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Stability and dynamics of two-dimensional fully nonlinear gravity–capillary solitary waves in deep water

Published online by Cambridge University Press:  15 November 2016

Z. Wang
Z. Wang*
Affiliation:
Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China
*
†Email address for correspondence: z.wang5@bath.ac.uk
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Abstract

The stability and dynamics of two-dimensional gravity–capillary solitary waves in deep water within the fully nonlinear water-wave equations are numerically studied. It is well known that there are two families of symmetric gravity–capillary solitary waves – depression waves and elevation waves – bifurcating from infinitesimal periodic waves at the minimum of the phase speed. The stability of both branches was previously examined by Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) by means of a numerical spectral analysis. Their results show that the depression solitary waves with single-valued profiles are stable, while the elevation branch experiences a stability exchange at a turning point on the speed–amplitude curve. In the present paper, we provide numerical evidence that the depression solitary waves with an overhanging structure are also stable. On the other hand, Dias et al. (Eur. J. Mech. B, vol. 15, 1996, pp. 17–36) numerically traced the elevation branch and discovered that its speed–amplitude bifurcation curve features a ‘snake-like’ behaviour with many turning points, whereas Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) only considered the stability exchange near the first turning point. Our results reveal that the stability exchange occurs again near the second turning point. A branch of asymmetric solitary waves is also considered and found to be unstable, even when the wave profile consists of a depression wave and a stable elevation one. The excitation of stable gravity–capillary solitary waves is carried out via direct numerical simulations. In particular, the stable elevation waves, which feature two troughs connected by a small dimple, can be excited by moving two fully localised, well-separated pressures on the free surface with the speed slightly below the phase speed minimum and removing the pressures simultaneously after a period of time.

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 809 , 25 December 2016 , pp. 530 - 552
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Copyright
© 2016 Cambridge University Press 

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References

Akers, B. F. 2012 The generation of capillary-gravity solitary waves by a surface pressure forcing. Math. Comput. Simul. 82 (6), 958–967.Google Scholar
Calvo, D. C. & Akylas, T. R. 2002 Stability of steep gravity-capillary waves in deep water. J. Fluid Mech. 452, 123–143.Google Scholar
Cho, Y. & Akylas, T. R. 2009 Forced waves near resonance at a phase-speed minimum. Stud. Appl. Maths 123, 1–15.Google Scholar
Dias, F. & Iooss, G. 1993 Capillary-gravity solitary waves with damped oscillations. Physica D 65, 399–423.Google Scholar
Dias, F., Menasce, D. & Vanden-Broeck, J.-M. 1996 Numerical study of capillary-gravity solitary waves. Eur. J. Mech. B 15, 17–36.Google Scholar
Diorio, J. D., Cho, Y., Duncan, J. H. & Akylas, T. R. 2009 Gravity-capillary lumps generated by a moving pressure source. Phys. Rev. Lett. 103, 214502.Google Scholar
Diorio, J. D., Cho, Y., Duncan, J. H. & Akylas, T. R. 2011 Resonantly forced gravity-capillary lumps on deep water. Part I. Experiment. J. Fluid Mech. 672, 268–287.Google Scholar
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. D. & Sakharov, V. E. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 14 (12), 73–79.CrossRefGoogle Scholar
Dyachenko, A. I., Lvov, Y. V. & Zakharov, V. E. 1995 Five-wave interaction on the surface of deep water. Physica D 87, 233–261.Google Scholar
Falcon, L., Laroche, C. & Fauve, S. 2007 Observation of gravity-capillary wave turbulence. Phys. Rev. Lett. 98, 094503.Google Scholar
Grimshaw, R., Maleewong, M. & Asavanant, J. 2009 Stability of gravity-capillary waves generated by a moving pressure disturbance in water of finite depth. Phys. Fluids 21, 082101.CrossRefGoogle Scholar
Keller, J. B. 1985 Soliton generation and nonlinear waves propagation. Phil. Trans. R. Soc. Lond. A 315, 367–377.Google Scholar
Kim, B. & Akylas, T. R. 2007 Transverse instability of gravity-capillary solitary waves. J. Engng Maths 58, 167–175.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Longuet-Higgins, M. S. 1989 Capillary-gravity waves of solitary type on deep water. J. Fluid Mech. 200, 451–478.Google Scholar
Longuet-Higgins, M. S. & Zhang, X. 1997 Experiments on capillary-gravity waves of solitary type on deep water. Phys. Fluids 123, 1963–1968.Google Scholar
Milewski, P. A. & Tabak, E. G. 1999 A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows. SIAM J. Sci. Comput. 21 (3), 1102–1114.CrossRefGoogle Scholar
Milewski, P. A. & Vanden-Broeck, J.-M. 1999 Time-dependent gravity-capillary flows past an obstacle. Wave Motion 29, 63–79.Google Scholar
Milewski, P. A., Vanden-Broeck, J.-M. & Wang, Z. 2010 Dynamics of steep two-dimensional gravity-capillary solitary waves. J. Fluid Mech. 664, 466–477.CrossRefGoogle Scholar
Părău, E. I., Vanden-Broeck, J.-M. & Cooker, J. 2005 Nonlinear three-dimensional gravity-capillary solitary waves. J. Fluid Mech. 536, 99–105.Google Scholar
Saffman, P. G. 1985 The superharmonic instability of finite amplitude water waves. J. Fluid Mech. 159, 169–174.CrossRefGoogle Scholar
Tanaka, M. 1985 The stability of solitary waves. Phys. Fluids 29 (3), 650–655.CrossRefGoogle Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. SIAM.Google 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.Google Scholar
Wang, Z., Vanden-Broeck, J.-M. & Milewski, P. A. 2014 Asymmetric gravity-capillary solitary waves on deep water. J. Fluid Mech. 759, R2.Google Scholar
Yang, T. S. & Akylas, T. R. 1997 On asymmetric gravity-capillary solitary waves. J. Fluid Mech. 330, 215–232.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.Google Scholar
Zhu, Y. & Yang, J. K. 2007 Universal fractal structures in the weak interaction of solitary waves in generalised nonlinear Schrödinger equations. Phys. Rev. E 75, 036605.Google Scholar
Zufiria, J. A. 1987 Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth. J. Fluid Mech. 184, 183–206.Google Scholar