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).
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Density and surface tension effects on vortex stability. Part 1. Curvature instability
The curvature instability of thin vortex rings is a parametric instability discovered from shortwavelength analysis by Hattori & Fukumoto ( Phys. Fluids , vol. 15, 2003, pp. 3151–3163). A fullwavelength 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 halfbandwidth 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 nonprincipal modes are also examined, and long waves are most unstable.
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 Award ID(s):
 1706934
 NSFPAR ID:
 10312404
 Date Published:
 Journal Name:
 Journal of Fluid Mechanics
 Volume:
 913
 ISSN:
 00221120
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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