Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T06:30:15.943Z Has data issue: false hasContentIssue false
  • English
  • Français

The Sadovskii vortex in strain

Published online by Cambridge University Press:  21 July 2017

Daniel V. Freilich  and
Stefan G. Llewellyn Smith Open the ORCID record for Stefan G. Llewellyn Smith [Opens in a new window]
Daniel V. Freilich
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
Stefan G. Llewellyn Smith*
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, University of California San Diego, 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

The point vortex is the simplest model of a two-dimensional vortex with non-zero circulation. The limitations introduced by its lack of core structure have been remedied by using desingularizations such as vortex patches and vortex sheets. We investigate steady states of the Sadovskii vortex in strain, a canonical model for a vortex in a general flow. The Sadovskii vortex is a uniform patch of vorticity surrounded by a vortex sheet. We recover previously known limiting cases of the vortex patch and hollow vortex, and examine the bifurcations away from these families. The result is a solution manifold spanned by two parameters. The addition of the vortex sheet to the vortex patch solutions immediately leads to splits in the solution manifold at certain bifurcation points. The more circular elliptical family remains attached to the family with a single pinch-off, and this family extends all the way to the simpler solution branch for the pure vortex sheet solutions. More elongated families below this one also split at bifurcation points, but these families do not exist in the vortex sheet limit.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 479 - 501
Check for updates
Copyright
© 2017 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

Ardalan, K., Meiron, D. I. & Pullin, D. I. 1995 Steady compressible vortex flows: the hollow-core vortex array. J. Fluid Mech. 301, 1–17.Google Scholar
Baker, G. & Nachbin, A. 1998 Stable methods for vortex sheet motion in the presence of surface tension. SIAM J. Sci. Comput. 19, 1737–1766.CrossRefGoogle Scholar
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, 469–476.Google Scholar
Batchelor, G. K. 1956a On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190.Google Scholar
Batchelor, G. K. 1956b A proposal concerning laminar wakes behind bluff bodies at large Reynolds number. J. Fluid Mech. 1, 388–398.CrossRefGoogle Scholar
Bunyakin, A. V., Chernyshenko, S. I. & Stepanov, G. Y. 1996 Inviscid Batchelor-model flow past an airfoil with a vortex trapped in a cavity. J. Fluid Mech. 323, 367–376.CrossRefGoogle Scholar
Bunyakin, A. V., Chernyshenko, S. I. & Stepanov, G. Y. 1998 High-Reynolds-number Batchelor-model asymptotics of a flow past an aerofoil with a vortex trapped in a cavity. J. Fluid Mech. 358, 283–297.CrossRefGoogle Scholar
Chernyshenko, S. I. 1998 Asymptotic theory of global separation. Appl. Mech. Rev. 51, 523–536.CrossRefGoogle Scholar
Chernyshenko, S. I., Galletti, B., Iollo, A. & Zannetti, L. 2003 Trapped vortices and a favourable pressure gradient. J. Fluid Mech. 482, 235–255.CrossRefGoogle Scholar
Crowdy, D. G. & Green, C. C. 2011 Analytical solutions for von Kármán streets of hollow vortices. Phys. Fluids 23, 126602.CrossRefGoogle Scholar
Deem, G. S. & Zabusky, N. J. 1978 Vortex waves: stationary ‘V states’, interactions, recurrence and breaking. Phys. Rev. Lett. 40, 859–862.CrossRefGoogle Scholar
Gallizio, F., Iollo, A., Protas, B. & Zannetti, L. 2010 On continuation of inviscid vortex patches. Physica D 239, 190–201.Google Scholar
Govaerts, W. 2000 Numerical Methods for Bifurcations of Dynamical Equilibria. Society for Industrial and Applied Mathematics.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
Kamm, J. R.1987 Shape and stability of two-dimensional uniform vorticity regions. PhD thesis, California Institute of Technology.Google Scholar
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Leppington, F. G. 2006 The field due to a pair of line vortices in a compressible fluid. J. Fluid Mech. 559, 45–55.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
Luzzatto-Fegiz, P. & Williamson, C. H. K. 2011 An efficient and general numerical method to compute steady uniform vortices. J. Comput. Phys. 230, 6495–6511.Google Scholar
Miyazaki, T., Imai, T. & Fukumoto, Y. 1995 Three-dimensional instability of Kirchhoff’s elliptic vortex. Phys. Fluids 7, 195–202.CrossRefGoogle Scholar
Moore, D. W. & Pullin, D. I. 1987 The compressible vortex pair. J. Fluid Mech. 185, 171–204.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.Google Scholar
Moore, D. W., Saffman, P. G. & Tanveer, S. 1988 The calculation of some Batchelor flows: the Sadovskii vortex and rotational corner flow. Phys. Fluids 31, 978–990.CrossRefGoogle Scholar
Overman, E. A. 1986 Steady-state solutions of the Euler equations in two dimensions II. Local analysis of limiting V-states. SIAM J. Appl. Maths 46, 765–800.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
Robinson, A. C. & Saffman, P. G. 1984 Three-dimensional stability of an elliptical vortex in a straining field. J. Fluid Mech. 142, 451–466.Google Scholar
Sadovskii, V. S. 1971 Vortex regions in a potential stream with a jump of Bernoulli’s constant at the boundary. Z. Angew. Math. Mech. 35, 729–735.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Saffman, P. G. & Tanveer, S. 1982 The touching pair of equal and opposite uniform vortices. Phys. Fluids 25, 1929–1930.CrossRefGoogle Scholar
Saffman, P. G. & Tanveer, S. 1984 Prandtl–Batchelor flow past a flat plate with a forward-facing flap. J. Fluid Mech. 143, 351–365.CrossRefGoogle Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys. 30, 96–106.Google Scholar
Zannetti, L., Ferlauto, M. & Llewellyn Smith, S. G. 2016 Hollow vortices in shear. J. Fluid Mech. 809, 705–715.CrossRefGoogle Scholar