EFFECT OF TURBULENCE ON THE GROWTH OF THE STOKES WAVE. PART 1. SLOW MOVING WAVES

Shahrdad G. Sajjadi (USA)

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 derived expressions for the momentum flux for slow wind-wave regime The 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.