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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

REFERENCES

Baker, G. R., Meiron, D. I. & Orszag, S. A. 1980 Vortex simulations of the Rayleigh–Taylor instability. Phys. Fluids 23, 1485–1490.CrossRefGoogle Scholar
Baumann, N., Joseph, D. D. & Mohr, P. 1992 Vortex rings of one fluid in another in free fall. Phys. Fluids A 4, 567–580.CrossRefGoogle Scholar
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
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14.CrossRefGoogle Scholar
Chang, C. & Llewellyn Smith, S. G. 2018 The motion of a buoyant vortex filament. J. Fluid Mech. 857, R1.CrossRefGoogle Scholar
Chang, C. & Llewellyn Smith, S. G. 2021 Density and surface tension effects on vortex stability. Part 2. Moore–Saffman–Tsai–Widnall instability. J. Fluid Mech. 913, A15.Google Scholar
Eloy, 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, 415–432.CrossRefGoogle Scholar
Krein, M. G. 1950 A generalization of several investigations of A. M. Liapunov on linear differential equations with periodic coefficients. Dokl. Akad. Nauk SSSR 73, 445–448.Google Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.CrossRefGoogle Scholar
MacKay, R. S. 1986 Stability of equilibria of Hamiltonian systems. In Nonlinear Phenomena and Chaos (ed. S. Sarkar), pp. 254–270. Adam Hilger.Google Scholar
Marten, K., Shariff, K., Psarakos, S. & White, D. J. 1996 Ring bubbles of dolphins. Sci. Am. 275, 82–87.CrossRefGoogle ScholarPubMed
Maxworthy, T. 1972 The structure and stability of vortex rings. J. Fluid Mech. 51, 15–32.CrossRefGoogle Scholar
Maxworthy, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465–495.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
Pedley, T. J. 1968 The toroidal bubble. J. Fluid Mech. 32, 97–112.CrossRefGoogle Scholar
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625–639.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google 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
Turner, J. S. 1957 Buoyant vortex rings. Proc. R. Soc. Lond. A 239, 61–75.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76–80.CrossRefGoogle Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663.CrossRefGoogle Scholar
Widnall, S. E. & Bliss, D. B. 1971 Slender-body analysis of the motion and stability of a vortex filament containing an axial flow. J. Fluid Mech. 50, 335–353.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
Widnall, S. E. & Sullivan, J. P. 1973 On the instability of thin vortex rings. Proc. R. Soc. Lond. A 332, 335–353.Google Scholar
Widnall, S. E. & Tsai, C. Y. 1977 The instability of the thin vortex ring of constant vorticity. Phil. Trans. R. Soc. Lond. A 287, 273–305.Google Scholar
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