A Model for Rayleigh-Taylor Mixing and Interface Turnover
A Model for Rayleigh-Taylor Mixing and Interface Turnover
复制标题
瑞利-泰勒混合和界面转换模型
DOI：
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发表时间：
2016
影响因子：
1.6
通讯作者：
S. Shkoller
中科院分区：
文献类型：
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作者：
R. Granero;S. Shkoller
本文首先建立了二维流体中受Rayleigh-Taylor （RT）不稳定性影响的两流体界面运动的数学模型，其最简单形式为：$ h_{tt}(alpha,t) = a g, Lambda h- frac{sigma}{
ho^++
ho^-} Lambda^3 h- a partial_alpha(h h_t h_t) $，其中$Lambda = h partial_alpha $, $ h $表示希尔伯特变换。在这个所谓的$h$模型中，$A$是阿特伍德数，$g$是加速度，$ sigma $是表面张力，$
ho^pm$表示两种流体的密度。在一定的稳定性条件下，我们证明了这个所谓的$h$-模型是局部和全局良定的。对$h$-模型的数值模拟表明，当较轻流体位于较重流体之上且加速度向下时，界面由于非线性而快速增长，然后趋于稳定。在较重流体被较轻流体支撑的不稳定情况下，我们发现混合层的生长与Read of young的“火箭钻机”实验数据吻合得很好。
We first develop a new mathematical model for two-fluid interface motion, subjected to the Rayleigh-Taylor (RT) instability in two-dimensional fluid flow, which in its simplest form, is given by $ h_{tt}(alpha,t) = A g, Lambda h - frac{sigma}{
ho^++
ho^-} Lambda^3 h - A partial_alpha(H h_t h_t) $, where $Lambda = H partial_ alpha $ and $H$ denotes the Hilbert transform. In this so-called $h$-model, $A$ is the Atwood number, $g$ is the acceleration, $ sigma $ is surface tension, and $
ho^pm$ denotes the densities of the two fluids. Under a certain stability condition, we prove that this so-called $h$-model is both locally and globally well-posed. Numerical simulations of the $h$-model show that the interface can quickly grow due to nonlinearity, and then stabilize when the lighter fluid is on top of the heavier fluid and acceleration is directed downward. In the unstable case of a heavier fluid being supported by the lighter fluid, we find good agreement for the growth of the mixing layer with experimental data in the "rocket rig" experiment of Read of Youngs.
We then derive an RT interface model with a general parameterization $z(alpha,t)$ such that $ z_{tt}= Lambdaigg{[}frac{A}{|partial_alpha z|^2}Hleft(z_tcdot (partial_alpha z)^perp H(z_tcdot (partial_alpha z)^perp)
ight)
+ A g z_2 igg{]} frac{(partial_alpha z)^perp}{|partial_alpha z|^2}
+z_tcdot (partial_alpha z)^perpleft(frac{(partial_alpha z_t)^perp}{|partial_alpha z|^2}-frac{(partial_alpha z)^perp 2(partial_alpha zcdot partial_alpha z_t)}{|partial_alpha z|^4}
ight)$.
This more general RT $z$-model allows for interface turn-over. Numerical simulations of the $z$-model show an even better agreement with the predicted mixing layer growth for the "rocket rig" experiment.