On the spatial linear growth of gravity-capillary water waves sheared by a laminar air flow

Abstract

The initial growth of mechanically generated small amplitude water waves below a laminar air stream was examined numerically and experimentally in order to explore the primary growth mechanism, that is, the interfacial instability of coupled laminar air and water flows. Measurements of the laminar velocity profile in the air over the water surface were found to be consistent with Lock’s [Q. J. Mech. Appl. Math. 4, 42 (1951)] theory. This profile was then used to calculate the spatial growth rates by solving the Orr-Sommerfeld equations. The simulation shows that the growth of the boundary layer affects the exponential growth of water waves along the fetch. The sensitivity of the growth rate is observed to vary by a factor of 2 for changes in the laminar velocity profile as small as 2% at the water surface. This indicates that the interfacial instability is strongly influenced by the wind-induced surface current. A laminar airflow was produced in the wind tunnel over mechanically generated monochromatic gravity-capillary water waves with the ka value in the order of 10−3. The novel experiment was designed to measure the minute changes in the wave slope and phase velocity simultaneously using a highly sensitive reflected twin laser beam technique. Agreement between linear theory and experiments for the spatial development of wave height and phase velocity suggests that the linear instability mechanism determines the initial stages of development of small-scale water waves.