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Inviscid and viscous global stability of vortex rings

Published online by Cambridge University Press:  04 September 2020

Naveen Balakrishna*
Department of Aerospace Engineering, Indian Institute of Science, Bengaluru560012, India
Joseph Mathew
Department of Aerospace Engineering, Indian Institute of Science, Bengaluru560012, India
Arnab Samanta
Department of Aerospace Engineering, Indian Institute of Science, Bengaluru560012, India Department of Aerospace Engineering, Indian Institute of Technology, Kanpur208016, India
Email address for correspondence:


We perform inviscid and viscous, global, linear stability analyses of vortex rings which are compared with asymptotic theories and numerical simulations. We find growth rates of rings to be very sensitive to the details of vorticity distribution, in a way not accounted for in asymptotic theories, clearly demonstrated in our analyses of equilibrated rings–ring base flows initially obtained from Gaussian rings evolved to a quasi-steady state before any instabilities set in. Such equilibrated rings with the same $\epsilon = a/R$, the ratio of core radius $a$ to ring radius $R$, but evolved with different viscosities, have inviscid growth rates differing by up to 9 %, though the differences in vorticity at any point are small. In contrast, the growth rates of rings with a Gaussian vorticity distribution are found to be up to 33 % smaller than the inviscid asymptotic theories over $0.4 > \epsilon > 0.05$. We attribute these differences to the nature of velocity fields at $O(\epsilon ^2)$, between equilibrated and Gaussian rings, where the former shows a good quantitative match with the asymptotic theories. Additionally, there are some differences with previous direct numerical simulations (DNS), but in very close quantitative agreement with our DNS results. Our calculations provide a new relation capturing the near-linear dependence of growth rates on the reciprocal of a strain rate-based Reynolds number $\widehat Re$. Importantly, our equilibrated ring calculations do tend to the inviscid limit of asymptotic theories, once corrections for ring radius evolution and equilibrated distribution are imposed, unlike for Gaussian rings.

JFM Papers
© The Author(s), 2020. Published by Cambridge University Press

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Archer, P. J., Thomas, T. G. & Coleman, G. N. 2008 Direct numerical simulation of vortex ring evolution from the laminar to the early turbulent regime. J. Fluid Mech. 598, 201226.CrossRefGoogle Scholar
Balay, S., Gropp, W. D., McInnes, L. C. & Smith, B. F. 1997 Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools in Scientific Computing (ed. Arge, E., Bruaset, A. M. & Langtangen, H. P.), pp. 163202. Birkhäuser.CrossRefGoogle Scholar
Balay, S. et al. 2015 a PETSc users manual. Tech. Rep. ANL-95/11 Rev. 3.6. Argonne National Laboratory.CrossRefGoogle Scholar
Balay, S. et al. 2015 b PETSc suite. Available at: Scholar
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14 (4), 529551.CrossRefGoogle Scholar
Bayliss, A., Class, A. & Matkowsky, B. J. 1995 Adaptive approximation of solutions to problems with multiple layers by Chebyshev pseudo-spectral methods. J. Comput. Phys. 116 (1), 160172.CrossRefGoogle Scholar
Bayliss, A. & Turkel, E. 1992 Mappings and accuracy for Chebyshev pseudo-spectral approximations. J. Comput. Phys. 101 (2), 349359.CrossRefGoogle Scholar
Blanco-Rodríguez, F. J. & Le Dizés, S. 2016 Elliptic instability of a curved batchelor vortex. J. Fluid Mech. 804, 224247.CrossRefGoogle Scholar
Blanco-Rodríguez, F. J. & Le Dizés, S. 2017 Curvature instability of a curved Batchelor vortex. J. Fluid Mech. 814, 397415.CrossRefGoogle Scholar
Blanco-Rodríguez, F. J., Le Dizés, S., Selçuk, C., Delbende, I. & Rossi, M. 2015 Internal structure of vortex rings and helical vortices. J. Fluid Mech. 785, 219247.CrossRefGoogle Scholar
Eloy, C. & Le Dizés, S. 2001 Stability of the rankine vortex in a multipolar strain field. Phys. Fluids 13 (3), 660676.CrossRefGoogle Scholar
Fabre, D., Sipp, D. & Jacquin, L. 2006 Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235274.CrossRefGoogle Scholar
Fukumoto, Y. & Hattori, Y. 2005 Curvature instability of a vortex ring. J. Fluid Mech. 526, 77115.CrossRefGoogle Scholar
Gargan-Shingles, C., Rudman, M. & Ryan, K. 2016 The linear stability of swirling vortex rings. Phys. Fluids 28 (11), 114106.CrossRefGoogle Scholar
Hattori, Y., Blanco-Rodríguez, F. J. & Le Dizés, S. 2019 Numerical stability analysis of a vortex ring with swirl. J. Fluid Mech. 878, 536.CrossRefGoogle Scholar
Hernandez, V., Roman, J. E. & Vidal, V. 2005 SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31 (3), 351362.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34 (1), 83113.CrossRefGoogle Scholar
Khorrami, M. R., Malik, M. R. & Ash, R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81 (1), 206229.CrossRefGoogle Scholar
Kundu, P. K., Cohen, I. M. & Dowling, D. R. 2012 Fluid Mechanics. Academic Press.Google Scholar
Laizet, S. & Lamballais, E. 2009 High-order compact schemes for incompressible flows: a simple and efficient method with quasi-spectral accuracy. J. Comput. Phys. 228 (16), 59896015.CrossRefGoogle Scholar
Laizet, S. & Li, N. 2011 Incompact3d: a powerful tool to tackle turbulence problems with up to O $(10^5)$ computational cores. Intl J. Numer. Meth. Fluids 67 (11), 17351757.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 (1646), 413425.Google Scholar
Naveen, B. 2021 On late stages of transition in round jets. PhD thesis, Indian Institute of Science.Google Scholar
Roman, J. E., Campos, C., Romero, E. & Tomas, A. 2017 SLEPc users manual. Tech. Rep. DSIC-II/24/02 Rev. 3.8. D. Sistemes Informàtics i Computació, Universitat Politècnica de València.Google Scholar
Saffman, P. G. 1970 The velocity of viscous vortex rings. Stud. Appl. Maths 49 (4), 371380.CrossRefGoogle Scholar
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84 (4), 625639.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2012 Stability and Transition in Shear Flows. Springer.Google Scholar
Shariff, K., Verzicco, R. & Orlandi, P. 1994 A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage. J. Fluid Mech. 279, 351375.CrossRefGoogle Scholar
Sipp, D., Jacquin, L. & Cosssu, C. 2000 Self-adaptation and viscous selection in concentrated two-dimensional vortex dipoles. Phys. Fluids 12 (2), 245248.CrossRefGoogle Scholar
Thomson, J. J. 1883 A Treatise on the Motion of Vortex Rings: An Essay to Which the Adams Prize was Adjudged in 1882, in the University of Cambridge. Macmillan.Google Scholar
Thomson, W. 1880 Vibrations of a columnar vortex. Lond. Edin. Dublin Phil. Mag. J. Sci. 10, 155168.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 (4), 721733.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 (1), 3547.CrossRefGoogle Scholar
Widnall, S. E. & Sullivan, J. P. 1973 On the stability of vortex rings. Proc. R. Soc. Lond. A 332, 335353.Google Scholar
Widnall, S. E. & Tsai, C. Y. 1977 The instability of the thin vortex ring of constant vorticity. Phil. Trans. R. Soc. Lond. 287 (1344), 273305.Google Scholar