Turbulent airflow over a Stokes water-wave train of small amplitude is studied numerically based on the two-equation closure model of Saffman & Wilcox (1974) together with appropriate boundary conditions on the wave surface. The model calculates, instead of assuming, the viscous sublayer flow, and it is found that the energy transfer between wind and waves depends significantly on the flow being hydraulically rough, transitional or smooth. Systematic computations have yielded a simple approximate formula for the fractional rate of growth per radian \[ \zeta = \delta_{\rm i}\frac{\rho}{\rho_{\rm w}}\left(\frac{U_{\lambda}}{c}-1 \right)^2, \] with δi = 0.04 for transitional or smooth flow and δi = 0.06 for rough flow, where ρ is density of air, ρw that of water, Uλ wind speed at one wavelength height and c the wave phase velocity. This formula is in good agreement with most existing data from field experiments and from wave-tank experiments. In the case of waves travelling against wind, the corresponding values are δi = −0.024 for transitional and smooth flow, and δi = −0.04 for rough flow.

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