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Topological fluid mechanics of the formation of the Kármán-vortex street

  • Matthias Heil (a1), Jordan Rosso (a2), Andrew L. Hazel (a1) and Morten Brøns (a3)
Abstract

We explore the two-dimensional flow around a circular cylinder with the aim of elucidating the changes in the topology of the vorticity field that lead to the formation of the Kármán vortex street. Specifically, we analyse the formation and disappearance of extremal points of vorticity, which we consider to be feature points for vortices. The basic vortex creation mechanism is shown to be a topological cusp bifurcation in the vorticity field, where a saddle and an extremum of the vorticity are created simultaneously. We demonstrate that vortices are first created approximately 100 diameters downstream of the cylinder, at a Reynolds number, $Re_{K}$ , which is slightly larger than the critical Reynolds number, $Re_{crit}\approx 46$ , at which the flow becomes time periodic. For $Re$ slightly above $Re_{K}$ , the newly created vortices disappear again a short distance further downstream. As $Re$ is further increased, the points of creation and disappearance move rapidly upstream and downstream, respectively, and the Kármán vortex street persists over increasingly large streamwise distances.

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Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Corresponding author
Email address for correspondence: M.Heil@maths.manchester.ac.uk
References
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Amestoy P. R., Duff I. S., Koster J. & L’Excellent J.-Y. 2001 A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Applics 23 (1), 1541.
Aref H. & Siggia E. D. 1981 Evolution and breakdown of a vortex street in two dimensions. J. Fluid Mech. 109, 435463.
Bakker P. G. 1991 Bifurcations in Flow Patterns. Klüver Academic.
Balci A., Andersen M., Thompson M. C. & Brøns M. 2015 Codimension three bifurcation of streamline patterns close to a no-slip wall: a topological description of boundary layer eruption. Phys. Fluids 27 (5), 053603.
Benard H. 1908a Étude cinématographique des remous et des rides produits par la translation d’un obstacle. C. R. Acad. Sci. 147, 970972.
Benard H. 1908b Formation des centres de giration à l’arriere d’un obstacle en mouvement. C. R. Acad. Sci. 147, 839842.
Brøns M. 2007 Streamline topology: patterns in fluid flows and their bifurcations. Adv. Appl. Mech. 41, 142.
Brøns M. & Bisgaard A. V. 2006 Bifurcation of vortex breakdown patterns in a circular cylinder with two rotating covers. J. Fluid Mech. 568, 329349.
Brøns M. & Bisgaard A. V. 2010 Topology of vortex creation in the cylinder wake. Theor. Comput. Fluid Dyn. 24 (1–4), 299303.
Brøns M., Jakobsen B., Niss K., Bisgaard A. V. & Voigt L. K. 2007 Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers. J. Fluid Mech. 584, 2343.
Cimbala J. M., Nagib H. M. & Roshko A. 1988 Large structure in the far wakes of two-dimensional bluff bodies. J. Fluid Mech. 190, 265298.
Cliffe K. A., Spence A. & Tavener S. J. 2000 The numerical analysis of bifurcation problems with application to fluid mechanics. Acta Numerica 9, 39131.
Coutanceau M. & Defaye J.-R. 1991 Circular cylinder wake configurations: a flow visualization survey. Appl. Mech. Rev. 44 (6), 255305.
Dušek J., Gal P. L. & Fraunié P. 1994 A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264, 5980.
Dynnikova G. Y., Dynnikov Y. A. & Guvernyuk S. V. 2016 Mechanism underlying Kármán vortex street breakdown preceding secondary vortex street formation. Phys. Fluids 28 (5), 054101.
Elman H. C., Silvester D. J. & Wathen A. J. 2005 Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Oxford University Press.
Gürcan F. & Bilgil H. 2013 Bifurcations and eddy genesis of Stokes flow within a sectorial cavity. Eur. J. Mech. (B/Fluids) 39, 4251.
Heil M. & Hazel A. L. 2006 oomph-lib – an object-oriented multi-physics finite-element library. In Fluid-Structure Interaction (ed. Schäfer M. & Bungartz H.-J.), pp. 1949. Springer; oomph-lib is available as open-source software at http://www.oomph-lib.org.
Henson V. E. & Yang U. M. 2002 BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Maths 41, 155177.
Iooss G. & Joseph D. D. 1990 Elementary Stability and Bifurcation Theory. Springer.
Jackson C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 2345.
Jeong J. & Hussain F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.
von Kármán T. 1912 Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfahrt. Nachr. Ges. Wiss. Göttingen 1912, 547556.
Kasten J., Reinighaus J., Hotz I., Hege H.-C., Noack B. R., Daviller G. & Morzyński M. 2016 Acceleration feature points of unsteady shear flows. Arch. Mech. 68 (1), 5580.
Kudela H. & Malecha Z. M. 2009 Eruption of a boundary layer induced by a 2 D vortex patch. Fluid Dyn. Res. 41 (5), 055502.
Lehoucq R. B., Sorensen D. C. & Yang C. 1998 ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, ISBN 978-0-89871-407-4.
Leweke T., Le Dizès S. & Williamson C. H. 2016 Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48 (1), 507541.
Mathis C., Provansal M. & Boyer L. 1984 The Benard-von Karman instability: an experimental study near the threshold. J. Physique Lett. 45, L-483L-491.
Noack B. R. & Eckelmann H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297330.
Orlandi P. 1990 Vortex dipole rebound from a wall. Phys. Fluids A 2 (8), 14291436.
Ponta F. & Aref H. 2006 Numerical experiments on vortex shedding from an oscillating cylinder. J. Fluids Struct. 22 (3), 327344.
Provansal M., Mathis C. & Boyer L. 1987 Bénard-von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 122.
Schnipper T., Andersen A. & Bohr T. 2009 Vortex wakes of a flapping foil. J. Fluid Mech. 633, 411423.
Taneda S. 1959 Downstream development of the wakes behind cylinders. J. Phys. Soc. Japan 14 (6), 843848.
Thompson M. C., Radi A., Rao A., Sheridan J. & Hourigan K. 2014 Low-Reynolds-number wakes of elliptical cylinders: from the circular cylinder to the normal flat plate. J. Fluid Mech. 751, 570600.
Wesfreid J. E., Goujon-Durand S. & Zielinska B. J. A. 1996 Global mode behaviour of the streamwise vorticity in wakes. J. Phys II France 6, 13431357.
Wiggins S. 1990 Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer.
Williamson C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28 (1), 477539.
Williamson C. H. K. & Roshko A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355381.
Zebib A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Maths 21 (2), 155165.
Zielinska B. J. A. & Wesfreid J. E. 1995 On the spatial structure of global modes in wake flow. Phys. Fluids 7, 14181424.
Zienkiewicz O. C. & Zhu J. Z. 1992 The superconvergent patch recovery and a posteriori error estimates. Part 1. The recovery technique. Intl J. Numer. Meth. Engng 33, 13311364.
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