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Exact free surfaces in constant vorticity flows

Published online by Cambridge University Press:  26 May 2020

Vera Mikyoung Hur Open the ORCID record for Vera Mikyoung Hur [Opens in a new window]  and
Miles H. Wheeler Open the ORCID record for Miles H. Wheeler [Opens in a new window]
Vera Mikyoung Hur*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Miles H. Wheeler*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
*
†Email addresses for correspondence: verahur@math.uiuc.edu, mw2319@bath.ac.uk
†Email addresses for correspondence: verahur@math.uiuc.edu, mw2319@bath.ac.uk
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Abstract

We present an exact solution for periodic travelling waves in two-dimensional, infinitely deep and constant vorticity flows, in the absence of the effects of gravity or surface tension. The shape of the free surface is the same as for Crapper’s celebrated capillary waves in an irrotational flow, but the flow beneath the wave, which is also explicit, is completely different. This confirms a conjecture made by Dyachenko & Hur (J. Fluid Mech., vol. 878, 2019b, pp. 502–521; Stud. Appl. Maths, vol. 142 (2), 2019c, pp. 162–189) and Hur & Vanden-Broeck (Eur. J. Mech. (B/Fluids), 2020, to appear), based on numerical and asymptotic evidence.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 896 , 10 August 2020 , R1
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Akers, B. F., Ambrose, D. M. & Wright, J. D. 2014 Gravity perturbed Crapper waves. Proc. R. Soc. Lond. Ser. A 470 (2161), 20130526.CrossRefGoogle Scholar
Chen, B. & Saffman, P. G. 1979 Steady gravity-capillary waves on deep water. I. Weakly nonlinear waves. Stud. Appl. Maths 60 (3), 183–210.CrossRefGoogle Scholar
Chen, B. & Saffman, P. G. 1980 Steady gravity-capillary waves on deep water. II. Numerical results for finite amplitude. Stud. Appl. Maths 62 (2), 95–111.CrossRefGoogle Scholar
Constantin, A. 2001 On the deep water wave motion. J. Phys. A 34 (7), 1405–1417.CrossRefGoogle Scholar
Constantin, A., Strauss, W. & Vărvărucă, E. 2016 Global bifurcation of steady gravity water waves with critical layers. Acta Math. 217 (2), 195–262.CrossRefGoogle Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532–540.CrossRefGoogle Scholar
Dyachenko, S. A. & Hur, V. M. 2019a Stokes waves in a constant vorticity flow. In Nonlinear Water Waves, pp. 71–86. Birkhäuser.CrossRefGoogle Scholar
Dyachenko, S. A. & Hur, V. M. 2019b Stokes waves with constant vorticity: folds, gaps and fluid bubbles. J. Fluid Mech. 878, 502–521.CrossRefGoogle Scholar
Dyachenko, S. A. & Hur, V. M. 2019c Stokes waves with constant vorticity: I. Numerical computation. Stud. Appl. Maths 142 (2), 162–189.CrossRefGoogle Scholar
Gerstner, F. 1802 Theorie der wellen. In Abhand. Koen. Boehmischen Gesel. Wiss.Google Scholar
Hur, V. M. & Vanden-Broeck, J.-M. 2020 A new application of Crapper’s exact solution to waves in constant vorticity flows. Eur. J. Mech. (B/Fluids) doi:10.1016/j.euromechflu.2020.04.015.CrossRefGoogle Scholar
King, F. W. 2009 Hilbert Transforms. Vol. 1, Encyclopedia of Mathematics and its Applications, vol. 124. Cambridge University Press.Google Scholar
Kinnersley, W. 1976 Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77 (2), 229–241.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J. Fluid Mech. 240, 659–679.CrossRefGoogle Scholar
Ribeiro, R. Jr, Milewski, P. A. & Nachbin, A. 2017 Flow structure beneath rotational water waves with stagnation points. J. Fluid Mech. 812, 792–814.CrossRefGoogle Scholar
Schwartz, L. W. & Vanden-Broeck, J.-M. 1979 Numerical solution of the exact equations for capillary-gravity waves. J. Fluid Mech. 95 (1), 119–139.CrossRefGoogle Scholar
Teles da Silva, A. F. & Peregrine, D. H. 1988 Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302.CrossRefGoogle Scholar
Simmen, J. A. & Saffman, P. G. 1985 Steady deep-water waves on a linear shear current. Stud. Appl. Maths 73 (1), 35–57.CrossRefGoogle Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Stokes, G. G. 1880 Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form. In Mathematical and Physical Papers, vol. 1, pp. 314–326. Cambridge University Press.Google Scholar
Wahlén, E. 2009 Steady water waves with a critical layer. J. Differ. Equ. 246 (6), 2468–2483.CrossRefGoogle Scholar