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Dynamics of steep two-dimensional gravity–capillary solitary waves

Published online by Cambridge University Press:  01 November 2010

PAUL A. MILEWSKI ,
J.-M. VANDEN-BROECK  and
ZHAN WANG
PAUL A. MILEWSKI*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA
J.-M. VANDEN-BROECK
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
ZHAN WANG
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA
*
†Email address for correspondence: milewski@math.wisc.edu
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Abstract

In this paper, the unsteady evolution of two-dimensional fully nonlinear free-surface gravity–capillary solitary waves is computed numerically in infinite depth. Gravity–capillary wavepacket-type solitary waves were found previously for the full Euler equations, bifurcating from the minimum of the linear dispersion relation. Small and moderate amplitude elevation solitary waves, which were known to be linearly unstable, are shown to evolve into stable depression solitary waves, together with a radiated wave field. Depression waves and certain large amplitude elevation waves were found to be robust to numerical perturbations. Two kinds of collisions are computed: head-on collisions whereby the waves are almost unchanged, and overtaking collisions which are either almost elastic if the wave amplitudes are both large or destroy the smaller wave in the case of a small amplitude wave overtaking a large one.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 664 , 10 December 2010 , pp. 466 - 477
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Akers, B. & Milewski, P. A. 2008 Model equations for gravity-capillary waves in deep water. Stud. Appl. Maths 121, 49–69.CrossRefGoogle Scholar
Akers, B. & Milewski, P. A. 2009 A model equation for wavepacket solitary waves arising from capillary-gravity flows. Stud. Appl. Maths 122, 249–274.CrossRefGoogle Scholar
Akers, B. & Milewski, P. A. 2010 Dynamics of three dimensional gravity-capillary solitary waves in deep water. SIAM J. Appl. Maths 70, 2390–2408.CrossRefGoogle Scholar
Berger, K. & Milewski, P. A. 2000 The generation and evolution of lump solitary waves in surface-tension-dominated flows. SIAM J. Appl. Maths 61, 731–750.CrossRefGoogle Scholar
Calvo, D. C., Yang, T. S. & Akylas, T. R. 2002 Stability of steep gravity-capillary waves in deep water. J. Fluid Mech. 452, 123–143.CrossRefGoogle Scholar
Dias, F. & Iooss, G. 1993 Capillary-gravity solitary waves with damped oscillations. Physica D 65, 399–423.CrossRefGoogle Scholar
Dias, F., Menasce, D. & Vanden-Broeck, J.-M. 1996 Numerical study of capillary-gravity solitary waves. Eur. J. Mech. B: Fluids 15, 17–36.Google Scholar
Diorio, J., Cho, Y., Duncan, J. H. & Akylas, T. R. 2009 Gravity-capillary lumps generated by a moving pressure source. Phys. Rev. Lett. 103, 214502.CrossRefGoogle ScholarPubMed
Dyachenko, A. L., Zakharov, V. E. & Kuznetsov, E. A. 1996 Nonlinear dynamics on the free surface of an ideal fluid. Plasma Phys. Rep. 22, 916–928.Google Scholar
Falcon, E., Laroche, C. & Fauve, S. 2002 Observation of depression solitary surface waves on a thin fluid layer. Phys. Rev. Lett. 89, 204501.CrossRefGoogle ScholarPubMed
Grimshaw, R., Maleewong, M. & Asanavant, J. 2009 Stability of gravity-capillary waves generated by a moving pressure disturbance in water of finite depth. Phys. Fluids 21, 082101.CrossRefGoogle Scholar
Groves, M. D. & Wahlén, E. 2010 On the existence and conditional energetic stability of solitary water waves with weak surface tension. C. R. Acad. Sci. Paris I 348, 397–402.Google Scholar
Iooss, G. & Kirrmann, P. 1996 Capillary gravity waves on the free surface of an inviscid fluid of infinite depth: existence of solitary waves. Arch. Rat. Mech. Anal. 136, 1–19.CrossRefGoogle Scholar
Kadomtsev, B. B. & Petviashvilli, V. I. 1970 On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539541.Google Scholar
Kim, B. & Akylas, T. R. 2007 Transverse instability of gravity-capillary solitary waves. J. Engng Maths 58, 167175.CrossRefGoogle Scholar
Li, Y. A., Hyman, R. J. M. & Choi, W. 2004 A numerical study of the exact evolution equations for surface waves in water of finite depth. Stud. Appl. Maths 113, 303–324.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1989 Capillary-gravity waves of solitary type on deep water. J. Fluid Mech. 200, 451–478.CrossRefGoogle Scholar
Malomed, B. & Vanden-Broeck, J.-M. 1996 Solitary wave interactions for the fifth-order KdV equation. Math. Probl. Theory Water Waves, Contemp. Maths 200, 133–143.CrossRefGoogle Scholar
Parau, E. I., Vanden-Broeck, J.-M. & Cooker, M. J. 2005 Nonlinear three-dimensional gravity-capillary solitary waves. J. Fluid Mech. 536, 99–105.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. & Dias, F. 1992 Gravity-capillary solitary waves in water of infinite depth and related free-surface flows. J. Fluid Mech. 240, 549–557.CrossRefGoogle Scholar
Zhang, X. 1995 Capillary-gravity and capillary waves generated in a wind wave tank: observations and theory. J. Fluid Mech. 289, 51–82.CrossRefGoogle Scholar