On the generation of surface waves by shear flows

TLDR

A mechanism for generating surface waves by a parallel shear flow U(y) is developed using the inviscid Orr‑Sommerfeld equation. The authors derive an approximate solution for a logarithmic velocity profile and compute the spectral distribution of the energy‑transfer coefficient as a function of wave speed. The model shows that energy transfer to a wave of speed c scales with the curvature of U(y) at U=c, predicts a minimum wind speed of ~100 cm/s for initiating deep‑water gravity waves, yields sheltering coefficients in rough agreement with experimental drag data, and agrees qualitatively with observations, while suggesting possible relevance to membrane flutter.

Abstract

A mechanism for the generation of surface waves by a parallel shear flow U(y) is developed on the basis of the inviscid Orr-Sommerfeld equation. It is found that the rate at which energy is transferred to a wave of speed c is proportional to the profile curvature -U\"(y) at that elevation where U = c. The result is applied to the generation of deep-water gravity waves by wind. An approximate solution to the boundary value problem is developed for a logarithmic profile and the corresponding spectral distribution of the energy transfer coefficient calculated as a function of wave speed. The minimum wind speed for the initiation of gravity waves against laminar dissipation in water having negligible mean motion is found to be roughly 100cm/sec. A spectral mean value of the sheltering coefficient, as defined by Munk, is found to be in order-of-magnitude agreement with total wave drag measurements of Van Dorn. It is concluded that the model yields results in qualitative agreement with observation, but truly quantitative comparisons would require a more accurate solution of the boundary value problem and more precise data on wind profiles than are presently available. The results also may have application to the flutter of membranes and panels.