格子Boltzmann方法数值模拟研究
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摘要
本文对格子Boltzmann方法的基本思想与发展历史进行了阐述,包括对格子Boltzmann方程与N-S方程的关系进行的推导与分析,其研究主要是围绕格子Boltzmann方法的理论与应用展开的。
采用格子Boltzmann方法需要构造粒子速度模型。文中首先对两种常用的二维粒子速度模型D2Q9与D2Q13进行了分析与比较,并基于这两种模型研制了对应的计算程序,成功地模拟出不同雷诺数情况下的二维驱动空腔流动;接着,对格子Boltzmann方法中涉及的各种边界处理方法做了详细的分析,结合非均匀网格的格子Boltzmann方法—TLLBM,给出了推导过程,然后用此种方法编程模拟了圆柱及二维翼型这两种曲面边界的绕流问题;最后,介绍了格子Boltzmann方法中常见的流动模型,其中重点对IEDDF热模型进行了分析,并选用二维封闭空腔自然对流进行了具体的数值模拟。
文中展示了上述典型流动问题的数值模拟结果,并在相关结果图中给出文献结果供比较。
Lattice Boltzmann method (LBM) is studied in this dissertation for solving fluid flow problems. The basic concept of the method is firstly described, and its history of development is briefly reviewed. In order to solve the viscous flow problems which are governed by the macroscopic Navier-Stokes Equations, a detailed investigation including the principle and application of LBM is presented.
When the flow problems are solved by LBM, the particle velocity model is usually required to accomplish the simulation. Therefore two particle velocity models named as D2Q9 and D2Q13 are firstly analyzed and compared, and then they are implemented in the framework of LBM to simulate 2D lid driven cavity flows; as the uniform grid is used by the traditional LBM, it encounters difficulties when dealing with complicated configurations represented by curved boundaries. In order to overcome this difficulty, a new method named as TLLBM which uses non-uniform grid is proposed. The new method is applied to solve the flows over a cylinder and NACA0012 airfoil in the present work; Furthermore, some popular models for LBM are described in the dissertation, and special attention is paid to the IEDDF thermal model that is implemented to simulate 2D cavity nature convection flows.
A comparison between the numerical results and the reference works is made, and they agree well with each other.
采用格子Boltzmann方法需要构造粒子速度模型。文中首先对两种常用的二维粒子速度模型D2Q9与D2Q13进行了分析与比较,并基于这两种模型研制了对应的计算程序,成功地模拟出不同雷诺数情况下的二维驱动空腔流动;接着,对格子Boltzmann方法中涉及的各种边界处理方法做了详细的分析,结合非均匀网格的格子Boltzmann方法—TLLBM,给出了推导过程,然后用此种方法编程模拟了圆柱及二维翼型这两种曲面边界的绕流问题;最后,介绍了格子Boltzmann方法中常见的流动模型,其中重点对IEDDF热模型进行了分析,并选用二维封闭空腔自然对流进行了具体的数值模拟。
文中展示了上述典型流动问题的数值模拟结果,并在相关结果图中给出文献结果供比较。
Lattice Boltzmann method (LBM) is studied in this dissertation for solving fluid flow problems. The basic concept of the method is firstly described, and its history of development is briefly reviewed. In order to solve the viscous flow problems which are governed by the macroscopic Navier-Stokes Equations, a detailed investigation including the principle and application of LBM is presented.
When the flow problems are solved by LBM, the particle velocity model is usually required to accomplish the simulation. Therefore two particle velocity models named as D2Q9 and D2Q13 are firstly analyzed and compared, and then they are implemented in the framework of LBM to simulate 2D lid driven cavity flows; as the uniform grid is used by the traditional LBM, it encounters difficulties when dealing with complicated configurations represented by curved boundaries. In order to overcome this difficulty, a new method named as TLLBM which uses non-uniform grid is proposed. The new method is applied to solve the flows over a cylinder and NACA0012 airfoil in the present work; Furthermore, some popular models for LBM are described in the dissertation, and special attention is paid to the IEDDF thermal model that is implemented to simulate 2D cavity nature convection flows.
A comparison between the numerical results and the reference works is made, and they agree well with each other.
引文
[1] S.chen and G.D.Doolen, Lattice Boltzmann Method For Fluid Flows, Annu.Rev. Fluid Mech ,1998,Vol.30:329-364。
[2]李元香,康立山,陈毓屏,格子气自动机,北京,清华大学出版社,1984。
[3] J.Neumann,Theory of Self-reproducing Automata,Urbana, University of Illionis, 1966。
[4] U.Frish, D.d’Humieres and Y.pomeau, Lattice–Gas automata for the Navier-Stokes Equation, Phy.Rev.e,1986,Vol.56:1505-1508。
[5] S.Wolfram, Cellular Automaton Fluid 1: Basic Theory, J.Stat.Phys.,1986,Vol.45:471-526。
[6] Higuera FJ, Jimenez J, Boltzmann approach to lattice gas simulations, Europhys.,1989,Vol.9:663-68。
[7] Benzi R, Succis S, Vergassola M, The lattice Boltzmann equation-theory and applications, 1992,Phys.Rep 222:145-97。
[8] P.L.Bhatnagar,E.P.Gross and M.Krook, A Model for Collision Processes in Gases, Phys.Rev.A.,1954,Vol.94:511-525。
[9] Qian YH, Lattice gas and Lattice kinetic theory applied to the Navier-stokes equations, 1990, phys thesis, Universite Pierre et. Marie Curie. Paris。
[10] Chen H, Chen S, Matthaeus WH, Recovery of the Navier-Stokes equations using a lattice-gasBoltzmann method, 1992, phys .rev .A. 45:R5339-42。
[11] Qian YH, d’Humieres D, Lallemand D, Lattice BGK models for Navier-Stokes equation,1992, Europhys,Vol.17:479-84。
[12] Qian YH, Orszag SA, lattice BGK models for Navier-Stokes equation: nonlinear deviation in compressible regimes, Euro.phys.Vol21:255-59。
[13]李克平,刘慕人,孔令江, 13速四方格子BGK热流体性质研究,广西师范大学学报, 1997年,15卷(4期):7-13。
[14] XIAO-YANG LU,CHAO-YING ZHANG, MU-REN LIU,LING-JIANG KONG,HUA-BING LI, THERMAL LATTICE BOLTZMANN simulation of viscous flow in a square cavity , International Journal of Modern Physics C , 2005,Vol. 16, No. 6: 867-877。
[15]郑忠,高小强,13-Bit多速格子气自动机模型的理论分析及模拟,重庆大学学报, 2000年, 23卷(6期):82-86。
[16] X.He and L-S.Lou, Lattice Boltzmann Model for the Incompressible Navier-Stokes Equation ,J,stat.Phys.,1997,Vol.88:927-944。
[17] J.D.Sterling and S.Chen, Stability Analysis of Lattice Boltzmann Methods,J.Comp.phys.,1996,Vol.123:196-206。
[18] F.Nanelli and S.Succi, The Lattice Boltzmann-Equation on Irregular lattices, J.Star.Phys., 1992,Vol.68(3-40:401-407。
[19] G.Amati, S.Succi and R.Benzi, Turbulence Channel Flow Simulations Using a Corse-GrainedExtension of the Lattice Boltzmann Method, Fluid.Dyn.Res., 1992,Vol.19:289-302。
[20] G.Peng, H.Xi and C.Duncan, Lattice Boltzmann Methods on Irrugular Meshes, Phys.Rev.E, 1998,Vol.58:4124-4127。
[21] O.Filippova and D.Hanel, Grid Refinement for Lattice-BGK models, J.comp.Phys., 1998,Vol.147:219-228。
[22] X.He and G.D.Doolen, Lattice Boltzmann Method on Curvilinear coordinates System: flow around a Circular Cylinder, J.Comp.Phys., 1997,Vol.134:306-315。
[23] X.He and G.D.Doolen, Lattice Boltzmann Method on Curvilinear coordinates System: shedding Behind a circular cylinder, Phys.Rev.E, 1997,Vol.56:434-440。
[24] A.Rodney, J.M.Worthing and S.Guy, Stability of Lattice Boltzmann Methods in Hydrodynamic regimes, Phys.Rev.E, 1997,Vol.56:2243-2253。
[25]郭照立,模拟不可压流体流动的格子Boltzmann方法研究,[博士学位论文],华中理工大学,2004年。
[26] Q.Zou, S.Hou and S.Chen etal, An Improvel Incompressible Lattice Boltzmann Model for Time-Independent Flows, J.Stat.Phys., 1995, Vol.54:6323-6330。
[27] Z.Lin, H.Fang and H.Tao, Improved Lattice Boltzmann Model for Incompressible Tow-Dimensional Steady Flows, Phys.Rev.E, 1996, Vol.54:6323-6330。
[28] Y.Chen and H.ohashe, Lattice-BGK Methods for Simelating Incompressible fluid Flows, Int.j.Mod.Phys.C, 1997,Vol.8:793-803。
[29] X.He, Q.Zou and L-S.Luo etal, Analytic Solutions of Simple Flows and Analysis of Nonslip Boundary Conditions for the Lattice Boltzmann BGK model, J.Stat.Phys., 1997, Vol.87:115-136。
[30] Alexander.F.j, Chen.S, Sterling.J.D, Lattice Boltzmann Thermo-hydrodynamics, Phys, Reve.E, 1993, Vol.47:R2249-2252。
[31] Chen.Y, Ohashi.H, akiyama.M, Thermal Lattice Bhatnagar-Cross-Kook Model without Nonlinear Deviations in Macro-dynamics Equations, Phys.Rev.E, 1994, Vol.50:2776-2783。
[32] Qian.Y, Succi.S, Orszag.S, Recent Advances in lattice Computing, Annu.Rev.comp.Phys, 1995, Vol.3:195-242。
[33] Chen.Y, Ohashi.H, akiyama.M, Tow Parameters Thermal Lattice BGK Model withControllable Prandtl Number, J.Sci.Comput, 1997, Vol.12:169-185。
[34] Mcnamara.GR, Garcia.AL, Alder.BJ, Stabilization of Thermal Lattice Boltzmann Models, j.Stat.phys., 1995, Vol.81:395-408。
[35] Shan.X, simulation of Rayleigh-Benard Convection using a Lattice-Boltzmann Method, Phys.Rev., 1997, Vol.55:2780-2788。
[36] He.X, Chen.S, Doolen.GD, A Novel Thermal Model for the Lattice Boltzmann Method in Incompressible Limit, J.Compat.phys., 1998, Vol.146:282-300。
[37] X.He, Q.Zou and L-S.Luo, etal, Analytic Solutions of Simple Flows and Analysis of Nonslip Boundary Conditions for the lattice Boltzmann BGKModel, J.Stat.Phys., 1997, Vol.87:115-136。
[38] P.A.Slordos, initial and Boundary Conditions for the Lattice Boltzmann Method, Phys.Rev.E, 1993, Vol.48:4823-4842。
[39] S.Noble, S.Chen and J.G.Georgiadis, etal, A Consistent Hydrodynamic Boundary Condition for the Lattice Boltzmann Method, Phys.Fluids, 1995, Vol.7:203-209。
[40] D.Ziegler, Boundary Conditions for Lattice Boltzmann Simulations, J. Stat. Phys., 1993, Voj.71:1171-1177。
[41] T.Inamuro, M.Yoshino and F.Ogino, A Non-Slip Boundary Condition for Lattice Boltzmann Simulations, Phys. Fluids, 1995, Vol.7:2928-2930。
[42] S.Chen, D.Martinez and R.Mei, On Boundary Conditions in Lattice Boltzmann Methods, Phys. Fluids, 1996,Vol.8:2527-2536。
[43] GUO. ZL , ZHENG. CG, SHI .BC, Non2equilibrium extrapolation method for velocity and pressure boundary conditions in lattice Boltzmann method,[J ] . Chinese Physics. , 2002 , Vol.4 :366-374。
[44] O. Filippova and D. Hanel, Grid refinement for lattice-BGKmodels, J.Comput.Phys., 1998, Vol.147: 219-228。
[45] Noble.DR, Georgiadis.JG, Buckius.RO, Comparison of accuracy and performance for Lattice Boltzmann and finite difference simulation of steady viscous flow, [J].Int J Numer Meth Fluids, 1996, Vol.23:1-18。
[46] Maier.RS, Bermard.RS, Grunau.DW, Boundary conditions for the Lattice Boltzmann Method, [J].Phys.Fluids,1996, Vol.8:1788-1801。
[47]程永光,Lattice Boltzmann方法极其在流场分析中应用的研究,武汉水利电力大学,1998年。
[48]王兴勇,索丽生,刘德有,程永光,Lattice Boltzmann方法理论和应用的新进展,河海大学,2004。
[49]梁宝社,格点Boltzmann方法及其应用,河北建筑科技学院学报, 1997年6月,第1卷第2期。
[50]徐辉,刘贵苏,吴宏伟,郑楚光,流体动力学的格子Boltzmann方法及其具体实现,燃烧科学与技术, 1996, Vol.2 No. 4。
[51] Chenghai SUN, Zujun XU,Baoguo WANG and Mengyu SHEN , Lattice-Boltzmann Models for Heat Transfer, Communications in Nonlinear Science&Numerical Simulation ,1997,Vol.2, No. 4。
[52] He.X, Luo.L, Dembo.M, some progress in Lattice Boltzmann Method , part∏: Reynolds number enhancement in simulation, [J], Phys.A, 1997, Vol.239:276-285。
[53]程永光,格子Boltzmann方法实用化及其在水利工程中应用的研究,博士后出站报告,南京河海大学,2000年。
[54] U.Ghia, L.N.Ghia, and C.T.Shin, High-Re solutions for Incompressible Flow Using the Navier-Stoles Equations and a Multigrid Method, J.Comp.Phys., 1982,Vol.48:387-411。
[55] R.Schreiber and H.B.Keller, Driven Cavity Flows by Efficient Numerical Techniques, J.Comp.Phys., 1982,Vol.49:310-333。
[56] C. SHU, Y. PENG and Y. T. CHEW, SIMULATION OF NATURAL CONVECTION IN A SQUARE CAVITY BY TAYLOR SERIES EXPANSION- AND LEAST SQUARES-BASED LATTICE BOLTZMANN METHOD, International Journal of Modern Physics C, 2002, Vol. 13, No. 10:1399-1414。
[57] Taro Imamura, Kojiro Suzuki, TakashiNakamura, Masahiro Yoshida, Flow simulation around an airfoil using Lattice Boltzmann method on generalized coordinates.,AIAA, 2004,Vol244:5– 8。
[58] Jung-Il Choi , Roshan C. Oberoi , Jack R. Edwards , Jacky A. Rosati, An immersed boundary method for complex incompressible flows, Journal Computation Physics, 2006。
[2]李元香,康立山,陈毓屏,格子气自动机,北京,清华大学出版社,1984。
[3] J.Neumann,Theory of Self-reproducing Automata,Urbana, University of Illionis, 1966。
[4] U.Frish, D.d’Humieres and Y.pomeau, Lattice–Gas automata for the Navier-Stokes Equation, Phy.Rev.e,1986,Vol.56:1505-1508。
[5] S.Wolfram, Cellular Automaton Fluid 1: Basic Theory, J.Stat.Phys.,1986,Vol.45:471-526。
[6] Higuera FJ, Jimenez J, Boltzmann approach to lattice gas simulations, Europhys.,1989,Vol.9:663-68。
[7] Benzi R, Succis S, Vergassola M, The lattice Boltzmann equation-theory and applications, 1992,Phys.Rep 222:145-97。
[8] P.L.Bhatnagar,E.P.Gross and M.Krook, A Model for Collision Processes in Gases, Phys.Rev.A.,1954,Vol.94:511-525。
[9] Qian YH, Lattice gas and Lattice kinetic theory applied to the Navier-stokes equations, 1990, phys thesis, Universite Pierre et. Marie Curie. Paris。
[10] Chen H, Chen S, Matthaeus WH, Recovery of the Navier-Stokes equations using a lattice-gasBoltzmann method, 1992, phys .rev .A. 45:R5339-42。
[11] Qian YH, d’Humieres D, Lallemand D, Lattice BGK models for Navier-Stokes equation,1992, Europhys,Vol.17:479-84。
[12] Qian YH, Orszag SA, lattice BGK models for Navier-Stokes equation: nonlinear deviation in compressible regimes, Euro.phys.Vol21:255-59。
[13]李克平,刘慕人,孔令江, 13速四方格子BGK热流体性质研究,广西师范大学学报, 1997年,15卷(4期):7-13。
[14] XIAO-YANG LU,CHAO-YING ZHANG, MU-REN LIU,LING-JIANG KONG,HUA-BING LI, THERMAL LATTICE BOLTZMANN simulation of viscous flow in a square cavity , International Journal of Modern Physics C , 2005,Vol. 16, No. 6: 867-877。
[15]郑忠,高小强,13-Bit多速格子气自动机模型的理论分析及模拟,重庆大学学报, 2000年, 23卷(6期):82-86。
[16] X.He and L-S.Lou, Lattice Boltzmann Model for the Incompressible Navier-Stokes Equation ,J,stat.Phys.,1997,Vol.88:927-944。
[17] J.D.Sterling and S.Chen, Stability Analysis of Lattice Boltzmann Methods,J.Comp.phys.,1996,Vol.123:196-206。
[18] F.Nanelli and S.Succi, The Lattice Boltzmann-Equation on Irregular lattices, J.Star.Phys., 1992,Vol.68(3-40:401-407。
[19] G.Amati, S.Succi and R.Benzi, Turbulence Channel Flow Simulations Using a Corse-GrainedExtension of the Lattice Boltzmann Method, Fluid.Dyn.Res., 1992,Vol.19:289-302。
[20] G.Peng, H.Xi and C.Duncan, Lattice Boltzmann Methods on Irrugular Meshes, Phys.Rev.E, 1998,Vol.58:4124-4127。
[21] O.Filippova and D.Hanel, Grid Refinement for Lattice-BGK models, J.comp.Phys., 1998,Vol.147:219-228。
[22] X.He and G.D.Doolen, Lattice Boltzmann Method on Curvilinear coordinates System: flow around a Circular Cylinder, J.Comp.Phys., 1997,Vol.134:306-315。
[23] X.He and G.D.Doolen, Lattice Boltzmann Method on Curvilinear coordinates System: shedding Behind a circular cylinder, Phys.Rev.E, 1997,Vol.56:434-440。
[24] A.Rodney, J.M.Worthing and S.Guy, Stability of Lattice Boltzmann Methods in Hydrodynamic regimes, Phys.Rev.E, 1997,Vol.56:2243-2253。
[25]郭照立,模拟不可压流体流动的格子Boltzmann方法研究,[博士学位论文],华中理工大学,2004年。
[26] Q.Zou, S.Hou and S.Chen etal, An Improvel Incompressible Lattice Boltzmann Model for Time-Independent Flows, J.Stat.Phys., 1995, Vol.54:6323-6330。
[27] Z.Lin, H.Fang and H.Tao, Improved Lattice Boltzmann Model for Incompressible Tow-Dimensional Steady Flows, Phys.Rev.E, 1996, Vol.54:6323-6330。
[28] Y.Chen and H.ohashe, Lattice-BGK Methods for Simelating Incompressible fluid Flows, Int.j.Mod.Phys.C, 1997,Vol.8:793-803。
[29] X.He, Q.Zou and L-S.Luo etal, Analytic Solutions of Simple Flows and Analysis of Nonslip Boundary Conditions for the Lattice Boltzmann BGK model, J.Stat.Phys., 1997, Vol.87:115-136。
[30] Alexander.F.j, Chen.S, Sterling.J.D, Lattice Boltzmann Thermo-hydrodynamics, Phys, Reve.E, 1993, Vol.47:R2249-2252。
[31] Chen.Y, Ohashi.H, akiyama.M, Thermal Lattice Bhatnagar-Cross-Kook Model without Nonlinear Deviations in Macro-dynamics Equations, Phys.Rev.E, 1994, Vol.50:2776-2783。
[32] Qian.Y, Succi.S, Orszag.S, Recent Advances in lattice Computing, Annu.Rev.comp.Phys, 1995, Vol.3:195-242。
[33] Chen.Y, Ohashi.H, akiyama.M, Tow Parameters Thermal Lattice BGK Model withControllable Prandtl Number, J.Sci.Comput, 1997, Vol.12:169-185。
[34] Mcnamara.GR, Garcia.AL, Alder.BJ, Stabilization of Thermal Lattice Boltzmann Models, j.Stat.phys., 1995, Vol.81:395-408。
[35] Shan.X, simulation of Rayleigh-Benard Convection using a Lattice-Boltzmann Method, Phys.Rev., 1997, Vol.55:2780-2788。
[36] He.X, Chen.S, Doolen.GD, A Novel Thermal Model for the Lattice Boltzmann Method in Incompressible Limit, J.Compat.phys., 1998, Vol.146:282-300。
[37] X.He, Q.Zou and L-S.Luo, etal, Analytic Solutions of Simple Flows and Analysis of Nonslip Boundary Conditions for the lattice Boltzmann BGKModel, J.Stat.Phys., 1997, Vol.87:115-136。
[38] P.A.Slordos, initial and Boundary Conditions for the Lattice Boltzmann Method, Phys.Rev.E, 1993, Vol.48:4823-4842。
[39] S.Noble, S.Chen and J.G.Georgiadis, etal, A Consistent Hydrodynamic Boundary Condition for the Lattice Boltzmann Method, Phys.Fluids, 1995, Vol.7:203-209。
[40] D.Ziegler, Boundary Conditions for Lattice Boltzmann Simulations, J. Stat. Phys., 1993, Voj.71:1171-1177。
[41] T.Inamuro, M.Yoshino and F.Ogino, A Non-Slip Boundary Condition for Lattice Boltzmann Simulations, Phys. Fluids, 1995, Vol.7:2928-2930。
[42] S.Chen, D.Martinez and R.Mei, On Boundary Conditions in Lattice Boltzmann Methods, Phys. Fluids, 1996,Vol.8:2527-2536。
[43] GUO. ZL , ZHENG. CG, SHI .BC, Non2equilibrium extrapolation method for velocity and pressure boundary conditions in lattice Boltzmann method,[J ] . Chinese Physics. , 2002 , Vol.4 :366-374。
[44] O. Filippova and D. Hanel, Grid refinement for lattice-BGKmodels, J.Comput.Phys., 1998, Vol.147: 219-228。
[45] Noble.DR, Georgiadis.JG, Buckius.RO, Comparison of accuracy and performance for Lattice Boltzmann and finite difference simulation of steady viscous flow, [J].Int J Numer Meth Fluids, 1996, Vol.23:1-18。
[46] Maier.RS, Bermard.RS, Grunau.DW, Boundary conditions for the Lattice Boltzmann Method, [J].Phys.Fluids,1996, Vol.8:1788-1801。
[47]程永光,Lattice Boltzmann方法极其在流场分析中应用的研究,武汉水利电力大学,1998年。
[48]王兴勇,索丽生,刘德有,程永光,Lattice Boltzmann方法理论和应用的新进展,河海大学,2004。
[49]梁宝社,格点Boltzmann方法及其应用,河北建筑科技学院学报, 1997年6月,第1卷第2期。
[50]徐辉,刘贵苏,吴宏伟,郑楚光,流体动力学的格子Boltzmann方法及其具体实现,燃烧科学与技术, 1996, Vol.2 No. 4。
[51] Chenghai SUN, Zujun XU,Baoguo WANG and Mengyu SHEN , Lattice-Boltzmann Models for Heat Transfer, Communications in Nonlinear Science&Numerical Simulation ,1997,Vol.2, No. 4。
[52] He.X, Luo.L, Dembo.M, some progress in Lattice Boltzmann Method , part∏: Reynolds number enhancement in simulation, [J], Phys.A, 1997, Vol.239:276-285。
[53]程永光,格子Boltzmann方法实用化及其在水利工程中应用的研究,博士后出站报告,南京河海大学,2000年。
[54] U.Ghia, L.N.Ghia, and C.T.Shin, High-Re solutions for Incompressible Flow Using the Navier-Stoles Equations and a Multigrid Method, J.Comp.Phys., 1982,Vol.48:387-411。
[55] R.Schreiber and H.B.Keller, Driven Cavity Flows by Efficient Numerical Techniques, J.Comp.Phys., 1982,Vol.49:310-333。
[56] C. SHU, Y. PENG and Y. T. CHEW, SIMULATION OF NATURAL CONVECTION IN A SQUARE CAVITY BY TAYLOR SERIES EXPANSION- AND LEAST SQUARES-BASED LATTICE BOLTZMANN METHOD, International Journal of Modern Physics C, 2002, Vol. 13, No. 10:1399-1414。
[57] Taro Imamura, Kojiro Suzuki, TakashiNakamura, Masahiro Yoshida, Flow simulation around an airfoil using Lattice Boltzmann method on generalized coordinates.,AIAA, 2004,Vol244:5– 8。
[58] Jung-Il Choi , Roshan C. Oberoi , Jack R. Edwards , Jacky A. Rosati, An immersed boundary method for complex incompressible flows, Journal Computation Physics, 2006。