Surface waves on shear currents: solution of the boundary-value problem
Surface waves on shear currents: solution of the boundary-value problem
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剪切流上的表面波：边值问题的求解
DOI：
10.1017/s002211209300388x
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发表时间：
1993
影响因子：
3.7
通讯作者：
V. Shrira
中科院分区：
文献类型：
--
作者：
V. Shrira
我们考虑一个经典的边界值问题的深水重力毛细波在剪切流，组成的瑞利方程和标准的线性化运动学和动力学无粘边界条件的自由表面。我们推导出了这个问题的精确解，在幂的一个无穷级数的某个参数e，它的特点是波动的偏差从潜在的运动的小。对于解的存在性和绝对收敛性，只要e小于1就足够了。该系列的截断和提供了具有先验规定精度的近似解。特别地，对于可以解释为Miles临界层型不稳定性的短波不稳定性，导出了增长率的显式近似表达式。在一定（非常狭窄）尺度范围内的增长率可以超过风引起的英里增量。薄的边界层上的色散关系的效果也进行了研究，使用渐近过程的基础上的产品的层厚度和波数的小。推导出了边界层影响可以忽略的条件和精度。
We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terms of an infinite series in powers of a certain parameter e, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that e be less than unity. The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind. The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived.