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Finite rotating and translating vortex sheets

Published online by Cambridge University Press:  29 July 2021

Bartosz Protas Open the ORCID record for Bartosz Protas [Opens in a new window] ,
Stefan G. Llewellyn Smith Open the ORCID record for Stefan G. Llewellyn Smith [Opens in a new window]  and
Takashi Sakajo Open the ORCID record for Takashi Sakajo [Opens in a new window]
Bartosz Protas*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada
Stefan G. Llewellyn Smith
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, La Jolla, CA 92093-0411, USA Scripps Institution of Oceanography, UCSD, La Jolla, CA 92093-0209, USA
Takashi Sakajo
Affiliation:
Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
*
†Email address for correspondence: [email protected]
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Abstract

We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyse equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite aspect ratio limit of the stability results known for Kirchhoff's ellipse (Love, Proc. Lond. Math. Soc., vol. s1–25, 1893, pp. 18–43; Mitchell & Rossi, Phys. Fluids, vol. 20, 2008, p. 054103) and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite aspect ratio limits of the analogous solutions known for vortex patches.

JFM classification

Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A23
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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