Flow in a deep turbulent boundary layer over a surface distorted by water waves

Abstract

Linearized equations for the mean flow and for the turbulent stresses over sinusoidal, travelling surface waves are derived using assumptions similar to those used by Bradshaw, Ferriss & Atwell (1967) to compute boundary-layer development. With the assumptions, the effects on the local turbulent stresses of advectal, vertical transport, generation and dissipation of turbulent energy can be assessed, and solutions of the equations are expected to resemble closely real flows with the same conditions. The calculated distributions of surface pressure indicate rates of wave growth (expressed as fractional energy gain during a radian advance of phase) of about 15(ρ_a/ρ_w) (τ_o/c²), where τ_o is the surface stress, c_o the phase velocity and ρ_a and ρ_w the densities of air and water, unless the wind velocity at height λ/2π is less than the phase velocity. The rates are considerably less than those measured by Snyder & Cox (1966), by Barnett & Wilkerson (1967) and by Dobson (1971), and arguments are presented to show that the linear approximation fails for wave slopes of order 0.1.