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The phase field model has become increasingly popular due to its underlying physics for describing two-phase interface dynamics. In this case, several lattice Boltzmann multiphase models have been constructed from the perspective of the phase field theory. All these models are composed of two distribution functions: one is used to solve the interface tracking equation and the other is adopted to solve the Navier-Stokes equations. It has been reported that to match the target equation, an additional interfacial force should be included in these models, but the scale of this force is found to be contradictory with the theoretical analysis. To solve this problem, in this paper an improved lattice Boltzmann model based on the Cahn-Hilliard phase-field theory is proposed for simulating two-phase flows. By introducing a novel and simple force distribution function, the improved model solves the problem that the scale of an additional interfacial force is not consistent with the theoretical one. The Chapman-Enskog analysis shows that the present model can accurately recover the Cahn-Hilliard equation for interface capturing and the incompressible Navier-Stokes equations, and the calculation of macroscopic velocity is also more efficient. A series of classic two-phase flow examples, including static drop test, droplets emerge, spinodal decomposition and Rayleigh-Taylor instability is simulated numerically. It is found that the numerical solutions agree well with the analytical solutions or the existing results, which verifies the accuracy and feasibility of the proposed model. In addition, the Rayleigh-Taylor instability with the imposed random perturbation is also simulated, where the influence of the Reynolds number on the evolution of the phase interface is analyzed. It is found that for the case of the high Reynolds number, a row of “mushroom” shape appears at the fluid interface in the early stages of evolution. At the later stages of evolution, the fluid interface presents a very complex chaotic topology. Unlike the case of the high Reynolds number, the fluid interface becomes relatively smooth at low Reynolds numbers, and no chaotic topology is observed at any of the later stages of evolution.
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Keywords:
- lattice Boltzmann method /
- interfacial force /
- two-phase flow /
- Rayleigh-Taylor instability
[1] Guo Z L, Zheng C G 2009 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) [郭照立, 郑楚光 2009 格子Boltzmann方法的原理及应用 (北京: 科学出版社)]
[2] Chen S, Doolen G D 1998 Annu. Rev. Fluid. Mech. 30 329
[3] He X, Chen S, Zhang R 1999 J. Comput. Phys. 152 642
[4] Zheng H W, Shu C, Chew Y T 2006 J. Comput. Phys. 218 353
[5] Lee T, Liu L 2010 J. Comput. Phys. 229 8045
[6] Zu Y Q, He S 2013 Phys. Rev. E 87 043301
[7] Liang H, Shi B C, Guo Z L, Chai Z H 2014 Phys. Rev. E 89 053320
[8] Liang H, Chai Z H, Shi B C, Guo Z L, Zhang T 2014 Phys. Rev. E 90 063311
[9] Liang H, Xu J R, Chen J X, Wang H L, Chai Z H, Shi B C, Chai Z H 2018 Phys. Rev. E 97 033309
[10] Liang H, Shi B C, Chai Z H 2016 Phys. Rev. E 93 013308
[11] Liang H, Li Q X, Shi B C, Chai Z H 2016 Phys. Rev. E 93 033113
[12] Liang H, Chai Z H, Shi B C 2016 Acta Phys. Sin. 65 204701 (in Chinese) [梁宏, 柴振华, 施保昌 2016 65 204701]
[13] Huang H, Hong N, Liang H, Shi B C, Chai Z H 2016 Acta Phys. Sin. 65 084702 (in Chinese) [黄虎, 洪宁, 梁宏, 施保昌, 柴振华 2016 65 084702]
[14] Lou Q, Guo Z L, Shi B C 2012 Europhys. Lett. 99 64005
[15] Li Q, Luo K H, Gao Y J, He Y L 2012 Phys. Rev. E 85 026704
[16] Wang Y, Shu C, Shao J Y, Wu J, Niu X D 2015 J. Comput. Phys. 290 336
[17] Yang K, Guo Z L 2016 Phys. Rev. E 723 043303
[18] Rayleigh L 1883 Proc. London Math. Soc. 14 1
[19] Taylor G 1950 Proc. Roy. Soc. London 201 192
[20] Zhou Y 2017 Phys. Rep. 91 013309
[21] Liang H, Li Y, Chen J X, Xu J R 2018 Int. J. Heat Mass. Tran. 130 1189
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[1] Guo Z L, Zheng C G 2009 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) [郭照立, 郑楚光 2009 格子Boltzmann方法的原理及应用 (北京: 科学出版社)]
[2] Chen S, Doolen G D 1998 Annu. Rev. Fluid. Mech. 30 329
[3] He X, Chen S, Zhang R 1999 J. Comput. Phys. 152 642
[4] Zheng H W, Shu C, Chew Y T 2006 J. Comput. Phys. 218 353
[5] Lee T, Liu L 2010 J. Comput. Phys. 229 8045
[6] Zu Y Q, He S 2013 Phys. Rev. E 87 043301
[7] Liang H, Shi B C, Guo Z L, Chai Z H 2014 Phys. Rev. E 89 053320
[8] Liang H, Chai Z H, Shi B C, Guo Z L, Zhang T 2014 Phys. Rev. E 90 063311
[9] Liang H, Xu J R, Chen J X, Wang H L, Chai Z H, Shi B C, Chai Z H 2018 Phys. Rev. E 97 033309
[10] Liang H, Shi B C, Chai Z H 2016 Phys. Rev. E 93 013308
[11] Liang H, Li Q X, Shi B C, Chai Z H 2016 Phys. Rev. E 93 033113
[12] Liang H, Chai Z H, Shi B C 2016 Acta Phys. Sin. 65 204701 (in Chinese) [梁宏, 柴振华, 施保昌 2016 65 204701]
[13] Huang H, Hong N, Liang H, Shi B C, Chai Z H 2016 Acta Phys. Sin. 65 084702 (in Chinese) [黄虎, 洪宁, 梁宏, 施保昌, 柴振华 2016 65 084702]
[14] Lou Q, Guo Z L, Shi B C 2012 Europhys. Lett. 99 64005
[15] Li Q, Luo K H, Gao Y J, He Y L 2012 Phys. Rev. E 85 026704
[16] Wang Y, Shu C, Shao J Y, Wu J, Niu X D 2015 J. Comput. Phys. 290 336
[17] Yang K, Guo Z L 2016 Phys. Rev. E 723 043303
[18] Rayleigh L 1883 Proc. London Math. Soc. 14 1
[19] Taylor G 1950 Proc. Roy. Soc. London 201 192
[20] Zhou Y 2017 Phys. Rep. 91 013309
[21] Liang H, Li Y, Chen J X, Xu J R 2018 Int. J. Heat Mass. Tran. 130 1189
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