Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T18:13:08.735Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bayly, B.J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.CrossRefGoogle ScholarPubMed
Blanco-Rodriguez, F.J. & Le Dizès, S. 2016 Elliptic instability of a curved Batchelor vortex. J. Fluid Mech. 804, 224–247.CrossRefGoogle Scholar
Blanco-Rodriguez, F.J. & Le Dizès, S. 2017 Curvature instability of a curved Batchelor vortex. J. Fluid Mech. 814, 397–415.CrossRefGoogle Scholar
Chang, C. & Llewellyn Smith, S.G. 2021 Density and surface tension effects on vortex ring stability. Part 1. Curvature instability. J. Fluid Mech. 913, A14.Google Scholar
Crow, S.C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.CrossRefGoogle Scholar
Dixit, H.N. & Govindarajan, R. 2011 Stability of a vortex in radial density stratification: role of wave interactions. J. Fluid Mech. 679, 582–615.CrossRefGoogle Scholar
Éloy, C. & Le Dizès, S. 2001 Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13, 660–676.CrossRefGoogle Scholar
Fukumoto, Y. 2003 The three-dimensional instability of a strained vortex tube revisited. J. Fluid Mech. 493, 287–318.CrossRefGoogle Scholar
Fukumoto, Y. & Hattori, Y. 2005 Curvature instability of a vortex ring. J. Fluid Mech. 526, 77–115.CrossRefGoogle Scholar
Hattori, Y., Blanco-Rodriguez, F.J. & Le Dizès, S. 2019 Numerical stability analysis of a vortex ring with swirl. J. Fluid Mech. 878, 5–36.CrossRefGoogle Scholar
Hattori, Y. & Fukumoto, Y. 2003 Short-wavelength stability analysis of thin vortex rings. Phys. Fluids 15, 3151–3163.CrossRefGoogle Scholar
Joly, L., Fontane, J. & Chassaing, P. 2005 The Rayleigh–Taylor instability of two-dimensional high-density vortices. J. Fluid Mech. 537, 415–431.CrossRefGoogle Scholar
Leweke, T. & Williamson, C.H.K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.CrossRefGoogle Scholar
Moore, D.W. & Saffman, P.G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence and its Detection (ed. J.H. Olsen, A. Goldburg & M. Rogers), pp. 339–354. Plenum.CrossRefGoogle Scholar
Moore, D.W. & Saffman, P.G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Saunders, P.M. 1973 The instability of a baroclinic vortex. J. Phys. Oceanogr. 3, 61–65.2.0.CO;2>CrossRefGoogle Scholar
Sipp, D., Fabre, D., Michelin, S. & Jacquin, L. 2005 Stability of a vortex with a heavy core. J. Fluid Mech. 526, 67–76.CrossRefGoogle Scholar
Tsai, C.Y. & Widnall, S.E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721–733.CrossRefGoogle Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76–80.CrossRefGoogle Scholar
Widnall, S.E., Bliss, D.B. & Tsai, C.Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35–47.CrossRefGoogle Scholar
Supplementary material: File

Chang and Smith supplementary material

Chang and Smith supplementary material

Download Chang and Smith supplementary material(File)
File 10.6 KB