In studies of the turbulent air flow over water waves it is usually assumed that the effect of viscosity near the water surface is negligible, i.e. the Reynolds number, Re = u∗λ/v, is considered to be high. However, for short waves or low wind speeds this assumption is not valid. Therefore, a second-order turbulence closure that takes into account viscous effects is used to simulate the air flow. The model shows reasonable agreement with laboratory measurements of wave-induced velocity profiles. Next, the dependence of the dimensionless energy flux from wind to waves, or growth rate, on Re is investigated. The growth rate of waves that are slow compared to the wind is found to increase strongly when Re decreases below 104, with a maximum around Re = 800. The numerical model predictions are in good agreement with analytical theories and laboratory observations. Results of the study are useful in field conditions for the short waves in the spectrum, which are particularly important for remote sensing applications.