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Spatio-temporal stability of the Kármán vortex street and the effect of confinement

  • Saviz Mowlavi (a1), Cristóbal Arratia (a1) (a2) and François Gallaire (a1)

Abstract

The instability of the Kármán vortex street is revisited under a spatio-temporal perspective that allows the taking into account of the advection of the vortices by the external flow. We analyse a simplified point vortex model and show through numerical simulations of its linear impulse response that the system becomes convectively unstable above a certain critical advection velocity. This critical velocity decreases as the aspect ratio approaches its specific value for temporal stability, and increases with the confinement induced by lateral walls. In the limiting unconfined case, direct application of the Briggs–Bers criterion to the dispersion relation gives results in excellent agreement with the numerical simulations. Finally, a direct numerical simulation of the $Re=100$ flow past a confined cylinder is performed, and the actual advection velocity of the resulting vortex street is found to be much larger than the critical advection velocity for convective instability given by our model. The Kármán vortex street is therefore strongly convectively unstable.

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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: cristobal.arratia@gmail.com

References

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Aref, H. & Siggia, E. D. 1981 Evolution and breakdown of a vortex street in two dimensions. J. Fluid Mech. 109, 435463.
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750756.
Biancofiore, L. & Gallaire, F. 2011 The influence of shear layer thickness on the stability of confined two-dimensional wakes. Phys. Fluids 23 (3), 034103.
Boniface, P.2014 Instabilité de Kelvin–Helmholtz et allée de Bénard–von Kármán en géométrie rectangulaire confinée. PhD thesis, Université Paris Diderot.
Brancher, P. & Chomaz, J.-M. 1997 Absolute and convective secondary instabilities in spatially periodic shear flows. Phys. Rev. Lett. 78 (4), 658.
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. Research Monograph 29. MIT Press.
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.
Deissler, R. J. 1987 The convective nature of instability in plane Poiseuille flow. Phys. Fluids 30 (8), 23032305.
Delbende, I., Chomaz, J.-M. & Huerre, P. 1998 Absolute/convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229254.
Healey, J. J. 2009 Destabilizing effects of confinement on homogeneous mixing layers. J. Fluid Mech. 623, 241.
Henderson, R. D. & Barkley, D. 1996 Secondary instability in the wake of a circular cylinder. Phys. Fluids 8 (6), 16831685.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22 (1), 473537.
Inoue, O. & Yamazaki, T. 1999 Secondary vortex streets in two-dimensional cylinder wakes. Fluid Dyn. Res. 25 (1), 118.
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.
Jiménez, J. 1987 On the linear stability of the inviscid Kármán vortex street. J. Fluid Mech. 178, 177194.
Jiménez, J. 1988 Linear stability of a non-symmetric, inviscid, Kármán street of small uniform vortices. J. Fluid Mech. 189, 337348.
Juniper, M. P. 2006 The effect of confinement on the stability of two-dimensional shear flows. J. Fluid Mech. 565, 171195.
Juniper, M. P. 2007 The full impulse response of two-dimensional jet/wake flows and implications for confinement. J. Fluid Mech. 590, 163185.
von Kármán, T. 1911 Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt. Göttingen Nachrichten, Math. Phys. Kl. 509517.
von Kármán, T. 1912 Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt. Göttingen Nachrichten, Math. Phys. Kl. 547556.
Kida, S. 1982 Stabilizing effects of finite core on kármán vortex street. J. Fluid Mech. 122, 487504.
Kumar, B. & Mittal, S. 2012 On the origin of the secondary vortex street. J. Fluid Mech. 711, 641666.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Mackay, R. S. 1987 Instability of vortex streets. Dyn. Stab. Syst. 2 (1), 5571.
Matsui, T. & Okude, M. 1983 Formation of the secondary vortex street in the wake of a circular cylinder. In Proceedings of the IUTAM Symposium on Structure of Complex Turbulent Shear Flow, pp. 156164. Springer.
Meiburg, E. 1987 On the role of subharmonic perturbations in the far wake. J. Fluid Mech. 177, 83.
Meiron, D. I., Saffman, P. G. & Schatzman, J. C. 1984 The linear two-dimensional stability of inviscid vortex streets of finite-cored vortices. J. Fluid Mech. 147, 187212.
Monkewitz, P. A. 1988 The absolute and convective nature of instability in two-dimensional wakes at low reynolds numbers. Phys. Fluids 31 (5), 9991006.
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407417.
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 122.
Rosenhead, L. 1929 The Kármán street of vortices in a channel of finite breadth. Phil. Trans. R. Soc. Lond. A 228 (659–669), 275329.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Saffman, P. G. & Schatzman, J. C. 1982a An inviscid model for the vortex-street wake. J. Fluid Mech. 122, 467486.
Saffman, P. G. & Schatzman, J. C. 1982b Stability of a vortex street of finite vortices. J. Fluid Mech. 117, 171185.
Sahin, M. & Owens, R. G. 2004 A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder. Phys. Fluids 16 (5), 13051320.
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.
Williamson, C. H. K. & Prasad, A. 1993 A new mechanism for oblique wave resonance in the ‘natural’ far wake. J. Fluid Mech. 256, 269313.
Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355381.
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Spatio-temporal stability of the Kármán vortex street and the effect of confinement

  • Saviz Mowlavi (a1), Cristóbal Arratia (a1) (a2) and François Gallaire (a1)

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