Turbulent airflow over water waves-a numerical study

Abstract

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.

Air-Sea Interactions under the Existence of Opposing Swell

“…As a reference, the relation for the growth rate of water waves due to a following wind It is interesting that the present relation for the decay rate of water waves due to opposing wind is, in a range 0.1< u * /C <1.0, quite similar to the relation for the growth rate of water waves due to following wind, except for the sign. This is different from the results reported by Al-Zanaidi and Hui (1984) and Donelan (1999), which show much a larger growth rate than decay rate. With regard to the results of Al-Zanaidi and Hui (1984), since they made several assumptions on the boundary layer over the wave surface in their theoretical computations, there may be a possibility that their assumptions are different from our experimental conditions.…”

“…3) Equation (A3) has a similar form to the theoretical relation derived by Al-Zanaidi and Hui (1984), but the present relation gives larger value of the growth rate (by a factor of about 1.7) than that given by their relation. From Eqs.…”

“…The relation Eq. (18) of Al-Zanaidi and Hui (1984) gives the smallest value among these relations and it gives a value one order of magnitude smaller than that given by the relation of Peirson et al (2003).…”

“…16 includes the relation Eq. (18) of Al-Zanaidi and Hui (1984) and the following relation for the growth rate of water waves by following wind (Mitsuyasu and Kusaba, 1988), Figure 16 shows that the relation Eq. (25) of Peirson et al (2003) gives the largest values among these relations and it gives about double the value that our present relation Eq.…”

Removing wave effects from the wind stress vector

“…Furthermore, the wave-induced stress must (nearly) equal the wind input to the waves. This wave growth term is theoretically and empirically well founded only for collinear winds and waves [Plant, 1982, Al-Zanaidi andHui, 1984]. Testing of various speculations about the angular dependence of this wave growth term is a logical subject for future study.…”