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A local model for the limiting configuration of interfacial solitary waves

Published online by Cambridge University Press:  25 June 2021

X. Guan ,
J.-M. Vanden-Broeck ,
Z. Wang Open the ORCID record for Z. Wang [Opens in a new window]  and
F. Dias Open the ORCID record for F. Dias [Opens in a new window]
X. Guan
Affiliation:
Institute of Mechanics, Chinese Academy of Sciences, Beijing100190, PR China Department of Mathematics, University College London, LondonWC1E 6BT, UK
J.-M. Vanden-Broeck
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
Z. Wang*
Affiliation:
Institute of Mechanics, Chinese Academy of Sciences, Beijing100190, PR China
F. Dias
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin4, Ireland
*
†Email address for correspondence: zwang@imech.ac.cn
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Abstract

The limiting configuration of interfacial solitary waves between two homogeneous fluids consisting of a sharp $120^{\circ }$ angle with an enclosed bubble of stagnant heavier fluid on top is investigated numerically. We use a boundary integral equation method to compute the almost limiting profiles which are nearly self-intersecting and thus extend the work of Pullin & Grimshaw (Phys. Fluids, vol. 31, 1988, pp. 3550–3559) by obtaining the overhanging solutions for very small density ratios. To further study the local configuration of the limiting profile, we propose a reduced model that replaces the $120^{\circ }$ angle with two straight solid walls intersecting at the bottom of the bubble. Using a series truncation method, a one-parameter family of solutions depending on the angle between the two solid walls (denoted by $\gamma$) is found. When $\gamma = {2{\rm \pi} }/{3}$, it is shown that the simplified model agrees well with the near-limiting wave profile if the density ratio is small, and thus provides a good local approximation to the assumed limiting configuration. Interesting solutions for other values of $\gamma$ are also explored.

JFM classification

Waves/Free-surface Flows: Solitary waves Geophysical and Geological Flows: Internal waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 921 , 25 August 2021 , A9
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Amick, C.J., Fraenkel, L.E. & Toland, J.F. 1982 On the stokes conjecture for the wave of extreme form. Acta Math. 148 (1), 193–214.CrossRefGoogle Scholar
Amick, C.J. & Turner, R.E.L. 1986 A global theory of internal solitary waves in two-fluid systems. Trans. Am. Math. Soc. 298 (2), 431–484.CrossRefGoogle Scholar
Benjamin, T.B. & Bridges, T.J. 1997 Reappraisal of the Kelvin–Helmholtz problem. Part 2. Interaction of the Kelvin–Helmholtz, superharmonic and Benjamin–Feir instabilities. J. Fluid Mech. 333, 327–373.CrossRefGoogle Scholar
Daboussy, D., Dias, F. & Vanden-Broeck, J.-M. 1998 Gravity flows with a free surface of finite extent. Eur. J. Mech. B/Fluids 17, 19–31.CrossRefGoogle Scholar
Funakoshi, M. & Oikawa, M. 1986 Long internal waves of large amplitude in a two-layer fluid. J. Phys. Soc. Japan 55, 128–144.CrossRefGoogle Scholar
Grimshaw, R.H.J. & Pullin, D.I. 1986 Extreme interfacial waves. Phys. Fluids 29 (9), 2802–2807.CrossRefGoogle Scholar
Havelock, T.H. 1918 Periodic irrotational waves of finite height. Proc. R. Soc. A 665, 38–51.Google Scholar
Hunter, J.K. & Vanden-Broeck, J.-M. 1983 Accurate computations for steep solitary waves. J. Fluid Mech. 136, 62–71.CrossRefGoogle Scholar
Kataoka, T. 2006 The stability of finite-amplitude interfacial solitary waves. Fluid Dyn. Res. 38, 831–867.CrossRefGoogle Scholar
Lenau, C.W. 1966 The solitary wave of maximum amplitude. J. Fluid Mech. 26, 309–320.CrossRefGoogle Scholar
Longuet-Higgins, M.S. 1973 On the form of the highest progressive and standing waves in deep water. Proc. R. Soc. A 331, 445–456.Google Scholar
Longuet-Higgins, M.S. & Tanaka, M. 1997 On the crest instabilities of steep surface waves. J. Fluid Mech. 336, 51–68.CrossRefGoogle Scholar
Maklakov, D.V. 2020 A note on the existence of pure gravity waves at the interface of two fluids. Physica D 401, 132157.CrossRefGoogle Scholar
Maklakov, D.V. & Sharipov, R.R. 2018 Almost limiting configurations of steady interfacial overhanging gravity waves. J. Fluid Mech. 856, 673–708.CrossRefGoogle Scholar
Meiron, D.I. & Saffman, P.G. 1983 Overhanging interfacial gravity waves of large amplitude. J. Fluid Mech. 129, 213–218.CrossRefGoogle Scholar
Okamoto, H. & Shoji, M. 2001 The Mathematical Theory of Permanent Progressive Water–waves. Advanced Series in Nonlinear Dynamics, vol. 20. World Scientific.CrossRefGoogle Scholar
Pullin, D.I. & Grimshaw, R.H.J. 1988 Finite-amplitude solitary waves at the interface between two homogeneous fluids. Phys. Fluids 31 (12), 3550–3559.CrossRefGoogle Scholar
Rusås, P.-O. & Grue, J. 2002 Solitary waves and conjugate flows in a three-layer fluid. Eur. J. Mech. B/Fluids 21, 185–206.CrossRefGoogle Scholar
Schwartz, L.W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553–578.CrossRefGoogle Scholar
Saffman, P.G. & Yuen, H.C. 1982 Finite-amplitude interfacial waves in the presence of a current. J. Fluid Mech. 123, 459–476.CrossRefGoogle Scholar
Sha, H. & Vanden-Broeck, J.-M. 1993 Two-layer flows past a semicircular obstruction. Phys. Fluids A 5, 2661–2668.CrossRefGoogle Scholar
Taylor, G.I. 1953 An experimental study of standing waves. Proc. R. Soc. A 218, 44–59.Google Scholar
Turner, R.E.L. & Vanden-Broeck, J.-M. 1986 The limiting configuration of interfacial gravity waves. Phys. Fluids 29 (2), 372–375.CrossRefGoogle Scholar
Turner, R.E.L. & Vanden-Broeck, J.-M. 1988 Broadening of interfacial solitary waves. Phys. Fluids 31 (9), 2486–2490.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 2010 Gravity-Capillary Free-Surface Flows. Cambridge University Press.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. & Schwartz, L.W. 1979 Numerical computation of steep gravity waves in shallow water. Phys. Fluids 22 (10), 1868–1871.CrossRefGoogle Scholar
Villermaux, E. & Pomeau, Y. 2010 Super free fall. J. Fluid Mech. 642, 147–157.CrossRefGoogle Scholar
Yamada, H. 1957 a Highest waves of permanent type on the surface of deep water. Rep. Res. Inst. Appl. Mech. Kyushu Univ. 5, 37–52.Google Scholar
Yamada, H. 1957 b On the highest solitary wave. Rep. Res. Inst. Appl. Mech. Kyushu Univ. 5, 53–67.Google Scholar