Abstract

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.

Highlights

  • 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 also report that the bubble amplitude first increases then decreases with the surface tension

  • Huang Hao -Wei Liang Hong † Xu Jiang -Rong (Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, China) ( Received 20 November 2020; revised manuscript received 22 January 2021 )

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Summary

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通过 Chapman-Enskog 多尺度分析 [13], 可以证明 本文采用的相场 LB 模型能够正确地恢复到 Cahn-Hilliard 方程和 Navier-Stokes 方程, 并且迁 移率 M 和运动学黏度 ν 的数学表达式为. 已有研究表 明 [36,37], 松弛参数的选取影响多松弛 LB 方法的数 值稳定性和精度. 在实际的模拟中, 松弛因子 sf 和 sg 可根据迁移率和流体黏性给定, sg4 = sg6 = 1.7 , 余下的松弛参数均取 1. 需要指出 的是, RT 不稳定性的界面演化可以用两个无量纲 参数来描述, 即雷诺数 (Reynolds number, Re) 和 阿特伍德数 (Atwood number, At ), 分别定义为: 114701-4 其中, 界面厚度 D固定为 4个格子网格. 需要指出 的是, RT 不稳定性的界面演化可以用两个无量纲 参数来描述, 即雷诺数 (Reynolds number, Re) 和 阿特伍德数 (Atwood number, At ), 分别定义为: 114701-4

At ρh ρh
Atwood 数下气泡与尖钉的后期增长率与表面张力
Atwood 数下的临界表面张力以及理论模型所预测

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