Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-09T21:39:21.293Z Has data issue: false hasContentIssue false
  • English
  • Français

A stationary potential-flow approximation for a breaking-wave crest

Published online by Cambridge University Press:  26 April 2006

Alastair D. Jenkins
Alastair D. Jenkins
Affiliation:
The IBM Environmental Sciences and Solutions Centre, Bergen, Norway Present address: Nansen Environmental and Remote Sensing Center, Edvard Griegs vei 3a, N-5037 Solheimsviken, Bergen, Norway
Get access Rights & Permissions [Opens in a new window]

Abstract

The flow in a breaking-wave crest is represented by a complex velocity potential on a Riemann surface, satisfying the Bernoulli condition on two free boundaries. The flow is assumed to be stationary in the reference frame which moves with the wave crest, and at large distances approximates Stokes corner flow in the main part of the fluid and a parabolic descending flow in the jet. The interaction of the jet with the rest of the fluid is neglected.

The solution is obtained by means of a conformal transformation from a bounded, teardrop-shaped domain, using a Faber polynomial expansion. The Bernoulli condition is applied at a number of discrete points on the boundaries, and the resulting nonlinear equations for the expansion coefficients are solved iteratively. The resulting surface form is similar to that obtained by laboratory experiments and time-dependent numerical simulations of waves up to the point of breaking, with a stagnation point at the top of the crest, an overturning loop with major axis $\ap 8g^{-\frac{1}{3}}\Psi^{\frac{2}{3}}$, and a maximum acceleration of ≈ 5.4 g, where g is the gravitational acceleration and ψ is the flux in the jet.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 280 , 10 December 1994 , pp. 335 - 347
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bervi, N. 1894 On motion of a fluid with the formation of a surface of separation under the influence of gravity. Proceedings of the Department of Physical Science of the Imperial Society of Natural Science, Moscow 7, 49–56.Google Scholar
Birkhoff, G. & Carter, D. 1956 Taylor instability. Appendices to report LA-1862. Los Alamos Rep. LA-1927.
Birkhoff, G. & Zarantonello, E. H. 1957 Jets, Wakes, and Cavities. Academic.
Bonmarin, P. 1989 Geometric properties of deep-water breaking waves. J. Fluid Mech. 209, 405–433.Google Scholar
Clarke, N. S. 1965 On two-dimensional inviscid flow in a waterfall. J. Fluid Mech. 22, 359–369.Google Scholar
Dias, F. & Christodoulides, P. 1991 Ideal jets falling under gravity. Phys. Fluids A 3, 1711–1717.Google Scholar
Dias, F, Keller, J. B. & Vanden-Broeck, J.-M. 1988 Flows over rectangular weirs. Phys. Fluids 31, 2071–2076.Google Scholar
Dias, F. & Tuck, E. O. 1993 A steady breaking wave. Phys. Fluids A 5, 277–279.Google Scholar
Dold, J. W. 1992 An efficient surface-integral algorithm applied to unsteady gravity waves. J. Comput. Phys. 103, 90–115.Google Scholar
Dold, J. W. & Peregrine, D. H. 1984 Steep unsteady waves: an efficient computational scheme. In Nineteenth Coastal Engineering Conf. (ed. B. L. Edge), pp. 955–967. American Society of Civil Engineers.
Dommermuth, D. G., Yue, D. K. P., 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, 423–342.Google Scholar
Feng, Z.-X. 1988 An alternative iteration algorithm for moving boundary free flow using boundary elements. In Boundary Elements X (ed. C. A. Brebbia), vol. 2, pp. 129–142. Computational Mechanics & Springer.
Geer, J. & Keller, J. B. 1979 Slender streams. J. Fluid Mech. 93, 97–115.Google Scholar
Goh, K. H. M. & Tuck, E. O. 1985 Thick waterfalls from horizontal slots. J. Engng Maths 19, 341–349.Google Scholar
Greenhow, M. 1983 Free-surface flows related to breaking waves. J. Fluid Mech. 134, 259–275.Google Scholar
Gurevich, M. I. 1966 The Theory of Jets in an Ideal Fluid. Pergamon.
Hasselmann, K. 1974 On the spectral dissipation of ocean waves due to white capping. Boundary-Layer Met. 6, 107–127.Google Scholar
Henrici, P. 1986 Applied and Computational Complex Analysis, vol. 3. Wiley.
Hwang, P. A., Poon, Y.-K. & Wu, J. 1991 Temperature effects on generation and entrainment of bubbles induced by a water jet. J. Phys. Oceanogr. 21, 1602–1605.Google Scholar
Jenkins, A. D. 1994 A quasi-stationary irrotational solution for a breaking wave crest. In The Air-Sea Interface (ed. M. Donelan, W. H. Hui & W. Plant). The University of Toronto Press.
John, F. 1953 Two-dimensional potential flows with a free boundary. Communs Pure Appl. Maths 6, 497–503.Google Scholar
Keller, J. B. & Geer, J. 1973 Flows of thin streams with free boundaries. J. Fluid Mech. 59, 417–432.Google Scholar
Lee, J. & Vanden-Broeck, J.-M. 1993 Two-dimensional jets falling from-funnels and nozzles. Phys. Fluids A 5, 2454–2460.Google Scholar
Longuet-Higgins, M. S. 1976 Self-similar, time-dependent flows with a free surface. J. Fluid Mech. 73, 603–620.Google Scholar
Longuet-Higgins, M. S. 1980a A technique for time-dependent, free-surface flows. Proc. R. Soc. Lond. A 371, 441–451.Google Scholar
Longuet-Higgins, M. S. 1980b On the forming of sharp corners at a free surface. Proc. R. Soc. Lond. A 371, 453–478.Google Scholar
Longuet-Higgins, M. S. 1981 On the overturning of gravity waves. Proc. R. Soc. Lond. A 376, 377–400.Google Scholar
Longuet-Higgins, M. S. 1982 Parametric solutions for breaking waves. J. Fluid Mech. 121, 403–424.Google Scholar
Longuet-Higgins, M. S. 1983 Rotating hyperbolic flow: particle trajectories and parametric representation. Q. J. Mech. Appl. Maths 36, 247–270.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
Longuet-Higgins, M. S. & Fox, M. J. H. 1977 Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80, 721–741.Google Scholar
Meiron, D. I., Orszag, S. A. & Israeli, M. 1981 Applications of numerical conformal mapping. J. Comput. Phys. 40, 345–360.Google Scholar
Menikoff, R. & Zemach, C. 1983 Rayleigh-Taylor instability and the use of conformal maps for ideal fluid flows. J. Comput. Phys. 51, 28–64.Google Scholar
Miller, R. L. 1957 Role of vortices in surf zone prediction: sedimentation and wave forces. In Beach and Nearshore Sedimentation (ed. R. A. Davis & R. L. Ethington), pp. 92–114. Society of Economic Palaeontologists and Mineralogists.
Nag 1990 The NAG Fortran Library Manual, Mark 14. Numerical Algorithms Group.
New, A. L. 1983 A class of elliptical free-surface flows. J. Fluid Mech. 130, 219–239.Google Scholar
New, A. L., McIver, P. & Peregrine, D. H. 1985 Computations of overturning waves. J. Fluid Mech. 150, 233–251.Google Scholar
Sawada, H. & Tomita, H. 1994 On the dynamical properties of plunging breakers in deep water. In The Air-Sea Interface (ed. M. Donelan, W. H. Hui & W. Plant). The University of Toronto Press.
Seo, S. N. & Dalrymple, R. A. 1990 An efficient model for periodic overturning waves. Engineering Analysis with Boundary Elements 7, 196–204.Google Scholar
Ting, L. & Keller, J. B. 1974 Planing of a flat plate at high Froude number. Phys. Fluids 17, 1080–1086.Google Scholar
Tuck, E. O. 1987 Efflux from a slit in a vertical wall. J. Fluid Mech. 176, 253–264.Google Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1982 Jets rising and falling under gravity. J. Fluid Mech. 124, 335–345.Google Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1986 Pouring flows. Phys. Fluids 29, 3958–3961.Google Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1989 Surfing on solitary waves. J. Fluid Mech. 198, 115–125.Google Scholar
Vinje, T. & Brevig, P. 1981 Breaking waves on finite water depths: a numerical study. Ship Res. Inst. of Norway, Rep. R-111.81.
Wegmann, R. 1978 An iterative method for conformal mapping. Numer. Math. 30, 453–466. (English translation: 1986 J. Comput. & Appl. Math. 14, 7–18.)Google Scholar
Zemach, C. 1986 A conformal map formula for difficult cases. J. Comput. & Appl. Math. 14, 207–215.Google Scholar