Turbulent flow over unsteady monochromatic and Stokes waves has been investigated for various wave steepness ka and wave age  using a Differential Second-Moment (DSM) closure turbulence model. Also presented is DSM simulation of flow over steady monochromatic waves of different wave steepness for comparison with Direct Numerical Simulation (DNS) reported by Sullivan et al. [38]. It is shown that there is a good agreement between results obtained from DNS and DSM simulations. For the simulations of unsteady waves, where the critical layer is no longer steady and has temporal dependency, the growth factor  is derived from a quadratic secular equation for complex phase speed c, obtained by coupling air and sea, whose solution leads to two wave speeds  and  The wave speed  corresponds to free-surface waves damped by viscous stresses in the boundary layers at the interface, whilst the wave speed  is associated with Tollmien-Schlichting instabilities in the shear airflow over the surface wave. It is shown that the wave growth throughthe complex phase speed  is finite and waves do not become sharp-crested, therefore waves are unlikely to break. However, when the complex wave speed  is adopted, for growth of Stokes waves, the waves rapidly peak and become sharp-crested. As the wave becomes relatively steep, a secondary vortical motion is induced around the wave crest with a region of high pressure behind and low pressure ahead of the crest. This pressure asymmetry will very likely cause the wave to break. It is also observed that as the waves grow and become steeper, the unsteady critical layer elevates from the inner to the middle region. Consequently, the cat’s-eye, and hence the unsteady critical layer, becomes increasingly asymmetrical, spatially dependent, and dramatically affects the flow field above the wave. Furthermore, at relatively high steepness, flow separation is observed in the lee of the wave. Also, in simulations reported here, the effect of non-separating sheltering is seen to be pronounced, since the boundary layer is perturbed and thickens on the leeside of the wave due to turbulent shear stress in the inner region. It is therefore argued that the wave growth is due to combined interactions between the unsteady elevated critical layer, the non-separated sheltering, and turbulence. From numerical results obtained, a new parameterization for the energy-transfer parameter is derived which agrees well with that obtained  by Belcher and Hunt [3] for slow moving waves and has the salient features of those evaluated numerically and asymptotically for intermediate as well as fast moving waves by Sajjadi et al. [34]. From the collective accounts of results of numerical simulations reported here, it is concluded that the unsteady critical layer has dynamical effect and plays a crucial role in shear flows over unsteady water waves.

air-sea interactions, growing waves, DNS, Reynolds stress closure.