Some Results for Surface Gravity Waves on Shear Flows
Some Results for Surface Gravity Waves on Shear Flows
复制标题
剪切流上的表面重力波的一些结果
DOI：
10.1093/imamat/12.1.1
复制
发表时间：
1973
影响因子：
1.2
通讯作者：
J. Fenton
中科院分区：
文献类型：
--
作者：
J. Fenton
这项工作试图填补二维流中稳定表面重力波的一些空白，其中速度随深度变化，就像波在流动的溪流中传播的情况一样。根据大多数以前的工作，该理论基本上是无粘性的，因为剪切被假定为由外部效应产生的：该理论考察了波扰动和剪切流之间的非粘性相互作用。特别地，我们得到了小波在任意速度分布流中的色散关系的一些结果，并将其推广到包括有限扰动到超临界流的衰减。针对等涡度流动的特殊情况，导出了所有表面重力波的精确算子方程；对这一问题进行了求解，得到了用通道流动不变量表示的孤立波和余弦波的一阶方程。在一些典型的剪切流上得到了小波的精确数值解，并说明了该理论是如何通过在最初为无旋转和超临界的流动中发展完全湍流的速度剖面来预测周期波在流上的增长的。本工作所有部分的结果表明，剪切是决定波和扰动传播行为的重要量。初级气流的微小变化可能会大大改变表面波的性质。事实上，它们可以将波从一种类型转换为另一种类型，对应于由速度剖面变化直接引起的超临界状态和亚临界状态之间的流动变化。
This work attempts to fill some gaps in the subject of steady surface gravity waves on two-dimensional flows in which the velocity varies with depth, as is the case for waves propagating on a flowing stream. Following most previous work the theory is basically inviscid, for the shear is assumed to be produced by external effects: the theory examines the non-viscous interaction between wave disturbances and the shear flow. In particular, some results are obtained for the dispersion relationship for small waves on a flow of arbitrary velocity distribution, and this is generalized to include the decay from finite disturbances into supercritical flows. An exact operator equation is developed for all surface gravity waves for the particular case of flow with constant vorticity; this is solved to give first-order equations for solitary and cnoidal waves in terms of channel flow invariants. Exact numerical solutions are obtained for small waves on some typical shear flows, and it is shown how the theory can predict the growth of periodic waves upon a stream by the development of a fully-turbulent velocity profile in flow which was originally irrotational and supercritical. Results from all sections of this work show that shear is an important quantity in determining the propagation behaviour of waves and disturbances. Small changes in the primary flow may alter the nature of the surface waves considerably. They may in fact transform the waves from one type to another, corresponding to changes in the flow between super- and sub-critical states directly caused by changes in the velocity profile.