Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-24T17:30:03.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Hollow vortices in shear

Published online by Cambridge University Press:  16 November 2016

Luca Zannetti ,
Michele Ferlauto  and
Stefan G. Llewellyn Smith
Luca Zannetti*
Affiliation:
Accademia delle Scienze di Torino, Via Accademia delle Scienze 6, 10123 Torino, Italy
Michele Ferlauto
Affiliation:
DIMEAS, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
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
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

An analytical method for determining the shape of hollow vortices in shear flows is presented in detail. In a non-dimensional formulation, it is shown that the problem has one degree of freedom represented by the free choice of the non-dimensionalized speed $\unicode[STIX]{x1D705}$ at the boundary of the vortex. The solutions form two families of shapes corresponding to vortex circulation and shear-flow vorticity having the opposite or same sign. When the signs are opposite, the shape family resembles that described by Llewellyn Smith & Crowdy (J. Fluid Mech., vol. 691, 2012, pp. 178–200) for hollow vortices in a potential flow with strain. As for that flow, there is a minimum value of $\unicode[STIX]{x1D705}$ below which there is no solution as the boundary of the vortex self-intersects, while, when the signs are the same, solutions exist for $0<\unicode[STIX]{x1D705}$.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 809 , 25 December 2016 , pp. 705 - 715
Check for updates
Copyright
© 2016 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

Baker, G. R., Saffman, P. G. & Sheffield, J. S. 1976 Structure of a linear array of hollow vortices of finite cross-section. J. Fluid Mech. 74, 1469–1476.CrossRefGoogle Scholar
Birkhoff, G. & Zarantonello, E. H. 1957 Jets, Wakes, and Cavities. Academic.Google Scholar
Crowdy, D. G. & Green, C. C. 2011 Analytical solutions for von Kármán streets of hollow vortices. Phys. Fluids 23, 126602.CrossRefGoogle Scholar
Crowdy, D. G., Llewellyn Smith, S. G. & Freilich, D. V. 2012 Translating hollow vortex pairs. Eur. J. Mech. (B/Fluids) 37, 180–186.CrossRefGoogle Scholar
Elcrat, A., Ferlauto, M. & Zannetti, L. 2014 Point vortex model for asymmetric inviscid wakes past bluff bodies. Fluid Dyn. Res. 46, 031407.CrossRefGoogle Scholar
Elcrat, A. & Zannetti, L. 2012 Models for inviscid wakes past a normal plate. J. Fluid Mech. 708, 377–396.CrossRefGoogle Scholar
Green, C. C. 2015 Analytical solutions for two hollow vortex configurations in an infinite channel. Eur. J. Mech. (B/Fluids) 54, 69–81.CrossRefGoogle Scholar
Gurevich, M. I. 1966 Theory of Jets in Ideal Fluids. Academic.Google Scholar
Hicks, W. M. 1883 On the steady motion of a hollow vortex. Proc. R. Soc. Lond. 35, 304–308.Google Scholar
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Lin, A. & Landweber, L. 1977 On a solution of the Lavrentiev wake model and its cascade. J. Fluid Mech. 79, 801–823.CrossRefGoogle Scholar
Llewellyn Smith, S. G. & Crowdy, D. G. 2012 Structure and stability of hollow vortex equilibria. J. Fluid Mech. 691, 178–200.CrossRefGoogle Scholar
Meleshko, V. V. & van Heijst, G. J. F. 1994 On Chaplygin’s investigations of two-dimensional vortex structures in an inviscid fluid. J. Fluid Mech. 272, 157–182.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. Olsen, J. A., Goldburg, A. & Rogers, M.), pp. 339–354. Plenum.CrossRefGoogle Scholar
Pocklington, H. C. 1895 The configuration of a pair of equal and opposite hollow straight vortices, of finite cross-section, moving steadily through fluid. Proc. Camb. Phil. Soc. 8, 178–187.Google Scholar
Schecter, D. A. & Dubin, D. H. E. 2001 Theory and simulations of two-dimensional vortex motion driven by a background vorticity gradient. Phys. Fluids 13 (6), 1704–1723.CrossRefGoogle Scholar
Shetty, S., Asay-Davis, X. S. & Marcus, P. S. 2010 On the interaction of Jupiter’s great red spot and zonal jet streams. J. Atmos. Sci. 64, 4432–4444.CrossRefGoogle Scholar
Tanga, P., Babiano, A., Dubrulle, B. & Provenzale, A. 1996 Forming planetesimals in vortices. Icarus 121, 158–170.CrossRefGoogle Scholar
Telib, H. & Zannetti, L. 2011 Hollow wakes past arbitrarily shaped obstacles. J. Fluid Mech. 669, 214–224.CrossRefGoogle Scholar
Zannetti, L. & Lasagna, D. 2013 Hollow vortices and wakes past Chaplygin cusps. Eur. J. Mech. (B/Fluids) 38, 78–84.CrossRefGoogle Scholar