflection boundary conditions for lattice Boltzmann models. Phys Rev E 2003;68:66614.
[65] Chun B, Ladd a. Interpolated boundary condition for lattice Boltzmann simulations of flows in narrow gaps. Phys Rev E 2007;75:66705. doi:10.1103/PhysRevE.75.066705.
[66] Verberg R, Ladd a. Lattice-Boltzmann Model with Sub-Grid-Scale Boundary Conditions. Phys Rev Lett 2000;84:2148–51. doi:10.1103/PhysRevLett.84.2148.
[67] Rohde M, Derksen JJ, Van Den Akker HEA. Volumetric method for calculating the flow around moving objects in lattice-Boltzmann schemes. Phys Rev E - Stat Nonlinear, Soft Matter Phys 2002;65:1–11. doi:10.1103/PhysRevE.65.056701.
[68] Kao P-H, Yang R-J. An investigation into curved and moving boundary treatments in the lattice Boltzmann method. J Comput Phys 2008;227:5671–90. doi:10.1016/j.jcp.2008.02.002.
[69] Filippova O, Hänel D. Grid refinement for lattice-BGK models. J Comput Phys 1998;147:219–28. doi:10.1006/jcph.1998.6089.
lattice Boltzmann method. Comput Fluids 2009;38:883–7. doi:10.1016/j.compfluid.2008.09.008.
[71] Kang J, Kang S, Suh YK. A dynamic boundary model for implementation of boundary conditions in lattice-Boltzmann method. J Mech Sci Technol 2008;22:1192–201. doi:10.1007/s12206-008-0307-y.
[72] Tiwari A, Vanka SP. A ghost fluid Lattice Boltzmann method for complex geometries. Int J Numer Methods Fluids 2012;69:481–98. doi:10.1002/fld.2573.
[73] Chen L, Yu Y, Hou G. Sharp-interface immersed boundary lattice Boltzmann method with reduced spurious-pressure oscillations for moving boundaries. Phys Rev E 2013;87:53306. doi:10.1103/PhysRevE.87.053306.
[74] Chen S, Bao S, Liu Z, Li J, Yi C, Zheng C. A heuristic curved-boundary treatment in lattice Boltzmann method. Europhys Lett 2010;92:54003. doi:10.1209/0295-5075/92/54003. [75] Chen H, Teixeira C, Molvig K. Realization of fluid boundary conditions via discrete
Boltzmann dynamics. Int J Mod Phys C 1998;9:1281–92.
doi:10.1142/S0129183198001151.
[76] Rohde M. Extending the lattice-Boltzmann method: Novel techniques for local grid refinement and boundary conditions. Delft University of Technology, 2004.
[77] Li Y. An improved volumetric LBM boundary approach and its extension for sliding mesh simulation. Iowa State University, 2011.
[78] Li Y, Zhang R, Shock R, Chen H. Prediction of vortex shedding from a circular cylinder using a volumetric Lattice-Boltzmann boundary approach. Eur Phys J Spec Top 2009;171:91–7. doi:10.1140/epjst/e2009-01015-9.
[79] Yu H, Chen X, Wang Z, Deep D, Lima E, Zhao Y, et al. Mass-conserved volumetric lattice Boltzmann method for complex flows with willfully moving boundaries. Phys Rev E 2014;89:63304. doi:10.1103/PhysRevE.89.063304.
[80] Zhang J, Johnson PC, Popel AS. An immersed boundary lattice Boltzmann approach to simulate deformable liquid capsules and its application to microscopic blood flows. Phys Biol 2007;4:285–95. doi:10.1088/1478-3975/4/4/005.
[81] Strack OE, Cook BK. Three-dimensional immersed boundary conditions for moving solids in the lattice-Boltzmann method. Int J Numer Methods Fluids 2007;55:103–25. doi:10.1002/fld.
[82] Shu C, Liu N, Chew Y, Lu Z. Numerical simulation of fish motion by using lattice Boltzmann-immersed boundary velocity correction method. J Mech Sci Technol 2007;21:1139–49.
[83] Shu C, Wu J. A new immersed boundary-lattice Boltzmann method and its application to incompressible flows. Mod Phys Lett B 2009;23:261–4. doi:10.1142/S0217984909018151. [84] Wu J, Shu C, Zhang YH. Simulation of incompressible viscous flows around moving objects by a variant of immersed boundary-lattice Boltzmann method. Int J Numer Methods Fluids 2010;62:327–54. doi:10.1002/fld.
[85] Tian F-B, Luo H, Zhu L, Liao JC, Lu X-Y. An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments. J Comput Phys 2011;230:7266–83. doi:10.1016/j.jcp.2011.05.028.
[86] Rojas R, Seta T, Hayashi K, Tomiyama A. Immersed boundary-finite difference lattice Boltzmann method using two relaxation times. In: CSIRO, editor. Ninth Int. Conf. CFD Miner. Process Ind., 2012, p. 1–6.
[87] Seta T, Rojas R, Hayashi K, Tomiyama A. Implicit-correction-based immersed boundary– lattice Boltzmann method with two relaxation times. Phys Rev E 2014;89:23307. doi:10.1103/PhysRevE.89.023307.
[88] Nishida H, Meichin Y. Seamless immersed boundary lattice Boltzmann method for incompressible flow simulation. Seventh Int. Conf. Comput. Fluid Dyn., Hawaii: ICCFD; 2012, p. 1–19.
[89] Peng Y, Shu C, Chew YT, Niu XD, Lu XY. Application of multi-block approach in the immersed boundary–lattice Boltzmann method for viscous fluid flows. J Comput Phys 2006;218:460–78. doi:10.1016/j.jcp.2006.02.017.
[90] Sui Y, Chew Y, Roy P, Low H. A hybrid immersed-•boundary and multi-•block lattice Boltzmann method for simulating fluid and moving-•boundaries interactions. … Numer Methods Fluids 2007;53:1727–54. doi:10.1002/fld.
[91] Dupuis A, Chatelain P, Koumoutsakos P. An immersed boundary–lattice-Boltzmann method for the simulation of the flow past an impulsively started cylinder. J Comput Phys 2008;227:4486–98. doi:10.1016/j.jcp.2008.01.009.
[92] Lu J, Han H, Shi B, Guo Z. Immersed boundary lattice Boltzmann model based on multiple relaxation times. Phys Rev E 2012;85:16711. doi:10.1103/PhysRevE.85.016711.
[93] Suzuki K, Inamuro T. A higher-order immersed boundary-lattice Boltzmann method using a smooth velocity field near boundaries. Comput Fluids 2013;76:105–15. doi:10.1016/j.compfluid.2013.01.029.
[94] Farnoush S, Manzari MT. Comparison of boundary slip for two variants of immersed boundary method in lattice Boltzmann framework. Phys A Stat Mech Its Appl 2014;404:200–16. doi:10.1016/j.physa.2014.02.010.
[95] De Rosis A, Falcucci G, Ubertini S, Ubertini F. Aeroelastic study of flexible flapping wings by a coupled lattice Boltzmann-finite element approach with immersed boundary method. J Fluids Struct 2014;49:516–33. doi:10.1016/j.jfluidstructs.2014.05.010.
[96] De Rosis A, Ubertini S, Ubertini F. A Comparison Between the Interpolated Bounce-Back Scheme and the Immersed Boundary Method to Treat Solid Boundary Conditions for Laminar Flows in the Lattice Boltzmann Framework. J Sci Comput 2014. doi:10.1007/s10915-014-9834-0.
[97] De Rosis A, Ubertini S, Ubertini F. A partitioned approach for two-dimensional fluid- structure interaction problems by a coupled lattice Boltzmann-finite element method with
immersed boundary. J Fluids Struct 2014;45:202–15.
doi:10.1016/j.jfluidstructs.2013.12.009.
[98] Li Z, Favier J, D’Ortona U, Poncet S. An immersed boundary-lattice Boltzmann method for single- and multi-component fluid flows. J Comput Phys 2015;304:424–40. doi:10.1016/j.jcp.2015.10.026.
[99] Pellerin N, Leclaire S, Reggio M. Equilibrium distributions for straight, curved, and immersed boundary conditions in the lattice Boltzmann method. Comput Fluids 2014:126– 35. doi:http://dx.doi.org/10.1016/j.compfluid.2014.06.007.
2016).
[101] Ghia U, Ghia K., Shin C. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J Comput Phys 1982;48:387–411. doi:10.1016/0021- 9991(82)90058-4.
[102] Erturk E, Corke TC, Gökçöl C. Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Int J Numer Methods Fluids 2005;48:747–74. doi:10.1002/fld.953.
[103] Higuera FJ, Succi S. Simulating the flow around a circular cylinder with a lattice Boltzmann equation. Europhys Lett 1989;8:517.
[104] Izham M, Fukui T, Morinishi K. Application of regularized lattice Boltzmann method for incompressible flow simulation at high Reynolds number and flow with curved boundary. J Fluid Sci Technol 2011;6:812–22. doi:10.1299/jfst.6.812.
[105] Pellerin N, Leclaire S, Reggio M. An implementation of the Spalart–Allmaras turbulence model in a multi-domain lattice Boltzmann method for solving turbulent airfoil flows. Comput Math with Appl 2015;70:3001–18. doi:10.1016/j.camwa.2015.10.006.
[106] Rohde M, Kandhai D, Derksen JJ, Akker HEA Van Den. A generic, mass conservative local grid refinement technique for lattice-Boltzmann schemes. Int J Numer Methods Fluids 2006:439–68.
[107] Eitel-Amor G, Meinke M, Schröder W. A lattice-Boltzmann method with hierarchically refined meshes. Comput Fluids 2013;75:127–39. doi:10.1016/j.compfluid.2013.01.013. [108] Geier M, Greiner A, Korvink JG. Bubble functions for the lattice Boltzmann method and
their application to grid refinement. Eur Phys J Spec Top 2009;171:173–9. doi:10.1140/epjst/e2009-01026-6.
[109] Spalart PR, Allmaras SR. A one-equation turbulence model for aerodynamic flows. 30th Aerosp. Sci. Meet. Exhib., Reno, NV: AIAA; 1992, p. 1–22.
[110] Lockard DP, Luo L, Milder SD, Singer BA. Evaluation of PowerFLOW for aerodynamic applications. J Stat Phys 2002;107:423–78. doi:https://doi.org/10.1023/A:1014539411062. [111] Mei R, Yu D, Shyy W, Luo L-S. Force evaluation in the lattice Boltzmann method involving
curved geometry. Phys Rev E 2002;65:41203. doi:10.1103/PhysRevE.65.041203.
[112] Pellerin N, Leclaire S, Reggio M. Solving incompressible fluid flows on unstructured meshes with the lattice Boltzmann flux solver. Eng Appl Comput Fluid Mech 2017;11:310– 27. doi:10.1080/19942060.2017.1292410.
[113] Peng G, Xi H, Duncan C, Chou S-H. Lattice Boltzmann method on irregular meshes. Phys Rev E 1998;58:R4124–7. doi:10.1103/PhysRevE.58.R4124.
[114] Peng G, Xi H, Duncan C, Chou S-H. Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys Rev E 1999;59:4675–82. doi:10.1103/PhysRevE.59.4675. [115] Zarghami A, Maghrebi MJ, Ghasemi J, Ubertini S. Lattice Boltzmann finite volume
formulation with improved stability. Commun Comput Phys 2012;12:42–64. doi:10.4208/cicp.151210.140711a.
[116] Li W, Luo L-S. Finite volume Lattice Boltzmann Method for Nearly Incompressible Flows on Arbitrary Unstructured Meshes. Commun Comput Phys 2016;20:301–24. doi:10.4208/cicp.211015.040316a.
[117] Shu C, Wang Y, Teo CJ, Wu J. Development of lattice Boltzmann flux solver for simulation of incompressible flows. Adv Appl Math Mech 2014;6:436–60. doi:10.4208/aamm.2014.4.s2.
[118] Wang Y, Yang L, Shu C. From lattice Boltzmann method to lattice Boltzmann flux solver. Entropy 2015:7713–35. doi:10.3390/e17117713.
[119] Geuzaine C, Remacle J-F. Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Methods Eng 2009;79:1309–31. doi:10.1002/nme.2579.
[120] Stiebler M, Tölke J, Krafczyk M. Advection–diffusion lattice Boltzmann scheme for hierarchical grids. Comput Math with Appl 2008;55:1576–84. doi:10.1016/j.camwa.2007.08.024.
[121] Succi S, Amati G, Benzi R. Challenges in lattice Boltzmann computing. J Stat Phys 1995;81:5–16. doi:10.1007/BF02179964.
ANNEXE A – LA MÉTHODE DE BOLTZMANN SUR RÉSEAU ET LES
ÉQUATIONS DE NAVIER-STOKES
Dans cette annexe nous démontrons, au moyen d’une expansion Chapman-Enskog, que la LBM avec opérateur de collision de type BGK est équivalente à résoudre les équations de Navier-Stokes en régime faiblement compressible.
L’équation de Boltzmann à vitesses finies s’écrit comme suit:
,
, , i i i i f t f t t t x c x x , (A.1)où ci xi t. Nous transformons cette équation en remplaçant les dérivées partielles par une dérivée totale, puis par une différence finie :
,
,
, i i i i f t t t f t t t x c x x (A.2) ou
,
,
, i i i i f x c t t t f x t t x t . (A.3)A.1 Expansion en série de Taylor
Les distributions de probabilités sont tout d’abord réécrites en séries de Taylor, autour du point de référence
x, t telles que :
* * 2 2 2 2 2 2 2 , , , , , , , 1 2 2 i i i i i i i f t f t f t f t t t f t f t f t t t t t x x x x x x x x x x x x x , (A.4)
2 2 2 2 2 2 2 , , , , , , , 1 2 2 i i i i i i i i i i i f t f t f + t t t f t t t t f t f t f t t t t t t t x x x c x c x x x x c c x x . (A.5)Pour simplifier la suite des dérivations, nous omettrons