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Potential energy in steep and breaking waves

Published online by Cambridge University Press:  26 April 2006

William W. Schultz ,
Jin Huh  and
Owen M. Griffin
William W. Schultz
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, Michigan 48109-2125, USA
Jin Huh
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, Michigan 48109-2125, USA Present address: Korea Atomic Energy Institute, Taejon, Korea
Owen M. Griffin
Affiliation:
Remote Sensing Division, Naval Research Laboratory, Washington, DC 20375-5000, USA
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Abstract

We find that the RMS wave height (square root of the potential energy) rather than peak-to-peak wave height is a better experimental and analytic criterion for determining when a regular, two-dimensional deep-water wave will break. A spectral algorithm for two-dimensional potential flow is developed and used to compare breaking onset criteria for energy input from (i) converging sidewalls, (ii) a submerged disturbance, and (iii) wave focusing. We also find that wave-breaking criteria (potential energy or the more classical peak-to-peak wave height) are a function of the rate of energy input. Large plunging waves occur when energy input rates are large. As energy input rates become smaller there is a smooth transition to smaller spilling waves. The various energy input methods show similar breaking trends in the limit as the energy input rate becomes small - waves break when the potential energy becomes approximately 52 % of the energy for the most energetic Stokes wave, with the formation of a singularity immediately before the crest. The effects of wave modulation and reflection are briefly discussed and shown not to affect the potential energy breaking criterion significantly. The experimental scatter of the RMS wave height is shown to be half that of wave steepness during incipient breaking in wave packets.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 278 , 10 November 1994 , pp. 201 - 228
Copyright
© 1994 Cambridge University Press

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References

Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501.Google Scholar
Banner, M. L. & Peregrine, D. H. 1993 Wave breaking in deep water. Ann. Rev. Fluid Mech. 25, 373–397.Google Scholar
Beck, R. F., Cao, Y. & Lee, T. 1993 Fully nonlinear water wave computations using the desingularized method. 6th Intl Conf. on Numerical Ship Hydrodynamics, The University of Iowa.
Bonmarin, P. 1989 Geometric properties of deep-water breaking waves. J. Fluid Mech. 209, 405–433.Google Scholar
Bonmarin, P. & Ramamonjiarisoa, A. 1985 Deformation to breaking of deep water gravity waves. Exp. Fluids 3, 11–16.Google Scholar
Boyd, J. P. 1989 Fourier and Chebyshev Spectral Methods. Springer.
Cao, Y., Schultz, W. W. & Beck, R. F. 1990 Three-dimensional, unsteady computations of nonlinear waves caused by underwater disturbances. 18th ONR Symp. on Naval Hydrodynamics (Proc.). National Academy Press, Washington DC.
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183–230.Google Scholar
Cokelet, E. D. 1979 Breaking waves—The plunging jet and the interior flow-field. In Mechanics of Wave-Induced forces on cylinders (ed. T. L. Shaw). Pitman.
Dold, J. W. & Peregrine, D. H. 1984 Steep unsteady water wave: An efficient computational scheme. School of Mathematics University of Bristol Internal Rep. AM-84–04.
Dold, J. W. & Peregrine, D. H. 1986 Water-wave modulation. School of Mathematics, Univ. of Bristol Rep. AM-86–03, presented at 20th Intl Conf. on Coastal Engng, Taipei.
Dommermuth, D. G., Yue, D. K., Lin, W. M., Rapp, R. J., Chan, E. S. & Melville, W. K. 1988 Deep-water plunging breakers: a comparison between potential theory and experiments. J. Fluid Mech. 189, 423–442.Google Scholar
Duncan, J. H. 1981 An experimental investigation of breaking waves produced by a towed hydrofoil. Proc. R. Soc. Lond. A 377, 331–348.Google Scholar
Duncan, J. H. 1983 The breaking and non-breaking wave resistance of a two-dimensional hydrofoil. J. Fluid Mech. 126, 507–520.Google Scholar
Duncan, J. H., Wallendorf, L. A. & Johnson, B. 1987 An experimental investigation of the kinematics of breaking waves. Rep. EW-7–87, US Naval Academy.
Huang, N. E., Bliven, L. F., Long, S. R. & Tung, C. C. 1986 An analytical model for oceanic whitecap coverage. J. Phys. Oceanogr. 16, 1597–1604.Google Scholar
Huh, J. 1991 A numerical study of capillary-gravity waves. PhD thesis, University of Michigan.
Kjeldsen, S. P. & Myrhaug, D. 1978 Kinematics and dynamics of breaking waves—Main report. VHL Rep., Norway.
Krasny, R. 1986 A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65–93.Google Scholar
Lin, W.-H., Newman, J. N. & Yue, D. K. 1985 Nonlinear forced motions of floating bodies, 15th ONR Symp. on Naval Hydrodynamics (Proc.), pp. 33–49. National Academy Press, Washington DC.
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water: II. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
Longuet-Higgins, M. S. 1981 A parametric flow for breaking waves. Hydrodynamics in Ocean Engineering, pp. 121–135. Norwegian Inst. Tech.
Longuet-Higgins, M. S. 1985 Accelerations in steep gravity waves. J. Phys. Oceanogr. 15, 1570–1579.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water: I. A numerical method of computation. Proc. R. Soc. Lond. A 350, 1–26.Google Scholar
Mciver, P. & Peregrine, D. H. 1981 Comparison of numerical and analytical results for waves that are starting to break. Intl Symp. on Hydrodynamics in Ocean Engineering, pp. 203–215. Norwegian Inst. of Tech.
Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165–185.Google Scholar
Melville, W. K. & Rapp, R. J. 1985 Momentum flux in breaking waves. Nature 317, 514–516.Google Scholar
Melville, W. K. & Rapp, R. J. 1988 The surface velocity field in steep and breaking waves. J. Fluid Mech. 189, 1–22.Google Scholar
Ochi, M. K. & Tsai, C.-H. 1983 Prediction and occurrence of breaking waves in deep water. J. Phys. Oceanogr. 13, 2008–2019.Google Scholar
Peltzer, R. D., Griffin, O. M. & Schultz, W. W. 1994 Steepness and energy properties of deep water breaking wave packets. Proc. Symp. on the Air-Sea Interface (ed. M. Donelan), Marseilles, June 1993. Kluwer, in press.
Ramberg, S. E., Barber, M. E. & Griffin, O. M. 1985 Laboratory studies of steep and breaking deep water waves in a convergent channel. Naval Research Laboratory Memorandum Rep. 5610. Washington, DC.
Ramberg, S. E. & Griffin, O. M. 1987 Laboratory studies of steep and breaking deep water waves. Proc. ASCE, J. Waterways, Port, Coastal and Ocean Engng 113, 493–506.Google Scholar
Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-water breaking waves. Phil. Trans. R. Soc. Land. A 331, 735–800.Google Scholar
Reed, A. M., Beck, R. F., Griffin, O. M. & Peltzer, R. D. 1991 Hydrodynamics of remotely sensed surface ship wakes. SNAME Trans. 98, 319–363.Google Scholar
Roberts, A. J. 1983 A stable and accurate numerical method to calculate the motion of a sharp interface between fluids. IMA J. Appl. Maths 31, 13–35.Google Scholar
Saad, Y. & Schultz, M. H. 1986 GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856–869.Google Scholar
Schultz, W. W., Griffin, O. M. & Ramberg, S. E. 1987 Steep and breaking deep water waves. 16th ONR Symp. on Naval Hydrodynamics (Proc.), pp. 106–115. National Academy Press, Washington DC.
Schultz, W. W. & Hong, S.-W. 1989 Solution of potential problems using an overdetermined complex boundary integral method. J. Comput. Phys. 84, 414–440.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553–78.Google Scholar
Schwartz, L. W. & Vanden-Broeck, J.-M. 1979 Numerical solution of the exact equations for capillary-gravity waves. J. Fluid Mech. 95, 119–139.Google Scholar
Srokosz, M. A. 1986 On the probability of wave breaking in deep water. J. Phys. Oceanogr. 16, 382–385.Google Scholar
Stokes, G. G. 1880 Supplement to a paper on the theory of oscillatory waves. In Mathematical and Physical Papers, vol. 1, pp. 314–326.
Vanden-Broeck, J.-M. 1980 Nonlinear stern waves. J. Fluid Mech. 96, 603–611.Google Scholar
Van Dorn, W. G. & Pazan, S. P. 1975 Laboratory investigation of wave breaking. II: Deep water waves. Scripps Inst. Oceanog. Rep. 75–21.
Vinje, T. & Brevig, P. 1981 Numerical simulation of breaking waves. Adv. Water Resources 4, 77–82.Google Scholar
Xü, H. & Yue, D. K. 1993 Computations of fully-nonlinear three-dimensional water waves. 19th Symp. on Naval Hydrodynamics, Seoul, Korea.