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arXiv 0806.0783physics.flu-dynphysics.comp-ph
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不混溶流体Rayleigh-Taylor不稳定性的相场模型

Phase-field model for the Rayleigh--Taylor instability of immiscible fluids

Antonio Celani, Andrea Mazzino, Paolo Muratore-Ginanneschi, Lara Vozella

摘要:

本文用相场描述法研究了小Atwood数极限下两种不互溶流体的Rayleigh-Taylor不稳定性。在这种方法中,尖锐的流体界面被薄的但有限的过渡层所取代,在过渡层中,界面力平滑地变化。这是通过引入一个序参量(相场)来实现的,该序参量的变化在界面层上是连续的,并且在体区域中是均匀的。相场模型服从Cahn-Hilliard方程,并与标准的纳维-斯托克斯方程双向耦合。从这个方程组出发,我们首先进行了线性分析,由此我们解析地重新推导出了已知的重力-毛细管色散关系在混合能量密度和毛细管宽度为零的极限。我们已经进行了数值模拟,并确定了一个区域的参数,在该区域中的线性相位(稳定和不稳定)的已知属性是复制在一个非常准确的方式。这已经在可忽略的粘度的情况下和在非零粘度的情况下完成。在后一种情况下,只有扰动增长率的上限和下限是已知的。最后,我们还研究了弱非线性阶段的扰动演化,并确定了一个政权的特点是由一个恒定的终极速度的气泡/尖峰。终极速度的测量值与现有的理论预测是完全一致的。相场法是一种很有价值的方法,它可以用来描述流体动力学湍流和波动湍流的动力学过程。

英文摘要:

The Rayleigh--Taylor instability of two immiscible fluids in the limit of small Atwood numbers is studied by means of a phase-field description. In this method the sharp fluid interface is replaced by a thin, yet finite, transition layer where the interfacial forces vary smoothly. This is achieved by introducing an order parameter (the phase field) whose variation is continuous across the interfacial layers and is uniform in the bulk region. The phase field model obeys a Cahn--Hilliard equation and is two-way coupled to the standard Navier--Stokes equations. Starting from this system of equations we have first performed a linear analysis from which we have analytically rederived the known gravity-capillary dispersion relation in the limit of vanishing mixing energy density and capillary width. We have performed numerical simulations and identified a region of parameters in which the known properties of the linear phase (both stable and unstable) are reproduced in a very accurate way. This has been done both in the case of negligible viscosity and in the case of nonzero viscosity. In the latter situation only upper and lower bounds for the perturbation growth-rate are known. Finally, we have also investigated the weakly-nonlinear stage of the perturbation evolution and identified a regime characterized by a constant terminal velocity of bubbles/spikes. The measured value of the terminal velocity is in perfect agreement with available theoretical prediction. The phase-field approach thus appears to be a valuable tecnhique for the dynamical description of the stages where hydrodynamic turbulence and wave-turbulence enter into play.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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