A note on the instabilities of a horizontal shear flow with a free surface
A note on the instabilities of a horizontal shear flow with a free surface
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关于自由表面水平剪切流不稳定性的注记
DOI：
10.1017/s0022112099007612
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发表时间：
2000
影响因子：
3.7
通讯作者：
L. Engevik
中科院分区：
文献类型：
--
作者：
L. Engevik
本文考虑了自由表面剪切流的不稳定性，特别着重于速度剖面为U* = U* 0 sech 2（by*）的剪切流。这个速度剖面，这是发现非常好的模型中的剪切流的水翼，已集中在以前的研究，例如由迪马斯和Triantyfallou谁做了一个纯粹的数值研究这个问题，并由Longuet-Higgins谁简化了问题，通过近似的速度剖面与分段线性剖面，使它服从于分析处理。然而，到目前为止，还没有人认识到这个问题实际上有一个非常简单的解决方案，可以通过分析找到;也就是说，稳定性边界，即波数（k）-弗劳德数（F）平面中稳定区域和不稳定区域之间的边界，由k和F中的简单代数方程给出。当包括表面张力时，这也适用。在没有表面张力的情况下，对于所有弗劳德数F > 0的值，存在两种不同的不稳定波区域。如果0 < F [Lt ] 1，则其中一个区域由0 < k <（1 − F2/6）给出，另一个由F−2 < k <9 F −2给出，这是k轴上非常扩展的区域。当F [Gt ] 1时，有一个小的不稳定区域接近k = 0，即0 < k < 9/（4F 2），另一个不稳定区域是（3/2）1/2F−1 < k < 2 + 27/（8 F2）。当包括表面张力时，取决于弗劳德数的值，可能存在一个、两个甚至三个不同的不稳定模式区域。当F较小时，只有一个不稳定区，当F为中间值时，有两个不稳定区，当F足够大时，有三个不同的不稳定区。
The instabilities of a free surface shear flow are considered, with special emphasis on the shear flow with the velocity profile U* = U*0sech2 (by*). This velocity profile, which is found to model very well the shear flow in the wake of a hydrofoil, has been focused on in previous studies, for instance by Dimas & Triantyfallou who made a purely numerical investigation of this problem, and by Longuet-Higgins who simplified the problem by approximating the velocity profile with a piecewise-linear profile to make it amenable to an analytical treatment. However, none has so far recognized that this problem in fact has a very simple solution which can be found analytically; that is, the stability boundaries, i.e. the boundaries between the stable and the unstable regions in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic equations in k and F. This applies also when surface tension is included. With no surface tension present there exist two distinct regimes of unstable waves for all values of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by 0 < k < (1 − F2/6), the other by F−2 < k < 9F−2, which is a very extended region on the k-axis. When F [Gt ] 1 there is one small unstable region close to k = 0, i.e. 0 < k < 9/(4F2), the other unstable region being (3/2)1/2F−1 < k < 2 + 27/(8F2). When surface tension is included there may be one, two or even three distinct regimes of unstable modes depending on the value of the Froude number. For small F there is only one instability region, for intermediate values of F there are two regimes of unstable modes, and when F is large enough there are three distinct instability regions.