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A global stability analysis of the steady and periodic cylinder wake

  • Bernd R. Noack (a1) and Helmut Eckelmann (a1)

Abstract

A global, three-dimensional stability analysis of the steady and the periodic cylinder wake is carried out employing a low-dimensional Galerkin method. The steady flow is found to be asymptotically stable with respect to all perturbations for Re < 54. The onset of periodicity is confirmed to be a supercritical Hopf bifurcation which can be modelled by the Landau equations. The periodic solution is observed to be only neutrally stable for 54 < Re < 170. While two-dimensional perturbations of the vortex street rapidly decay, three-dimensional perturbations with long spanwise wavelengths neither grow nor decay. The periodic solution becomes unstable at Re = 170 by a perturbation with the spanwise wavelength of 1.8 diameters. This instability is shown to be a supercritical Hopf bifurcation in the spanwise coordinate and leads to a three-dimensional periodic flow. Finally the transition scenario for higher Reynolds numbers is discussed.

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Ahlborn, B. & Lefrançois, M. 1994 The clockwork of vortex shedding. Phys. Fluids A (submitted).
Albarède, P. & Monkewitz, P. A. 1992 A model for the formation of oblique shedding patterns and ‘chevrons’ in cylinder wakes. Phys. Fluids A 4, 744756.
Albarède, P., Provansal, M. & Boyer, L. 1990 Modélisation par l’équation de Ginzburg-Landau du sillage tridimensionnel d’un obstacle allonge. C.R. Acad. Sci. Paris II 310, 459464.
Brede, M., Eckelmann, H., König, M. & Noack, B. R. 1994 Discrete shedding modes of the cylinder wake in a jet with a homogeneous core. Phys. Fluids A (submitted).
Busse, F. H. 1991 Numerical analysis of secondary and tertiary states of fluid flow and their stability properties. Appl. Sci. Res. 48, 341351.
Clever, R. M. & Busse, F. H. 1990 Convection at very low Prandtl numbers. Phys. Fluids A 2, 334339.
Detemple-Laake, E. & Eckelmann, H. 1989 Phenomenology of Kármán vortex streets in oscillatory flow. Exps. Fluids 7, 217227.
Gerrard, J. H. 1966 The three-dimensional structure of the wake of a circular cylinder. J. Fluid Mech. 25, 143164.
Hama, F. R. 1957 Three-dimensional vortex pattern behind a circular cylinder. J. Aero. Sci. 24, 156158.
Hammache, M. & Gharib, M. 1991 An experimental study of the parallel and oblique vortex shedding from circular cylinders. J. Fluid Mech. 232, 567590.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 32, 473537.
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.
Jordan, D. W. & Smith, P. 1988 Nonlinear Ordinary Differential Equations. Clarendon.
Karniadakis, G. E. & Triantafyllou, G. S. 1992 Three-dimensional dynamics and transition to turbulence in the wake of bluff bodies. J. Fluid Mech. 238, 130.
König, M. 1993 Experimentelle Untersuchung des dreidimensionalen Nachlaufs zylindrische Körper bei kleinen Reynoldszahlen. PhD thesis, Georg-August-Universität, Göttingen.
König, M., Eisenlohr, H. & Eckelmann, H. 1990 The fine structure in the Strouhal-Reynolds number relationship of the laminar wake of a circular cylinder. Phys. Fluids A 2, 16071614.
König, M., Eisenlohr, H. & Eckelmann, H. 1992 Visualization of the spanwise cellular structure of the laminar wake of wall-bounded circular cylinders. Phys. Fluids A 4, 869872.
König, M., Noack, B. R. & Ecklemann, H. 1993 Descrite shedding modes in the von Kármán vortex street. Phys. Fluids A 5, 18461848.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.
Leweke, T., Provansal, M. & Boyer, L. 1993 Sillage tridimensionnel d’un obstacle torique et modélisation par l’équation de Ginzburg-Landau. C.R. Acad. Sci. Paris II, 316, 287292.
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130141.
Morzyński, M. & Thiele, F. 1991 Numerical stability analysis of a flow about a cylinder. Z. Angew. Math. Mech. 71, T424T428.
Noack, B. R. 1990 Untersuchung chaotischer Phänomene in der Nachlaufströmung. Rep. 111/1990. Max-Planck-Institut für Strömungsforschung, Göttingen.
Noack, B. R. 1992 Theoretische Untersuchung der Zylinderumströmung mit einem niedrig-dimensionalen Galerkin-Verfahren. Rep. 25/1992. Mitteilungen des Max-Planck-Instituts für Strömungsforschung, Göttingen.
Noack, B. R. & Eckelmann, H. 1991 Two-dimensional, viscous, incompressible flow around a circular cylinder. Rep. 104/1991. Max-Planck-Institut für Strömungsforschung, Göttingen.
Noack, B. R. & Eckelmann, H. 1992 On chaos in wakes. Physica D 56, 151164.
Noack, B. R. & Eckelmann, H. 1993 Theoretical investigation of the cylinder wake with a low-dimensional Galerkin method. In IUTAM-Symposium on Bluff Body Wakes, Dynamics, and Instability, Göttingen, 7–11 September 1992 (ed. H. Eckelmann, J. M. R. Graham, P. Huerre & P. A. Monkewitz), pp 143146. Springer.
Noack, B. R. & Eckelmann, H. 1994a A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder. Phys. Fluids 6, 124143.
Noack, B. R. & Eckelmann, H. 1994b Theoretical investigation of the bifurcations and the turbulence attractor of the cylinder wake. Z. Angew Math. Mech. 74, T396T397.
Noack, B. R., König, M. & Eckelmann, H. 1993 Three-dimensional stability analysis of the periodic flow around a circular cylinder. Phys. Fluids A 5, 12791281.
Noack, B. R. & Obermeier, F. 1991 A chaos-theoretical investigation of the wake behind a cylinder. Z. Angew. Math. Mech. 71, T259T261.
Noack, B. R., Ohle, F. & Eckelmann, H. 1991 On cell formation in vortex streets. J. Fluid Mech. 227, 293308.
Oertel, H. 1990 Wakes behind blunt bodies. Ann. Rev. Fluid Mech. 22, 539546, 1990.
Papangelou, A. 1992 Vortex shedding from slender cones at low Reynolds numbers. J. Fluid Mech. 242, 299321.
Patel, V. A. 1978 Kármán vortex street behind a circular cylinder by series truncation method. J. Comput. Phys. 28, 1442.
Press, W. H., Flamery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes, The Art of Scientific Computing. Cambridge University Press.
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard-von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 122.
Roussopoulos, K. 1993 Feedback control of vortex shedding at low Reynolds numbers. J. Fluid Mech. 248, 267296.
Sreenivasan, K. R., Strykowski, P. J. & Olinger, D. J. 1987 Hopf bifurcation, Landau equation, and vortex shedding behind circular cylinders. In Forum on Unsteady Flow Separation (ed. K. N. Ghia), p. 1. ASME FED vol. 52.
Sreenivasan, K. R., Strykowski, P. J. & Olinger, D. J. 1991 On the Hopf bifurcation and Landau-Stuart constants associated with vortex ‘shedding’ behind circular cylinders. Unpublished.
Tomboulides, A. G., Triantafyllou, G. S. & Karniadakis, G. E. 1992 A mechanism of period doubling in free shear flows. Phys. Fluids A 4, 13291332.
Triantafyllou, G. S. 1990 Three-dimensional flow patterns in two-dimensional wakes. In Intl Symp. on Nonsteady Fluid Dynamics, p 395402. ASME.
Williamson, C. H. K. 1988a Defining a universal and continuous Strouhal-Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Phys. Fluids 31, 27422744.
Williamson, C. H. K. 1988b The existence of two stages in the transition to three-dimensionality of a cylinder wake. Phys. Fluids 31, 31653167.
Williamson, C. H. K. 1989 Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 206, 579627.
Zebib, A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Maths 21, 155165.
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A global stability analysis of the steady and periodic cylinder wake

  • Bernd R. Noack (a1) and Helmut Eckelmann (a1)

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