Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-06T14:02:56.868Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential asymptotics for steady parasitic capillary ripples on steep gravity waves

Published online by Cambridge University Press:  30 March 2022

Josh Shelton Open the ORCID record for Josh Shelton [Opens in a new window]  and
Philippe H. Trinh Open the ORCID record for Philippe H. Trinh [Opens in a new window]
Josh Shelton*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Philippe H. Trinh*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
*
†Email addresses for correspondence: j.shelton@bath.ac.uk, p.trinh@bath.ac.uk
†Email addresses for correspondence: j.shelton@bath.ac.uk, p.trinh@bath.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we develop an asymptotic theory for steadily travelling gravity–capillary waves under the small-surface tension limit. In an accompanying work (Shelton et al., J. Fluid Mech., vol. 922, 2021) it was demonstrated that solutions associated with a perturbation about a leading-order gravity wave (a Stokes wave) contain surface-tension-driven parasitic ripples with an exponentially small amplitude. Thus, a naive Poincaré expansion is insufficient for their description. Here, we develop specialised methodologies in exponential asymptotics for derivation of the parasitic ripples on periodic domains. The ripples are shown to arise in conjunction with Stokes lines and the Stokes phenomenon. The resultant analysis associates the production of parasitic ripples to the complex-valued singularities associated with the crest of a steep Stokes wave. A solvability condition is derived, showing that solutions of this type do not exist at certain values of the Bond number. The asymptotic results are compared with full numerical solutions and show excellent agreement. The work provides corrections and insight of a seminal theory on parasitic capillary waves first proposed by Longuet-Higgins (J. Fluid Mech., vol. 16, issue 1, 1963, pp. 138–159).

JFM classification

Interfacial Flows (free surface): Capillary flows Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 939 , 25 May 2022 , A17
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Berry, M.V. 1989 Uniform asymptotic smoothing of Stokes's discontinuities. Proc. R. Soc. Lond. A 422 (1862), 7–21.Google Scholar
Boyd, J.P. 1998 Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics. Kluwer Academic.CrossRefGoogle Scholar
Chandler, G.A. & Graham, I.G. 1993 The computation of water waves modelled by Nekrasov's equation. SIAM J. Numer. Anal. 30 (4), 1041–1065.CrossRefGoogle Scholar
Chapman, S.J. 1999 On the role of Stokes lines in the selection of Saffman–Taylor fingers with small surface tension. Eur. J. Appl. Maths 10 (6), 513–534.CrossRefGoogle Scholar
Chapman, S.J., King, J.R. & Adams, K.L. 1998 Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Proc. R. Soc. Lond. A 454, 2733–2755.CrossRefGoogle Scholar
Chapman, S.J. & Mortimer, D.B. 2005 Exponential asymptotics and Stokes lines in a partial differential equation. Proc. R. Soc. Lond. A 461 (2060), 2385–2421.Google Scholar
Chapman, S.J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.CrossRefGoogle Scholar
Crapper, G.D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2 (6), 532–540.CrossRefGoogle Scholar
Crew, S.C. & Trinh, P.H. 2016 New singularities for Stokes waves. J. Fluid Mech. 798, 256–283.CrossRefGoogle Scholar
Deike, L., Popinet, S. & Melville, W.K. 2015 Capillary effects on wave breaking. J. Fluid Mech. 769, 541–569.CrossRefGoogle Scholar
Dingle, R.B. 1973 Asymptotic Expansions: Their Derivation and Interpretation. Academic.Google Scholar
Fedorov, A.V. & Melville, W.K. 1998 Nonlinear gravity–capillary waves with forcing and dissipation. J. Fluid Mech. 354, 1–42.CrossRefGoogle Scholar
Gao, T., Wang, Z. & Vanden-Broeck, J.-M. 2017 Investigation of symmetry breaking in periodic gravity–capillary waves. J. Fluid Mech. 811, 622–641.CrossRefGoogle Scholar
Grant, M.A. 1973 The singularity at the crest of a finite amplitude progressive Stokes wave. J. Fluid Mech. 59 (2), 257–262.CrossRefGoogle Scholar
Hung, L.-P. & Tsai, W.-T. 2009 The formation of parasitic capillary ripples on gravity–capillary waves and the underlying vortical structures. J. Phys. Oceanogr. 39 (2), 263–289.CrossRefGoogle Scholar
Longuet-Higgins, M.S. 1963 The generation of capillary waves by steep gravity waves. J. Fluid Mech. 16, 138–159.CrossRefGoogle Scholar
Longuet-Higgins, M.S. 1995 Parasitic capillary waves: a direct calculation. J. Fluid Mech. 301, 79–107.CrossRefGoogle Scholar
Longuet-Higgins, M.S. & Fox, M.J.H. 1977 Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80 (04), 721–741.CrossRefGoogle Scholar
Longuet-Higgins, M.S. & Fox, M.J.H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85 (04), 769–786.CrossRefGoogle Scholar
Lustri, C.J., Pethiyagoda, R. & Chapman, S.J. 2019 Three-dimensional capillary waves due to a submerged source with small surface tension. J. Fluid Mech. 863, 670–701.CrossRefGoogle Scholar
Lustri, C.J. 2013 Exponential asymptotics in unsteady and three-dimensional flows. PhD thesis, Oxford University.Google Scholar
Mailybaev, A.A. & Nachbin, A. 2019 Explosive ripple instability due to incipient wave breaking. J. Fluid Mech. 863, 876–892.CrossRefGoogle Scholar
Murashige, S. & Choi, W. 2017 A numerical study on parasitic capillary waves using unsteady conformal mapping. J. Comput. Phys. 328, 234–257.CrossRefGoogle Scholar
Olde Daalhuis, A.B., Chapman, S.J., King, J.R., Ockendon, J.R. & Tew, R.H. 1995 Stokes phenomenon and matched asymptotic expansions. SIAM J. Appl. Maths 55 (6), 1469–1483.CrossRefGoogle Scholar
Perlin, M., Lin, H. & Ting, C.-L. 1993 On parasitic capillary waves generated by steep gravity waves: an experimental investigation with spatial and temporal measurements. J. Fluid Mech. 255, 597–620.CrossRefGoogle Scholar
Perlin, M. & Schultz, W.W. 2000 Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32 (1), 241–274.CrossRefGoogle Scholar
Shelton, J., Milewski, P. & Trinh, P.H. 2021 On the structure of steady parasitic gravity–capillary waves in the small surface tension limit. J. Fluid Mech. 922, A16.CrossRefGoogle Scholar
Tanveer, S. & Xie, X. 2003 Analyticity and nonexistence of classical steady Hele-Shaw fingers. Commun. Pure Appl. Maths 56 (3), 353–402.CrossRefGoogle Scholar
Trinh, P.H. 2017 On reduced models for gravity waves generated by moving bodies. J. Fluid Mech. 813, 824–859.CrossRefGoogle Scholar
Trinh, P.H., Chapman, S.J. & Vanden-Broeck, J.-M. 2011 Do waveless ships exist? Results for single-cornered hulls. J. Fluid Mech. 685, 413–439.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 1986 Steep gravity waves: Havelock's method revisited. Phys. Fluids 29 (9), 3084–3085.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 2010 Gravity-Capillary Free-Surface Flows. Cambridge University Press.CrossRefGoogle Scholar
Wilkening, J. & Zhao, X. 2021 Quasi-periodic travelling gravity–capillary waves. J. Fluid Mech. 915, A7.CrossRefGoogle Scholar
Wilton, J.R. 1915 On ripples. Phil. Mag. 29 (173), 688–700.CrossRefGoogle Scholar
Yang, T.-S. & Akylas, T.R. 1996 Weakly nonlocal gravity–capillary solitary waves. Phys. Fluids 8 (6), 1506–1514.CrossRefGoogle Scholar
Yang, T.-S. & Akylas, T.R. 1997 On asymmetric gravity–capillary solitary waves. J. Fluid Mech. 330, 215–232.CrossRefGoogle Scholar
Zufiria, J.A. 1987 a Non-symmetric gravity waves on water of infinite depth. J. Fluid Mech. 181, 17–39.CrossRefGoogle Scholar
Zufiria, J.A. 1987 b Symmetry breaking in periodic and solitary gravity–capillary waves on water of finite depth. J. Fluid Mech. 184, 183–206.CrossRefGoogle Scholar