Surface-wave generation: a viscoelastic model

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Abstract
The Reynolds-averaged equations for turbulent flow over a deep-water sinusoidal gravity wave, z = acoskx ≡ h0(x), are formulated in the wave-following coordinates ζ, η, where x = ζ, z =η + h(ζ, η), h(ζ, 0) = h0(ζ) and h is exponentially small for kη [Gt ] 1, and closed by a viscoelastic consitutive equation (a mixing-length model with relaxation). This closure is derived from Townsend's boundary-layer-evolution equation on the assumptions that: the basic velocity profile is logarithmic in η + z0, where z0 is a roughness length determined by Charnock's similarity relation; the lateral transport of turbulent energy in the perturbed flow is negligible; the dissipation length is proportional to η + z0. A counterpart of the Orr-Sommerfeld equation for the complex amplitude of the perturbation stream function is derived and used to construct a quadratic functional for the energy transfer to the wave. A corresponding Galerkin approximation that is based on independent variational approximations for outer (quasi-laminar) and inner (shear-stress) domains yields an energy-transfer parameter β that is comparable in magnitude with that of the quasi-laminar model (Miles 1957) and those calculated by Townsend (1972) and Gent & Taylor (1976) through numerical integration of the Reynolds-averaged equations. The calculated limiting values of β for very slow waves, with Charnock's relation replaced by kz0 = constant, are close to those inferred from observation but about three times the limiting values obtained through extrapolation of Townsend's results.