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‘H-states’: exact solutions for a rotating hollow vortex

Published online by Cambridge University Press:  01 March 2021

D.G. Crowdy Open the ORCID record for D.G. Crowdy [Opens in a new window] ,
R.B. Nelson Open the ORCID record for R.B. Nelson [Opens in a new window]  and
V.S. Krishnamurthy Open the ORCID record for V.S. Krishnamurthy [Opens in a new window]
D.G. Crowdy*
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
R.B. Nelson
Affiliation:
Department of Earth Science and Engineering, Imperial College London, LondonSW7 2AZ, UK
V.S. Krishnamurthy
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria
*
†Email address for correspondence: d.crowdy@imperial.ac.uk
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Abstract

Exact solutions are found for an $N$-fold rotationally symmetric, steadily rotating hollow vortex where a continuous real parameter governs its deformation from a circular shape and $N \ge 2$ is an integer. The vortex shape is found as part of the solution. Following the designation ‘V-states’ assigned to steadily rotating vortex patches (Deem & Zabusky, Phys. Rev. Lett., vol. 40, 1978, pp. 859–862) we call the analogous rotating hollow vortices ‘H-states’. Unlike V-states where all but the $N=2$ solution – the Kirchhoff ellipse – must be found numerically, it is shown that all H-state solutions can be written down in closed form. Surface tension is not present on the boundaries of the rotating H-states but the latter are shown to be intimately related to solutions for a non-rotating hollow vortex with surface tension on its boundary (Crowdy, Phys. Fluids, vol. 11, 1999a, pp. 2836–2845). It is also shown how the results here relate to recent work on constant-vorticity water waves (Hur & Wheeler, J. Fluid Mech., vol. 896, 2020, R1) where a connection to classical capillary waves (Crapper, J. Fluid Mech., vol. 2, 1957, pp. 532–540) is made.

JFM classification

Vortex Flows: Vortex dynamics
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , R5
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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