Derivation of lattice Boltzmann equation via analytical characteristic integral
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摘要
A lattice Boltzmann(LB) theory, the analytical characteristic integral(ACI) LB theory, is proposed in this paper.ACI LB theory takes the Bhatnagar–Gross–Krook(BGK)-Boltzmann equation as the exact kinetic equation behind Navier–Stokes continuum and momentum equations and constructs an LB equation by rigorously integrating the BGK-Boltzmann equation along characteristics. It is a general theory, supporting most existing LB equations including the standard lattice BGK(LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of an LB equation is collision number, depicting the particle-interaction intensity in the time span of the LB equation, instead of the traditionally assumed relaxation time, and the over-relaxation time problem is merely a manifestation of the temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove this.
A lattice Boltzmann(LB) theory, the analytical characteristic integral(ACI) LB theory, is proposed in this paper.ACI LB theory takes the Bhatnagar–Gross–Krook(BGK)-Boltzmann equation as the exact kinetic equation behind Navier–Stokes continuum and momentum equations and constructs an LB equation by rigorously integrating the BGK-Boltzmann equation along characteristics. It is a general theory, supporting most existing LB equations including the standard lattice BGK(LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of an LB equation is collision number, depicting the particle-interaction intensity in the time span of the LB equation, instead of the traditionally assumed relaxation time, and the over-relaxation time problem is merely a manifestation of the temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove this.
A lattice Boltzmann(LB) theory, the analytical characteristic integral(ACI) LB theory, is proposed in this paper.ACI LB theory takes the Bhatnagar–Gross–Krook(BGK)-Boltzmann equation as the exact kinetic equation behind Navier–Stokes continuum and momentum equations and constructs an LB equation by rigorously integrating the BGK-Boltzmann equation along characteristics. It is a general theory, supporting most existing LB equations including the standard lattice BGK(LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of an LB equation is collision number, depicting the particle-interaction intensity in the time span of the LB equation, instead of the traditionally assumed relaxation time, and the over-relaxation time problem is merely a manifestation of the temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove this.
引文
[1]Liu H,Zou C,Shi B,Tian Z,Zhang L and Zheng C 2006 Int.J.Heat Mass Tran.49 4672
[2]Shi B and Guo Z 2003 Int.J.Mod.Phys.B 17 173
[3]Ladd A J C and Verberg R 2001 J.Stat.Phys.104 1191
[4]Sheikholeslami M,Gorji-Bandpy M and Ganji D D 2014 Powder Technol.254 82
[5]Liu X and Cheng P 2013 Int.J.Heat Mass Tran.64 1041
[6]Semma E,El Ganaoui M,Bennacer R and Mohamad A A 2008 Int.J.Therm.Sci.47 201
[7]Mountrakis L,Lorenz E,Malaspinas O,Alowayyed S,Chopard B and Hoekstra A G 2015 J.Comput.Sci.9 45
[8]Meng X and Guo Z 2016 Int.J.Heat Mass Tran.100 767
[9]Liu Q and He Y L 2017 Physica A 465 742
[10]Zhang Q,Sun D and Zhu M 2017 Chin.Phys.B 26 084701
[11]Wu X,Liu H and Chen F 2017 Acta Phys.Sin.-Ch.Ed.66 224702
[12]Sun D,Jiang D,Xiang N,Chen K and Ni Z 2013 Chin.Phys.Lett.30074702
[13]Frisch U,Hasslacher B and Pomeau Y 1986 Phys.Rev.Lett.56 1505
[14]Bhatnagar P L,Gross E P and Krook M 1954 Phys.Rev.94 511
[15]Dellar P J 2013 Comput.Math.Appl.65 129
[16]He X,Chen S and Doolen G D 1998 J.Comput.Phys.146 282
[17]Hwang Y H 2016 J.Comput.Phys.322 52
[18]Yong W A,Zhao W and Luo L S 2016 Phys.Rev.E 93 033310
[19]Bosch F and Karlin I V 2013 Phys.Rev.Lett.111 090601
[20]He X Y and Luo L S 1997 Phys.Rev.E 56 6811
[21]Shan X 2010 Phys.Rev.E 81 36702
[22]Sterling J D and Chen S 1996 J.Comput.Phys.123 196
[23]Chapman S and Cowling T G 1970 The Mathematical Theory of NonUniform Gases:An Account of the Kinetic Theory of Viscosity,Thermal Conduction and Diffusion in Gases(Cambridge:Cambridge University Press)
[24]Latt J,Chopard B,Malaspinas O,Deville M and Michler A 2008 Phys.Rev.E 77 56703
[25]Zou Q and He X 1997 Phys.Fluids 9 1591
[26]Inamuro T,Yoshino M and Ogino F 1995 Phys.Fluids 7 2928
[27]Ghia U,Ghia K N and Shin C T 1982 J.Comput.Phys.48 387
[2]Shi B and Guo Z 2003 Int.J.Mod.Phys.B 17 173
[3]Ladd A J C and Verberg R 2001 J.Stat.Phys.104 1191
[4]Sheikholeslami M,Gorji-Bandpy M and Ganji D D 2014 Powder Technol.254 82
[5]Liu X and Cheng P 2013 Int.J.Heat Mass Tran.64 1041
[6]Semma E,El Ganaoui M,Bennacer R and Mohamad A A 2008 Int.J.Therm.Sci.47 201
[7]Mountrakis L,Lorenz E,Malaspinas O,Alowayyed S,Chopard B and Hoekstra A G 2015 J.Comput.Sci.9 45
[8]Meng X and Guo Z 2016 Int.J.Heat Mass Tran.100 767
[9]Liu Q and He Y L 2017 Physica A 465 742
[10]Zhang Q,Sun D and Zhu M 2017 Chin.Phys.B 26 084701
[11]Wu X,Liu H and Chen F 2017 Acta Phys.Sin.-Ch.Ed.66 224702
[12]Sun D,Jiang D,Xiang N,Chen K and Ni Z 2013 Chin.Phys.Lett.30074702
[13]Frisch U,Hasslacher B and Pomeau Y 1986 Phys.Rev.Lett.56 1505
[14]Bhatnagar P L,Gross E P and Krook M 1954 Phys.Rev.94 511
[15]Dellar P J 2013 Comput.Math.Appl.65 129
[16]He X,Chen S and Doolen G D 1998 J.Comput.Phys.146 282
[17]Hwang Y H 2016 J.Comput.Phys.322 52
[18]Yong W A,Zhao W and Luo L S 2016 Phys.Rev.E 93 033310
[19]Bosch F and Karlin I V 2013 Phys.Rev.Lett.111 090601
[20]He X Y and Luo L S 1997 Phys.Rev.E 56 6811
[21]Shan X 2010 Phys.Rev.E 81 36702
[22]Sterling J D and Chen S 1996 J.Comput.Phys.123 196
[23]Chapman S and Cowling T G 1970 The Mathematical Theory of NonUniform Gases:An Account of the Kinetic Theory of Viscosity,Thermal Conduction and Diffusion in Gases(Cambridge:Cambridge University Press)
[24]Latt J,Chopard B,Malaspinas O,Deville M and Michler A 2008 Phys.Rev.E 77 56703
[25]Zou Q and He X 1997 Phys.Fluids 9 1591
[26]Inamuro T,Yoshino M and Ogino F 1995 Phys.Fluids 7 2928
[27]Ghia U,Ghia K N and Shin C T 1982 J.Comput.Phys.48 387