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Bubbles, breaking waves and hyperbolic jets at a free surface

Published online by Cambridge University Press:  20 April 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, and Institute of Oceanographic Sciences, Wormley, Surrey
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Abstract

Experiments have shown that bubbles approaching an air–water interface give rise to axisymmetric jets projected upwards into the air. Similar jets occur during the collapse of cavitation bubbles near a solid surface. In this paper we show that such jets are well modelled by a Dirichlet hyperboloid, a hyperbolic form of the better-known ellipsoid. The vertex angle of the hyperboloid is calculated as a function of time and found to agree with the observations of Blake & Gibson (1981) and others.

The jet is initiated, according to this model, when the vertex angle passes through 2 arctan √2, or 109·47°, at which instant the fluid accelerations become large. This compares with a vertical angle of 90° in the corresponding two-dimensional flow.

Further experiments demonstrate that an axisymmetric standing wave, when driven beyond its maximum amplitude, can break by throwing up a jet of the same hyperbolic form. Hyperbolic jets may occur commonly in free-surface flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 127 , February 1983 , pp. 103 - 121
Copyright
© 1983 Cambridge University Press

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References

Benjamin, T. B. & Ellis, A. T. 1966 The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries.Phil. Trans. R. Soc. Lond A 260, 221–240.
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion.Proc. R. Soc. Lond A 225, 505–515.
Birkhoff, G., Macdougall, D. P., Pugh, E. M. & Taylor, G. I. 1948 Explosives with lined cavities J. Appl. Phys. 19, 563–582.Google Scholar
Blake, J. R. & Gibson, D. C. 1981 Growth and collapse of a vapour cavity near a free surface J. Fluid Mech. 111, 123–140.Google Scholar
Blanchard, D. C. & Woodcock, A. H. 1980 The production, concentration and vertical distribution of the sea-salt aerosol Ann. New York Acad. Sci. 338, 330–347.Google Scholar
Bowden, F. P. 1966 The formation of microjets in liquids under the influence of impact or shock.Phil. Trans. R. Soc. Lond A 260, 94–95.
Dirichlet, P. L. 1860 Untersuchungen über ein Problem der Hydrodynamik Abh. Kön. Ges. Wiss. Göttingen 8, 3–42.Google Scholar
Edge, R. D. & Walters, G. 1964 Period of standing gravity waves of largest amplitude on water J. Geophys. Res. 69, 1674–1675.Google Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles on vibrating elastic surfaces. Phil. Trans. R. Soc. Lond., pp. 299–340.
Fultz, D. & Murty, T. S. 1963 Experiments on the frequency of finite-amplitude axisymmetric gravity waves in a circular cylinder J. Geophys. Res. 68, 1457–1462.Google Scholar
Haubrich, R. A., Munk, W. H. & Snodgrass, F. E. 1963 Comparative spectra of microseisms and swell Bull. Seism. Soc. Am. 53, 27–38.Google Scholar
Lamb, H. 1932 Introduction to Hydrodynamics. 6th edn. Cambridge University Press.
Longuet-Higgins, M. S. 1950 A theory of the origin of microseisms.Phil. Trans. R. Soc. Lond A 243, 1–35.
Longuet-Higgins, M. S. 1972 A class of exact, time-dependent, free-surface flows J. Fluid Mech. 55, 529–543.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. 1980 On the forming of sharp corners at a free surface.Proc. R. Soc. Lond A 371, 453–478.
Longuet-Higgins, M. S. 1983 Rotating hyperbolic flow: particle trajectories and parametric representation. Q. J. Mech. Appl. Math. 36 (May 1983).
Longuet-Higgins, M. S. & Ursell, F. 1948 Sea waves and microseisms Nature 162, 700.Google Scholar
Macintyre, F. 1968 Bubbles: a boundary-layer ‘microtome’ for micron-thick samples of a liquid surface. J. Phys. Chem. 72, 589–592.
Mciver, P. & Peregrine, D. H. 1981 Comparison of numerical and analytical results for waves that are starting to break. In Proc. Symp. on Hydrodynamics in Ocean Engineering, Trondheim, Norway, August 1981, pp. 203–215. University of Trondheim.
Mack, L. R. 1962 Periodic, finite-amplitude, axisymmetric gravity waves J. Geophys. Res. 67, 829–843.Google Scholar
Miche, R. 1944 Mouvements ondulatoires de la mer en profondeur constante ou décroissante. Ann. Ponts et Chaussées 114, 27–78, 131–164, 270–292, 369–406.
Penney, W. G. & Price, A. T. 1952 Finite periodic stationary gravity waves in a perfect fluid.Phil. Trans. R. Soc. Lond A 244, 254–284.
Plesset, M. S. & Chapman, R. B. 1971 Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary J. Fluid Mech. 47, 283–290.Google Scholar
Taylor, G. I. 1953 An experimental study of standing waves.Proc. R. Soc. Lond A 218, 44–59.
Worthington, A. M. 1908 A Study of Splashes. Longmans, Green & Co.