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Density and surface tension effects on vortex stability. Part 1. Curvature instability

Published online by Cambridge University Press:  23 February 2021

Ching Chang Open the ORCID record for Ching Chang [Opens in a new window]  and
Stefan G. Llewellyn Smith Open the ORCID record for Stefan G. Llewellyn Smith [Opens in a new window]
Ching Chang*
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
Stefan G. Llewellyn Smith
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA Scripps Institution of Oceanography, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0213, USA
*
†Email address for correspondence: [email protected]
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Abstract

The curvature instability of thin vortex rings is a parametric instability discovered from short-wavelength analysis by Hattori & Fukumoto (Phys. Fluids, vol. 15, 2003, pp. 3151–3163). A full-wavelength analysis using normal modes then followed in Fukumoto & Hattori (J. Fluid Mech., vol. 526, 2005, pp. 77–115). The present work extends these results to the case with different densities inside and outside the vortex core in the presence of surface tension. The maximum growth rate and the instability half-bandwidth are calculated from the dispersion relation given by the resonance between two Kelvin waves of $m$ and $m+1$, where $m$ is the azimuthal wavenumber. The result shows that vortex rings are unstable to resonant waves in the presence of density and surface tension. The curvature instability for the principal modes is enhanced by density variations in the small axial wavenumber regime, while the asymptote for short wavelengths is close to the constant density case. The effect of surface tension is marginal. The growth rates of non-principal modes are also examined, and long waves are most unstable.

JFM classification

Instability: Parametric instability Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A14
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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