A numerical model of the air flow above water waves

A numerical model is proposed for the flow in a deep turbulent boundary layer over water waves. The momentum equations are closed by the use of an isotropic eddy viscosity and the turbulent energy equation. For small amplitudes the results are similar to those of Townsend's (1972) linear model, but nonlinear effects become important as the ratio of wave height to wavelength increases. With uniform surface roughness z o , the predicted fractional rate of energy input per radian advance in phase, ζ, decreases slightly with increasing amplitude and is of the same order of magnitude as in Miles’ (1957, 1959) and Townsend's linear theories. If z o is allowed to vary with position along the wave, however, the fractional rate of energy input can be significantly increased for small amplitude waves. If the variation in z o is half the mean value and the maximum wave slope z ak is 0.01, we find ζ ≈ 60 (ρair/ρwater) (u o /c) 2 , where u o is the friction velocity and c the wave phase speed. Comparison is also made with recent laboratory and field data.