Wave motion induced by turbulent shear flows over growing Stokes waves

Abstract

The recent analytical of multi-layer analyses proposed by Sajjadi, Hunt and Drullion (2014) (SHD14 therein) is solved numerically for atmospheric turbulent shear flows blowing over growing (or unsteady) Stokes (bimodal) water waves, of low to moderate steepness. For unsteady surface waves the amplitude a(t) ∝ e kcit , where kc i is the wave growth factor, k is the wavenumber, and c i is the complex part of the wave phase speed, and thus the waves begin to grow as more energy is transferred to them by the wind. This will then display the critical height to a point where the thickness of the inner layer k i become comparable to the critical height kz c , where the mean wind shear velocity U (z) equals the real part of the wave speed c r . It is demonstrated that as the wave steepens further the inner layer exceeds the critical layer and beneath the cat's-eye there is a strong reverse flow which will then affect the surface drag, but at the surface the flow adjusts itself to the orbital velocity of the wave. We show that in the limit as c r /U * is very small, namely slow moving waves (i.e. for waves traveling with a speed c r which is much less than the friction velocity U * ), the energy-transfer rate to the waves, β (being proportional to momentum flux from wind to waves), computed here using an eddy-viscosity model, agrees with the asymptotic steady state analysis of Belcher and Hunt (1993) and the earlier model of Townsend (1980). The nonseparated sheltering flow determines the drag and the energy-transfer and not the weak critical shear layer within the inner shear layer. Computations for the cases when the waves are traveling faster (i.e. when c r > U * ) and growing

Turbulent windprint on a liquid surface

“…The situation is different for larger amplitude, close to the wave onset, for which a feedback of the wave motion on the air turbulence is expected, requiring a full two-phase flow approach. This approach, much more demanding computationally speaking, has been investigated recently (Belcher & Hunt 1998;Kudryavtsev & Makin 2002;Lin et al 2008;Druzhinin et al 2012;Kudryavtsev et al 2014;Sajjadi et al 2017). Because of the high computational cost, the range of physical parameters covered by these studies remains limited, and no scaling relation has yet been obtained for the wave onset.…”