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Highly accelerated, free-surface flows

Published online by Cambridge University Press:  26 April 2006

Michael S. Longuet-Higgins
Michael S. Longuet-Higgins
Affiliation:
Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA 92093-0402, USA
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Abstract

Accelerations exceeding 20g in surface waves have been observed both in experiments and in numerically computed flows with a free surface. The present paper describes a family of analytic solutions which display such behaviour. They are expressible in parametric form as z = F sinh ω + iG cosh ω + γω + iH, where F, G and H are functions of the time t only, and γ is linear in t. ω is a complex parameter which is real at the free surface. The functions F(t) and G(t) satisfy two nonlinear, coupled ODEs, which can be solved numerically. Typically the solutions pass through an ‘inertial shock’, or singularity in the time, where the displacements vary as t2/3, the velocities as t1/3 and the accelerations as t-4/3. In this class of solution the free surface develops a cusp as t → ∞. In a special case, F and G vary as t4/7 and the cusp is reached in finite time. Gravity is neglected, but plays a part in setting up the initial conditions for the highly accelerated flow.

In future papers it will be shown that more general solutions exist in which the acceleration is momentarily large but bounded.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 248 , March 1993 , pp. 449 - 475
Copyright
© 1993 Cambridge University Press

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References

Cooker, M. J. & Peregrine, D. H. 1990 Violent water motion at breaking-wave impact. In Coastal Engineering (Proc. ICCE Conf., Delft, Holland), vol. 1, pp. 164– 176. ASCE.
John, F. 1953 Two-dimensional potential flows with a free boundary. Commun. Pure Appl. Maths 6, 497– 503.Google Scholar
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. 1983a On the forming of sharp corners at a free surface. Proc. R. Soc. Lond. A 371, 453– 478.Google Scholar
Longuet-Higgins, M. S. 1983b Towards the analytic description of overturning waves. In Nonlinear Waves (ed. L. Debnath), pp. 1– 24. Cambridge University Press, 360 pp.
Longuet-Higgins, M. S. 1983c Bubbles, breaking waves and hyperbolic jets at a free surface. J. Fluid Mech. 127, 103– 121.Google Scholar
Longuet-Higgins, M. S. 1983a Highly-accelerated, free-surface flows II. Space-periodic solutions. (To be submitted).
Longuet-Higgins, M. S. 1993b Highly-accelerated, free-surface flows. III. Inertial shocks of bounded intensity. (To be submitted).
Nishimura, H. & Takewaka, S. 1990 Experimental and numerical study on solitary wave breaking. In Coastal Engineering (Proc. ICCE Conf., Delft, Holland), vol. 1, pp. 1033– 1045. ASCE.
Peregrine, D. H. & Cooker, M. J. 1991 Violent motion as near-breaking waves meet a wall. Proc. IUTAM Symp. on Breaking Waves, Sydney, Australia, July 1991.