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In this paper, we numerically investigate the late-time growth of high-Reynolds-number single-mode Rayleigh-Taylor instability in a long pipe by using an advanced phase-field lattice Boltzmann multiphase method. We mainly analyze the influence of surface tension on interfacial dynamic behavior and the development of the bubble front and spike front. The numerical experiments indicate that increasing surface tension can significantly reduce the complexity of formed interfacial structure and also prevents the breakup of phase interfaces. The interface patterns in the instability process cannot always preserve the symmetric property under the extremely small surface tension, but they do maintain the symmetries with respect to the middle line as the surface tension is increased. We also report that the bubble amplitude first increases then decreases with the surface tension. There are no obvious differences between the curves of spike amplitudes for low surface tensions. However, when the surface tension increases to a critical value, it can slow down the spike growth significantly. When the surface tension is lower than the critical value, the development of the high-Reynolds-number Rayleigh-Taylor instability can be divided into four different stages, i.e. the linear growth, saturated velocity growth, reacceleration, and chaotic mixing. The bubble and spike velocities at the second stage show good agreement with those from the modified potential flow theory that takes the surface tension effect into account. After that, the bubble front and spike front are accelerated due to the formation of Kelvin-Helmholtz vortices in the interfacial region. At the late time, the bubble velocity and spike velocity become unstable and slightly fluctuate over time. To determine the nature of the late-time growth, we also measure the bubble and spike normalized accelerations at various interfacial tensions and Atwood numbers. It is found that both the spike and bubble growth rates first increase then decrease with the surface tension in general. Finally, we deduce a theoretical formula for the critical surface tension, below which the Rayleigh-Taylor instability takes place and above which tension it does not occur. It is shown that the critical surface tension increases with the Atwood number and also the numerical predictions by the lattice Boltzmann method are also in accord well with the theoretical results.
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Keywords:
- Rayleigh-Taylor instability /
- lattice Boltzmann method /
- surface tension /
- late-time growth /
- critical value
[1] Zhou Y 2017 Phys. Rep. 720–722 1Google Scholar
[2] Rayleigh L 1883 Proc. London Math. Soc. 14 170
[3] Taylor G I 1950 Proc. R. Soc. London, Ser. A 201 192Google Scholar
[4] Lewis D J 1950 Proc. R. Soc. London, Ser. A 202 81Google Scholar
[5] Sharp D H 1984 Physica D 12 3Google Scholar
[6] Waddell J T, Niederhaus C E, Jacobs J W 2001 Phys. Fluids 13 1263Google Scholar
[7] Glimm J, Li X L, Lin A D 2002 Acta Math. Appl. Sin. 18 1
[8] Wilkinson J P, Jacobs J W 2007 Phys. Fluids 19 124102Google Scholar
[9] Goncharov V N 2002 Phys. Rev. Lett. 88 134502Google Scholar
[10] Ramaprabhu P, Dimonte G, Woodward P, Fryer C, Rockefeller G, Muthuraman K, Lin P H, Jayaral J 2012 Phys. Fluids 24 074107Google Scholar
[11] Wei T, Livescu D 2012 Phys. Rev. E 86 046405Google Scholar
[12] Lai H L, Xu A G, Zhang G, Gan Y B, Jun Y, Succi S 2016 Phys. Rev. E 94 023106Google Scholar
[13] Liang H, Shi B C, Guo Z L, Chai Z H 2014 Phys. Rev. E 89 053320Google Scholar
[14] Liang H, Li Q X, Shi B C, Chai Z H 2016 Phys. Rev. E 93 033113Google Scholar
[15] 胡晓亮, 梁宏, 王会利 2020 物理学报 69 044701Google Scholar
Hu X L, Liang H, Wang H L 2020 Acta Phys. Sin. 69 044701Google Scholar
[16] Sohn S I, Baek S 2017 Phys. Lett. A 381 3812Google Scholar
[17] Cherfils C, Mikaelian K O 1996 Phys. Fluids 8 522Google Scholar
[18] Dimonte G, Schneider M 2000 Phys. Fluids 12 304Google Scholar
[19] Garnier J, Cherfils-Clérouin C, Holstein P A 2003 Phys. Rev. E 68 036401Google Scholar
[20] Chertkov M, Kolokolov I, Lebedev V 2005 Phys. Rev. E 71 055301(RGoogle Scholar
[21] Daly B J 1969 Phys. Fluids 12 1340Google Scholar
[22] Zhang R Y, He X Y, Chen S Y 2000 Comput. Phys. Commun. 129 121Google Scholar
[23] Young Y N, Ham F E 2006 J. Turbul. 7 71Google Scholar
[24] Sohn S I 2009 Phys. Rev. E 80 055302(RGoogle Scholar
[25] 夏同军, 董永强, 曹义刚 2013 物理学报 62 214702Google Scholar
Xia T J, Dong Y Q, Cao Y G 2013 Acta Phys. Sin. 62 214702Google Scholar
[26] Zufiria J A 1988 Phys. Fluids 31 440Google Scholar
[27] Li M J, Zhu Q F, Li G B 2016 Appl. Math. Mech. 37 1607Google Scholar
[28] Guo H Y, Wang L F, Ye W H, Wu J F, Zhang W Y 2017 Chin. Phys. Lett. 34 045201Google Scholar
[29] Guo Z L, Shu C 2013 Lattice Boltzmann Method and its Applications in Engineering (Singapore: World Scientific), pp239–284
[30] Wang H L, Yuan X L, Liang H, Chai Z H, Shi B C 2019 Capillarity 2 33Google Scholar
[31] Liang H, Xu J R, Chen J X, Wang H L, Chai Z H, Shi B C 2018 Phys. Rev. E 97 033309Google Scholar
[32] Liang H, Hu X L, Huang X F, Xu J R 2019 Phys. Fluids 31 112104Google Scholar
[33] Jacqmin D 1999 J. Comput. Phys. 155 96Google Scholar
[34] Wei Y K, Wang Z D, Dou H S, Qian Y H 2017 Comput. Fluids 156 97Google Scholar
[35] Wei Y K, Yang H, Lin Z, Wang Z D, Qian Y H 2018 Appl. Math. Comput. 339 556
[36] Lallemand P, Luo L S 2000 Phys. Rev. E 61 6546Google Scholar
[37] Chai Z H, Shi B C, Lu J H, Guo Z L 2010 Comput. Fluids 39 2069Google Scholar
[38] Liang H, Shi B C, Chai Z H 2016 Phys. Rev. E 93 013308Google Scholar
[39] He X, Zou Q, Luo L S, Dembo M 1997 J. Stat. Phys. 87 115Google Scholar
[40] Bian X, Aluie H, Zhao D X, Zhang H S, Livescu D 2020 Physica D 403 132250Google Scholar
[41] Cabot W H, Cook A W 2006 Nat. Phys. 2 562Google Scholar
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图 1
$ Re = 10^4 $ ,$ A_t = 0.7 $ 时, 表面张力对高雷诺数非混相RT不稳定相界面演化的影响 (a)$ \sigma = 5\times10^{-6} $ ; (b)$\sigma = $ $ 5\times10^{-4}$ ; (c)$ \sigma = 5\times10^{-3} $ ; (d)$ \sigma = 1\times10^{-2} $ Figure 1. Effect of the surface tension on the evolution of phase interface in the immiscible RT instability with
$ Re = 10^4 $ ,$ A_t = 0.7 $ : (a)$ \sigma = 5\times10^{-6} $ ; (b)$ \sigma = 5\times10^{-4} $ ; (c)$ \sigma = 5\times10^{-3} $ ; (d)$ \sigma = 1\times10^{-2} $ .图 3 表面张力对随时间演化的气泡和尖钉增长速度的影响. 蓝色和虚线分别表示势能模型预测气泡与尖钉速度在
$ \sigma = 5\times10^{-3} $ 和$ \sigma = 1\times10^{-2} $ 时的解析解Figure 3. Influence of surface tension on the time evolutions of bubble and spike growth velocities. The blue and yellow dotted lines represent the analytical solutions of the bubble and spike velocities from potential flow model at
$ \sigma = 5\times10^{-3} $ and$ \sigma = 1\times10^{-2} $ . -
[1] Zhou Y 2017 Phys. Rep. 720–722 1Google Scholar
[2] Rayleigh L 1883 Proc. London Math. Soc. 14 170
[3] Taylor G I 1950 Proc. R. Soc. London, Ser. A 201 192Google Scholar
[4] Lewis D J 1950 Proc. R. Soc. London, Ser. A 202 81Google Scholar
[5] Sharp D H 1984 Physica D 12 3Google Scholar
[6] Waddell J T, Niederhaus C E, Jacobs J W 2001 Phys. Fluids 13 1263Google Scholar
[7] Glimm J, Li X L, Lin A D 2002 Acta Math. Appl. Sin. 18 1
[8] Wilkinson J P, Jacobs J W 2007 Phys. Fluids 19 124102Google Scholar
[9] Goncharov V N 2002 Phys. Rev. Lett. 88 134502Google Scholar
[10] Ramaprabhu P, Dimonte G, Woodward P, Fryer C, Rockefeller G, Muthuraman K, Lin P H, Jayaral J 2012 Phys. Fluids 24 074107Google Scholar
[11] Wei T, Livescu D 2012 Phys. Rev. E 86 046405Google Scholar
[12] Lai H L, Xu A G, Zhang G, Gan Y B, Jun Y, Succi S 2016 Phys. Rev. E 94 023106Google Scholar
[13] Liang H, Shi B C, Guo Z L, Chai Z H 2014 Phys. Rev. E 89 053320Google Scholar
[14] Liang H, Li Q X, Shi B C, Chai Z H 2016 Phys. Rev. E 93 033113Google Scholar
[15] 胡晓亮, 梁宏, 王会利 2020 物理学报 69 044701Google Scholar
Hu X L, Liang H, Wang H L 2020 Acta Phys. Sin. 69 044701Google Scholar
[16] Sohn S I, Baek S 2017 Phys. Lett. A 381 3812Google Scholar
[17] Cherfils C, Mikaelian K O 1996 Phys. Fluids 8 522Google Scholar
[18] Dimonte G, Schneider M 2000 Phys. Fluids 12 304Google Scholar
[19] Garnier J, Cherfils-Clérouin C, Holstein P A 2003 Phys. Rev. E 68 036401Google Scholar
[20] Chertkov M, Kolokolov I, Lebedev V 2005 Phys. Rev. E 71 055301(RGoogle Scholar
[21] Daly B J 1969 Phys. Fluids 12 1340Google Scholar
[22] Zhang R Y, He X Y, Chen S Y 2000 Comput. Phys. Commun. 129 121Google Scholar
[23] Young Y N, Ham F E 2006 J. Turbul. 7 71Google Scholar
[24] Sohn S I 2009 Phys. Rev. E 80 055302(RGoogle Scholar
[25] 夏同军, 董永强, 曹义刚 2013 物理学报 62 214702Google Scholar
Xia T J, Dong Y Q, Cao Y G 2013 Acta Phys. Sin. 62 214702Google Scholar
[26] Zufiria J A 1988 Phys. Fluids 31 440Google Scholar
[27] Li M J, Zhu Q F, Li G B 2016 Appl. Math. Mech. 37 1607Google Scholar
[28] Guo H Y, Wang L F, Ye W H, Wu J F, Zhang W Y 2017 Chin. Phys. Lett. 34 045201Google Scholar
[29] Guo Z L, Shu C 2013 Lattice Boltzmann Method and its Applications in Engineering (Singapore: World Scientific), pp239–284
[30] Wang H L, Yuan X L, Liang H, Chai Z H, Shi B C 2019 Capillarity 2 33Google Scholar
[31] Liang H, Xu J R, Chen J X, Wang H L, Chai Z H, Shi B C 2018 Phys. Rev. E 97 033309Google Scholar
[32] Liang H, Hu X L, Huang X F, Xu J R 2019 Phys. Fluids 31 112104Google Scholar
[33] Jacqmin D 1999 J. Comput. Phys. 155 96Google Scholar
[34] Wei Y K, Wang Z D, Dou H S, Qian Y H 2017 Comput. Fluids 156 97Google Scholar
[35] Wei Y K, Yang H, Lin Z, Wang Z D, Qian Y H 2018 Appl. Math. Comput. 339 556
[36] Lallemand P, Luo L S 2000 Phys. Rev. E 61 6546Google Scholar
[37] Chai Z H, Shi B C, Lu J H, Guo Z L 2010 Comput. Fluids 39 2069Google Scholar
[38] Liang H, Shi B C, Chai Z H 2016 Phys. Rev. E 93 013308Google Scholar
[39] He X, Zou Q, Luo L S, Dembo M 1997 J. Stat. Phys. 87 115Google Scholar
[40] Bian X, Aluie H, Zhao D X, Zhang H S, Livescu D 2020 Physica D 403 132250Google Scholar
[41] Cabot W H, Cook A W 2006 Nat. Phys. 2 562Google Scholar
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