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Stokes waves with constant vorticity: folds, gaps and fluid bubbles

Published online by Cambridge University Press:  17 September 2019

Sergey A. Dyachenko Open the ORCID record for Sergey A. Dyachenko [Opens in a new window]  and
Vera Mikyoung Hur Open the ORCID record for Vera Mikyoung Hur [Opens in a new window]
Sergey A. Dyachenko*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
Vera Mikyoung Hur
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
*
†Email address for correspondence: sdyachen@math.uiuc.edu
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Abstract

The Stokes wave problem in a constant vorticity flow is formulated, by virtue of conformal mapping techniques, as a nonlinear pseudodifferential equation, involving the periodic Hilbert transform, which becomes the Babenko equation in the irrotational flow setting. The associated linearized operator is self-adjoint, whereby the modified Babenko equation is efficiently solved by the Newton-conjugate gradient method. For strong positive vorticity, a ‘fold’ appears in the wave speed versus amplitude plane, and a ‘gap’ as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself at the trough line, enclosing a bubble of air. More folds and gaps follow for stronger vorticity. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely or, equivalently, gravitational acceleration vanishes, while the profile encloses a circular bubble of fluid in rigid body rotation at the ends of the gaps. Touching waves at the beginnings of the second gaps tend to the circular vortex wave on top of the limiting Crapper wave in the infinite vorticity limit, or the zero gravity limit, and the circular vortex wave on top of itself at the ends of the gaps. Touching waves for higher gaps accommodate more circular bubbles of fluid.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 878 , 10 November 2019 , pp. 502 - 521
Copyright
© 2019 Cambridge University Press 

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