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Surface-wave generation: a viscoelastic model

Published online by Cambridge University Press:  26 April 2006

John Miles
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093-0225, USA
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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.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 322 , 10 September 1996, pp. 131 - 145
Copyright
© 1996 Cambridge University Press

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References

Abrams, J. & Hanratty, T. 1985 Relaxation effects observed for turbulent flow over a wavy surface. J. Fluid Mech. 151, 443–455.Google Scholar
Belcher, S. E. & Hunt, J. C. R. 1993 Turbulent flow over slowly moving waves. J. Fluid Mech. 251, 109–148.Google Scholar
Benjamin, T. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.Google Scholar
Benney, D. J. & Bergeron, R. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181–204.Google Scholar
Bradshaw, P., Ferriss, D. & Atwell, N. 1967 Calculation of boundary-layer development using the turbulent energy equation. J. Fluid Mech. 28, 593–616.Google Scholar
Charnock, H. 1955 Wind stress on a water surface. Q. J. R. Met. Soc. 81, 639–640.Google Scholar
Crow, S. C. 1968 Viscoelastic properties of fine-grained incompressible turbulence. J. Fluid Mech. 33, 1–20.Google Scholar
Davis, R. E. 1972 On prediction of the turbulent flow over a wavy boundary. J. Fluid Mech. 52, 287–306.Google Scholar
Duin, C. A. Van & Janssen, P. A. E. M. 1992 An analytical model for the generation of surface gravity waves by turbulent air flow. J. Fluid Mech. 236, 197–215.Google Scholar
Gent, P. R. & Taylor, P. 1976 A numerical model of the flow above water waves. J. Fluid Mech. 77, 105–128.Google Scholar
Jacobs, S. J. 1987 An asymptotic theory for the turbulent flow over a progressive water wave. J. Fluid Mech. 174, 69–80.Google Scholar
Knight, D. 1977 Turbulent flow over a wavy boundary. Boundary-Layer Met. 11, 209–222.Google Scholar
Komen, G. J., Cavaleri, L., Donelan, M. et al. 1994 Dynamics and Modelling of Ocean Waves. Cambridge University Press.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lighthill, M. J. 1962 Physical interpretation of the theory of wind generated waves. J. Fluid Mech. 14, 385–397.Google Scholar
Manton, M. J. 1972 On the generation of sea waves by a turbulent wind. Boundary-Layer Met. 2, 348–364.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204 (referred to herein as M 57).Google Scholar
Miles, J. W. 1959a On the generation of surface waves by shear flows. Part 2. J. Fluid Mech. 6, 568–582.Google Scholar
Miles, J. W. 1959b On the generation of surface waves by shear flows. Part 3. Kelvin-Helmholtz instability. J. Fluid Mech. 6, 583–598.Google Scholar
Miles, J. W. 1967 On the generation of surface waves by shear flows. Part 5. J. Fluid Mech. 30, 163–175.Google Scholar
Miles, J. 1993 Surface-wave generation revisited. J. Fluid Mech. 256, 427–442 (referred to herein as M93).Google Scholar
Miles, J. 1996 On viscoelastic eddy viscosity for flow over a wavy surface. Boundary-Layer Met. (in the press).Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.
Plant, W. J. 1982 A relationship between wind stress and wave slope. J. Geophys. Res. 87, 1961–1967.Google Scholar
Taylor, P. A. 1980 Some recent results from a numerical model of surface boundary layer flow over hills. Workshop on the Planetary Boundary Layer (ed. J. C. Wyngaard). American Meteorological Society, Boston.
Townsend, A. A. 1972 Flow in a deep turbulent boundary layer over a surface distorted by water waves. J. Fluid Mech. 55, 719–735.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.
Zeman, O. & Jensen, N. 1987 Modification of turbulence in flow over hills. Q. J. R. Met. Soc. 113, 55–88.Google Scholar