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

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’ assig...

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Bibliographic Details
Published in:Journal of fluid mechanics 2021-03, Vol.913, Article R5
Main Authors: Crowdy, D.G., Nelson, R.B., Krishnamurthy, V.S.
Format: Article
Language:English
Subjects:
Capillary waves
Computational fluid dynamics
Deformation
Exact solutions
Fluid mechanics
Fluids
HuR protein
JFM Rapids
Rotation
Shape
Surface tension
Vortices
Vorticity
Water waves
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Summary: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.
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2021.55