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Density and surface tension effects on vortex stability. Part 2. Moore–Saffman–Tsai–Widnall 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, CA92093-0411, USA
Stefan G. Llewellyn Smith
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, 9500 Gilman Drive, La Jolla, CA92093-0411, USA Scripps Institution of Oceanography, UCSD, 9500 Gilman Drive, La Jolla, CA92093-0213, USA
*
†Email address for correspondence: [email protected]
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Abstract

The Moore–Saffman–Tsai–Widnall (MSTW) instability is a parametric instability that arises in strained vortex columns. The strain is assumed to be weak and perpendicular to the vortex axis. In this second part of our investigation of vortex instability including density and surface tension effects, a linear stability analysis for this situation is presented. The instability is caused by resonance between two Kelvin waves with azimuthal wavenumber separated by two. The dispersion relation for Kelvin waves and resonant modes are obtained. Results show that the stationary resonant waves for $m=\pm 1$ are more unstable when the density ratio $\rho_2/\rho_1$, the ratio of vortex to ambient fluid density, approaches zero, whereas the growth rate is maximised near $\rho _2/\rho _1 =0.215$ for the resonance $(m,m+2)=(0,2)$. Surface tension suppresses the instability, but its effect is less significant than that of density. As the azimuthal wavenumber $m$ increases, the MSTW instability decays, in contrast to the curvature instability examined in Part 1 (Chang & Llewellyn Smith, J. Fluid Mech. vol. 913, 2021, A14).

JFM classification

Instability: Parametric instability Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A15
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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