Nonlinear evolution of waves on a vertically falling film
Nonlinear evolution of waves on a vertically falling film
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DOI：
10.1017/s0022112093001521
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发表时间：
1993-05
影响因子：
3.7
通讯作者：
Hsueh-Chia Chang;E. Demekhin;D. Kopelevich
中科院分区：
文献类型：
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作者：
Hsueh-Chia Chang;E. Demekhin;D. Kopelevich
落膜上的波浪形成是一种有趣的流体力学现象，涉及丰富多样的空间和时间结构之间的转换。在初始区域之外，可以观察到短的、近正弦的毛细管波。再往下游，出现长而近孤立的波，有大的泪滴峰，前面出现短而超前的毛细波。这两种波都向下游缓慢发展，在大约10个波长内，它们类似于以恒定速度和形状传播的静止波。我们利用这种准稳态性质来研究垂直下落薄膜上的波演化和选择。从运动方程的边界层近似出发，用数值方法构造了具有与努塞尔平面膜相同平均厚度的所有有限振幅平稳波。与先前的近临界分析一致，发现了两个行波族，每个行波族都由波长或速度参数化。一类γ - 1比同样波长的无限小波传播得慢，而另一类γ - 2及其杂波传播得快。在任意波长的三维扰动下，对这些波的稳定性分析表明，在慢族γ - 1上存在一个独特的波数αs（或α2）的近正弦波，其增长率最低。该波略短于波数αm增长最快的线性模式，接近低雷诺数下流量最大的γ - 1上的波。然而，在快速γ - 2族中，发现以αf为界的近孤立波的多个波段对二维扰动是稳定的。这种稳定带的多样性可以解释为孤立波类相干结构之间有利的相互作用形成周期性序列的结果。（所有的波对于小增长率的三维扰动都不稳定。）所提出的选择机制与文献数据和我们的数值实验一致，表明波在开始之后立即减速，当它们接近波数为α2的慢γ1家族的短毛细管波。然后，它们接近下游γ - 2族的长稳定波，因此加速并发展成独特的孤波形状，然后屈服于缓慢发展的横向扰动。
Wave formation on a falling film is an intriguing hydrodynamic phenomenon involving transitions among a rich variety of spatial and temporal structures. Immediately beyond an inception region, short, near-sinusoidal capillary waves are observed. Further downstream, long, near-solitary waves with large tear-drop humps preceded by short, front-running capillary waves appear. Both kinds of waves evolve slowly downstream such that over about ten wavelengths, they resemble stationary waves which propagate at constant speeds and shapes. We exploit this quasi-steady property here to study wave evolution and selection on a vertically falling film. All finite-amplitude stationary waves with the same average thickness as the Nusselt flat film are constructed numerically from a boundary-layer approximation of the equations of motion. As is consistent with earlier near-critical analyses, two travelling wave families are found, each parameterized by the wavelength or the speed. One family γ1 travels slower than infinitesimally small waves of the same wavelength while the other family γ2 and its hybrids travel faster. Stability analyses of these waves involving three-dimensional disturbances of arbitrary wavelength indicate that there exists a unique nearly sinusoidal wave on the slow family γ1 with wavenumber αs (or α2) that has the lowest growth rate. This wave is slightly shorter than the fastest growing linear mode with wavenumber αm and approaches the wave on γ1 with the highest flow rate at low Reynolds numbers. On the fast γ2 family, however, multiple bands of near-solitary waves bounded below by αf are found to be stable to two-dimensional disturbances. This multiplicity of stable bands can be interpreted as a result of favourable interaction among solitary-wave-like coherent structures to form a periodic train. (All waves are unstable to three-dimensional disturbances with small growth rates.) The suggested selection mechanism is consistent with literature data and our numerical experiments that indicate waves slow down immediately beyond inception as they approach the short capillary wave with wavenumber α2 of the slow γ1 family. They then approach the long stable waves on the γ2 family further downstream and hence accelerate and develop into the unique solitary wave shapes, before they succumb to the slowly evolving transverse disturbances.