Numerical simulation of three-phase Rayleigh-Taylor instability based on phase field model using lattice Boltzmann method

YANG Xuguang; WANG Xin; YUAN Xiaolei

doi:10.7498/aps.75.20251095

Abstract

This paper develops a regularized lattice Boltzmann method for efficiently simulating the flow of N-phase immiscible incompressible fluids based on the phase field model that satisfies conservation and compatibility. By designing auxiliary moments, this method can accurately recover the second-order Allen-Cahn equation and the modified momentum equation. The correctness and effectiveness of the developed N-phase regularized lattice Boltzmann method are validated through numerical simulations of three-phase liquid lens spreading and Kelvin-Helmholtz instability phenomena. Finally, numerical simulations and analyses of three-phase Rayleigh-Taylor instabilities (RTI) are conducted, focusing on the evolution of the phase interface within the Reynolds number range of $ 500 \leqslant Re \leqslant 20000 $ (particularly under high Reynolds number condition of $ Re = 20000 $). Quantitative analyses are performed on the amplitude variations of bubbles and spikes at the two interfaces, as well as the changes in dimensionless velocity. We find that as the Reynolds number increases, the phase interface curls up at multiple locations due to Kelvin-Helmholtz instability, making the fluid more prone to dispersion and fragmentation. This study also simulates the evolutionary processes of RTI under different interface perturbations. These results demonstrate that RTI first develops at the perturbed interface, with its subsequent evolution inducing instability at a secondary interface.

Keywords:

Authors and contacts

1.
School of Mathematics and Statistics, Hunan First Normal University, Changsha 410205, China
2.
College of Mathematics and Information Science, Hebei University, Baoding 071002, China
3.
Hebei Key Laboratory of Machine Learning and Computational Intelligence, Hebei University, Baoding 071002, China

Corresponding author: YUAN Xiaolei, yuanxl@hbu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12202130, 12372286), the Science and Technology Innovation Program of Hunan Province, China (Grant No. 2024RC3228), the Excellent Youth Research Innovation Team of Hebei University, China (Grant No. QNTD202414), the Excellent Youth Foundation of Changsha Scientific Committee, China (Grant No. kq2306023), and the Aeronautical Science Foundation of China (Grant No. 20220054069001).

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References

[1]	Barber J L, Kadau K, Germann T C, Alder B J 2008 Eur. Phys. J. B 64 271 Google Scholar
[2]	Celani A, Mazzino A, Bjorkholm J E, Vozella L 2006 Phys. Rev. Lett. 96 134504 Google Scholar
[3]	Moin P 1991 Comput. Methods Appl. Mech. Eng. 87 329 Google Scholar
[4]	Guo Z L, Zheng C G 2009 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) (in Chinese) pp156–200 [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) 第156—200页] Guo Z L, Zheng C G 2009 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) (in Chinese) pp156–200
[5]	He Y L, Wang Y, Li Q 2009 Lattice Boltzmann Method: Theory and Applications (Beijing: Science Press) (in Chinese) pp31–55 [何雅玲, 王勇, 李庆 2009 格子 Boltzmann 方法的理论及应用 (北京: 科学出版社) 第31—55页] He Y L, Wang Y, Li Q 2009 Lattice Boltzmann Method: Theory and Applications (Beijing: Science Press) (in Chinese) pp31–55
[6]	He X Y, Chen S Y, Zhang R Y 1999 J. Comput. Phys. 152 642 Google Scholar
[7]	Liang H, Shi B C, Guo Z L, Chai Z H 2014 Phys. Rev. E 89 053320 Google Scholar
[8]	Liang H, Li Q X, Shi B C, Chai Z H 2016 Phys. Rev. E 93 033113
[9]	Liang H, Xia Z H, Huang H W 2021 Phys. Fluids 33 082103 Google Scholar
[10]	李洋, 苏婷, 梁宏, 徐江荣 2018 物理学报 67 224701 Li Y, Su T, Liang H, Xu J R 2018 Acta Phys. Sin. 67 224701
[11]	马聪, 刘斌, 梁宏 2022 物理学报 71 044701 Google Scholar Ma C, Liu B, Liang H 2022 Acta Phys. Sin. 71 044701 Google Scholar
[12]	李德梅, 赖惠林, 许爱国, 张广财, 林传栋, 甘延标 2018 物理学报 67 080501 Google Scholar Li D M, Lai H L, Xu A G, Zhang G C, Lin C D, Gan Y B 2018 Acta Phys. Sin. 67 080501 Google Scholar
[13]	Zhang R Y, He X Y, Chen S Y 2000 Comput. Phys. Commun. 129 121 Google Scholar
[14]	胡晓亮, 梁宏, 王会利 2020 物理学报 69 044701 Google Scholar Hu X L, Liang H, Wang H L 2020 Acta Phys. Sin. 69 044701 Google Scholar
[15]	Zhan C J, Liu X, Chai Z H, Shi B C 2024 Commun. Comput. Phys. 36 850 Google Scholar
[16]	Kalantarpour R, Ebadi A, Hosseinalipour S M, Liang H 2020 Comput. Fluids 204 104480
[17]	Boyer F, Lapuerta C 2006 ESAIM: Math. Model. Numer. Anal. 40 653 Google Scholar
[18]	Boyer F, Lapuerta C, Minjeaud S, et al. 2010 Transp. Porous Media 82 463 Google Scholar
[19]	Dong S 2018 J. Comput. Phys. 361 1 Google Scholar
[20]	Zheng L, Zheng S, Zhai Q L 2020 Phys. Rev. E 101 043302 Google Scholar
[21]	Mirjalili S, Mani A 2024 J. Comput. Phys. 498 112657 Google Scholar
[22]	Xia Q, Yang J X, Li Y B 2023 Phys. Fluids 35 012120 Google Scholar
[23]	Latt J, Chopard B 2006 Math. Comput. Simulat. 72 165 Google Scholar
[24]	Montessori A, Falcucci G, Prestininzi P, et al. 2014 Phys. Rev. E 89 053317 Google Scholar
[25]	Liu X, Chen Y, Chai Z H, Shi B C 2024 Phys. Rev. E 109 025301 Google Scholar
[26]	Huang Y H, Chen X M, Chai Z H, Shi B C 2025 Adv. Appl. Math. Mech. 17 1370 Google Scholar
[27]	黄皓伟, 梁宏, 徐江荣 2021 物理学报 70 114701 Google Scholar Huang H W, Liang H, Xu J R 2021 Acta Phys. Sin. 70 114701 Google Scholar
[28]	李春熠, 郭照立 2025 物理学报 74 064702 Google Scholar Li C Y, Guo Z L 2025 Acta Phys. Sin. 74 064702 Google Scholar
[29]	Huang Z Y, Lin G, Ardekani A M 2021 J. Comput. Phys. 434 110229 Google Scholar
[30]	Mirjalili S, Mani A 2021 J. Comput. Phys. 426 109918 Google Scholar
[31]	Qian Y H, d'Humières D, Lallemand P 1992 Europhys. Lett. 17 479 Google Scholar
[32]	Yang X F, Zhao J, Wang Q, Shen J 2017 Math. Models Methods Appl. Sci. 27 1993 Google Scholar
[33]	Hu Y, Li D C, He Q 2020 Int. J. Multiph. Flow 132 103432 Google Scholar
[34]	Yuan X L, Shi B C, Zhan C J, Chai Z H 2022 Phys. Fluids 34 023311 Google Scholar
[35]	Wu J W, Yang J X, Tan Z J 2022 Comput. Methods Appl. Mech. Eng. 398 115291 Google Scholar
[36]	章诗婷, 肖鸿威, 周锦翔, 牛小东 2022 空气动力学学报 40 75 Zhang S T, Xiao H W, Zhou J X, Niu X D 2022 Acta Aerodyn. Sin. 40 75
[37]	Fakhari A, Lee T 2013 Phys. Rev. E 87 023304 Google Scholar
[38]	Fakhari A, Geier M, Lee T 2016 J. Comput. Phys. 315 434 Google Scholar
[39]	Zhou X, Dong B, Li W Z 2020 Int. J. Aerosp. Eng. 2020 8885226
[40]	Ramaprabhu P, Dimonte G, Young Y N, Calder A C, Fryxell B 2006 Phys. Rev. E 74 066308

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图 1 三相液体透镜示意图

Figure 1. Schematic of the spreading of a liquid lens.

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图 2 不同界面张力比下的三相液体透镜平衡态　(a) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:1:1 $; (b) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:\sqrt{2}:1 $; (c) $ \sigma_{12}:\sigma_{13}: $$ \sigma_{23}=1:\sqrt{3}:1 $

Figure 2. The equilibrium shapes of liquid by ternary fluids: (a) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:1:1 $; (b) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:\sqrt{2}:1 $; (c) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:\sqrt{3}:1 $.

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图 3 Kelvin-Helmholtz不稳定性示意图

Figure 3. Schematic of the spreading of Kelvin-Helmholtz instability.

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图 4 不同时刻下的密度分布　(a) $ t=2000 $; (b) $ t=3000 $; (c) $ t=5000 $; (d) $ t=9000 $

Figure 4. Density distribution at different times: (a) $ t=2000 $; (b) $ t=3000 $; (c) $ t=5000 $; (d) $ t=9000 $.

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图 5 不同时刻下的涡场图　(a) $ t=2000 $; (b) $ t=3000 $; (c) $ t=5000 $; (d) $ t=9000 $

Figure 5. Vorticity field at different times: (a) $ t=2000 $; (b) $ t=3000 $; (c) $ t=5000 $; (d) $ t=9000 $.

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图 6 Rayleigh-Taylor不稳定性示意图

Figure 6. Schematic of the Rayleigh-Taylor instability.

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图 7 两相情况相界面扰动演化过程　(a) $ Re = 30 $; (b) $ Re = 150 $; (c) $ Re = 3000 $; (d) $ Re = 30 $^[7]; (e) $ Re = 150 $^[7]; (f)$ Re = 3000 $^[7]

Figure 7. Phase interface disturbance evolution process in two-phase situation: (a) $ Re=30 $; (b)$ Re=150 $; (c) $ Re=3000 $; (d) $ Re= $$ 30 $^[7]; (e) $ Re=150 $^[7]; (f) $ Re=3000 $^[7].

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图 8 两相情况尖钉与气泡振幅随时间的变化　(a) 尖钉振幅; (b) 气泡振幅

Figure 8. Variation of spike and bubble amplitudes with time in two-phase situation: (a)$ H_{{\rm{s}}} $; (b)$ H_{{\rm{b}}} $.

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图 9 较低雷诺数对R-T不稳定性中相界面演化过程的影响　(a) $ Re=500 $; (b) $ Re=1000 $; (c) $ Re=2000 $

Figure 9. The effect of lower Reynolds numbers on the evolution of the phase interface in RTI: (a) $ Re=500 $; (b) $ Re=1000 $; (c) $ Re=2000 $.

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图 10 较高雷诺数对R-T不稳定性中相界面演化过程的影响　(a) $ Re=5000 $; (b)$ Re=10000 $; (c) $ Re=20000 $

Figure 10. The effect of higher Reynolds numbers on the evolution of the phase interface in RTI: (a) $ Re=5000 $; (b)$ Re=10000 $; (c) $ Re=20000 $.

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图 11 雷诺数对气泡与尖钉振幅随时间演化的影响　(a) 界面一尖钉振幅; (b) 界面二尖钉振幅; (c) 界面一气泡振幅; (d) 界面二气泡振幅

Figure 11. Effect of Reynolds number on the temporal evolution of bubble and spike amplitudes: (a)$ H_{{\rm{s1}}} $; (b)$ H_{{\rm{s2}}} $; (c)$ H_{{\rm{b1}}} $; (d)$ H_{{\rm{b2}}} $.

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图 12 雷诺数对无量纲化的气泡和尖钉演化速度的影响　(a) 界面一尖钉; (b) 界面二尖钉; (c) 界面一气泡; (d)界面二气泡

Figure 12. Effect of Reynolds number on the normalized growth rate of bubbles and spikes: (a)$ Fr_{{\rm{s1}}} $; (b)$ Fr_{{\rm{s2}}} $; (c)$ Fr_{{\rm{b1}}} $; (d)$ Fr_{{\rm{b2}}} $.

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图 13 上层相界面扰动相界面演化过程　(a) $ Re=500 $; (b) $ Re=1000 $; (c) $ Re=5000 $; (d) $ Re=20000 $

Figure 13. Upper interfacial perturbation and evolution: (a) $ Re=500 $; (b) $ Re=1000 $; (c) $ Re=5000 $; (d) $ Re=20000 $.

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图 14 上层相界面扰动时气泡与尖钉振幅随时间的变化　(a) 界面一尖钉振幅; (b) 界面二尖钉振幅; (c) 界面一气泡振幅; (d) 界面二气泡振幅

Figure 14. Bubble and spike amplitude evolution during upper interface perturbation: (a) $ H_{{\rm{s1}}} $; (b) $ H_{{\rm{s2}}} $; (c) $ H_{{\rm{b1}}} $; (d) $ H_{{\rm{b2}}} $.

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图 15 下层相界面扰动相界面演化过程　(a) $ Re=500 $; (b) $ Re=1000 $; (c) $ Re=5000 $; (d) $ Re=20000 $

Figure 15. Lower interfacial perturbation and evolution: (a) $ Re=500 $; (b)$ Re=1000 $; (c) $ Re=5000 $; (d) $ Re=20000 $.

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图 16 下层相界面扰动时气泡与尖钉振幅随时间的变化　(a) 界面一尖钉振幅; (b) 界面二尖钉振幅; (c) 界面一气泡振幅; (d) 界面二气泡振幅

Figure 16. Bubble and spike amplitude evolution during lower interface perturbation: (a) $ H_{{\rm{s1}}} $; (b) $ H_{{\rm{s2}}} $; (c) $ H_{{\rm{b1}}} $; (d) $ H_{{\rm{b2}}} $.

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表 1 不同表面张力比下液体透镜的长度d和高度$ h_1 $, $ h_2 $

Table 1. The length d, and height $ h_1 $, $ h_2 $ at equilibrium state with different surface tension ratios.

$ \sigma_{12}:\sigma_{13}:\sigma_{23} $	解析解			数值解			相对误差
$ \sigma_{12}:\sigma_{13}:\sigma_{23} $	d	$ h_1 $	$ h_2 $	d	$ h_1 $	$ h_2 $	d	$ h_1 $	$ h_2 $
$ 1:1:1 $	83.10	23.99	23.99	84.26	24.45	24.42	1.40%	1.92%	1.80%
$ 1:\sqrt{2}:1 $	72.67	36.34	15.05	74.02	37.03	15.34	1.86%	1.90%	1.93%
$ 1:\sqrt{3}:1 $	55.05	47.67	7.38	55.83	48.42	7.48	1.42%	1.57%	1.36%

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[1]	Barber J L, Kadau K, Germann T C, Alder B J 2008 Eur. Phys. J. B 64 271 Google Scholar
[2]	Celani A, Mazzino A, Bjorkholm J E, Vozella L 2006 Phys. Rev. Lett. 96 134504 Google Scholar
[3]	Moin P 1991 Comput. Methods Appl. Mech. Eng. 87 329 Google Scholar
[4]	Guo Z L, Zheng C G 2009 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) (in Chinese) pp156–200 [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) 第156—200页] Guo Z L, Zheng C G 2009 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) (in Chinese) pp156–200
[5]	He Y L, Wang Y, Li Q 2009 Lattice Boltzmann Method: Theory and Applications (Beijing: Science Press) (in Chinese) pp31–55 [何雅玲, 王勇, 李庆 2009 格子 Boltzmann 方法的理论及应用 (北京: 科学出版社) 第31—55页] He Y L, Wang Y, Li Q 2009 Lattice Boltzmann Method: Theory and Applications (Beijing: Science Press) (in Chinese) pp31–55
[6]	He X Y, Chen S Y, Zhang R Y 1999 J. Comput. Phys. 152 642 Google Scholar
[7]	Liang H, Shi B C, Guo Z L, Chai Z H 2014 Phys. Rev. E 89 053320 Google Scholar
[8]	Liang H, Li Q X, Shi B C, Chai Z H 2016 Phys. Rev. E 93 033113
[9]	Liang H, Xia Z H, Huang H W 2021 Phys. Fluids 33 082103 Google Scholar
[10]	李洋, 苏婷, 梁宏, 徐江荣 2018 物理学报 67 224701 Li Y, Su T, Liang H, Xu J R 2018 Acta Phys. Sin. 67 224701
[11]	马聪, 刘斌, 梁宏 2022 物理学报 71 044701 Google Scholar Ma C, Liu B, Liang H 2022 Acta Phys. Sin. 71 044701 Google Scholar
[12]	李德梅, 赖惠林, 许爱国, 张广财, 林传栋, 甘延标 2018 物理学报 67 080501 Google Scholar Li D M, Lai H L, Xu A G, Zhang G C, Lin C D, Gan Y B 2018 Acta Phys. Sin. 67 080501 Google Scholar
[13]	Zhang R Y, He X Y, Chen S Y 2000 Comput. Phys. Commun. 129 121 Google Scholar
[14]	胡晓亮, 梁宏, 王会利 2020 物理学报 69 044701 Google Scholar Hu X L, Liang H, Wang H L 2020 Acta Phys. Sin. 69 044701 Google Scholar
[15]	Zhan C J, Liu X, Chai Z H, Shi B C 2024 Commun. Comput. Phys. 36 850 Google Scholar
[16]	Kalantarpour R, Ebadi A, Hosseinalipour S M, Liang H 2020 Comput. Fluids 204 104480
[17]	Boyer F, Lapuerta C 2006 ESAIM: Math. Model. Numer. Anal. 40 653 Google Scholar
[18]	Boyer F, Lapuerta C, Minjeaud S, et al. 2010 Transp. Porous Media 82 463 Google Scholar
[19]	Dong S 2018 J. Comput. Phys. 361 1 Google Scholar
[20]	Zheng L, Zheng S, Zhai Q L 2020 Phys. Rev. E 101 043302 Google Scholar
[21]	Mirjalili S, Mani A 2024 J. Comput. Phys. 498 112657 Google Scholar
[22]	Xia Q, Yang J X, Li Y B 2023 Phys. Fluids 35 012120 Google Scholar
[23]	Latt J, Chopard B 2006 Math. Comput. Simulat. 72 165 Google Scholar
[24]	Montessori A, Falcucci G, Prestininzi P, et al. 2014 Phys. Rev. E 89 053317 Google Scholar
[25]	Liu X, Chen Y, Chai Z H, Shi B C 2024 Phys. Rev. E 109 025301 Google Scholar
[26]	Huang Y H, Chen X M, Chai Z H, Shi B C 2025 Adv. Appl. Math. Mech. 17 1370 Google Scholar
[27]	黄皓伟, 梁宏, 徐江荣 2021 物理学报 70 114701 Google Scholar Huang H W, Liang H, Xu J R 2021 Acta Phys. Sin. 70 114701 Google Scholar
[28]	李春熠, 郭照立 2025 物理学报 74 064702 Google Scholar Li C Y, Guo Z L 2025 Acta Phys. Sin. 74 064702 Google Scholar
[29]	Huang Z Y, Lin G, Ardekani A M 2021 J. Comput. Phys. 434 110229 Google Scholar
[30]	Mirjalili S, Mani A 2021 J. Comput. Phys. 426 109918 Google Scholar
[31]	Qian Y H, d'Humières D, Lallemand P 1992 Europhys. Lett. 17 479 Google Scholar
[32]	Yang X F, Zhao J, Wang Q, Shen J 2017 Math. Models Methods Appl. Sci. 27 1993 Google Scholar
[33]	Hu Y, Li D C, He Q 2020 Int. J. Multiph. Flow 132 103432 Google Scholar
[34]	Yuan X L, Shi B C, Zhan C J, Chai Z H 2022 Phys. Fluids 34 023311 Google Scholar
[35]	Wu J W, Yang J X, Tan Z J 2022 Comput. Methods Appl. Mech. Eng. 398 115291 Google Scholar
[36]	章诗婷, 肖鸿威, 周锦翔, 牛小东 2022 空气动力学学报 40 75 Zhang S T, Xiao H W, Zhou J X, Niu X D 2022 Acta Aerodyn. Sin. 40 75
[37]	Fakhari A, Lee T 2013 Phys. Rev. E 87 023304 Google Scholar
[38]	Fakhari A, Geier M, Lee T 2016 J. Comput. Phys. 315 434 Google Scholar
[39]	Zhou X, Dong B, Li W Z 2020 Int. J. Aerosp. Eng. 2020 8885226
[40]	Ramaprabhu P, Dimonte G, Young Y N, Calder A C, Fryxell B 2006 Phys. Rev. E 74 066308

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