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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: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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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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