Abstract: A generalization of
the laminar model is constructed by averaging the linearized equations of motion
for a turbulent shear flow in the direction parallel to the crest of Stokes
wave. It is shown that the resulting mean momentum transfer comprises (i) a
singular part, which is proportional to product of the velocity-profile
curvature and the mean square of the wave-induced vertical velocity in the
critical layer, where the mean wind speed is equal to the wave speed; (ii) a
vertical integral of mean product of the vertical velocity and the vorticity w, where w is the wave-induced
perturbation in the total velocity along a streamline of the y-averaged
motion; (iii) the perturbation in the mean turbulent shear stress at the
air-water interface. A closure model, based on Townsend [The Structure of
Turbulent Shear Flow, 2nd ed., Cambridge University Press, 1976], is constructed
for the specification of turbulent Reynolds stresses. The resulting equation
together with its corresponding boundary conditions is solved numerically using
a multigrid algorithm. The growth rate of Stokes wave is then calculated from
the derivedexpressions for the momentum flux for slow wind-wave
regimeThe result of calculations for the energy transfer
parameter agrees well with the numerical integration of the Reynolds-stress
transport equations over Stokes wave (Sajjadi[A numerical study for the growth of a fully non-linear
Stokes wave by turbulent shear flow, CHL Technical Report, CHL-HPC-01-3, 2001]),
also with the numerical calculations of Ierley and Miles [J. Fluid Mech. 435
(2001), 175], and provides further evidence to support the earlier postulation
of Belcher and Hunt [J. Fluid Mech. 251 (1993), 109] for rapid distortion theory
of turbulence over water waves.
Keywords and phrases: wind-wave interaction, turbulence, Stokes water waves.
