Symmetry breaking of two-dimensional time-periodic wakes

A number of two-dimensional time-periodic flows, for example the Kármán street wake of a symmetrical bluff body such as a circular cylinder, possess a spatio-temporal symmetry: a combination of evolution by half a period in time and a spatial reflection leaves the flow invariant. Floquet analyses for the stability of these flows to three-dimensional perturbations have in the past been based on the Poincaré map, without attempting to exploit the spatio-temporal symmetry. Here, Floquet analysis based on the half-period-flip map provides a comprehensive interpretation of the symmetry-breaking bifurcations.

[1]  Oscillatory modes in an enclosed swirling flow , 2001, Journal of Fluid Mechanics.

[2]  Hugh Maurice Blackburn,et al.  The onset of three-dimensional standing and modulated travelling waves in a periodically driven cavity flow , 2003, Journal of Fluid Mechanics.

[3]  Bifurcation theory for three-dimensional flow in the wake of a circular cylinder , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  Bau-Sen Du.  On the nature of chaos , 2006 .

[5]  John Guckenheimer,et al.  Multiple Bifurcation Problems of Codimension Two , 1984 .

[6]  J. Lasheras,et al.  Three-dimensional instability and vorticity patterns in the wake of a flat plate , 2003, Journal of Fluid Mechanics.

[7]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[8]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[9]  J. Lamb,et al.  Bifurcation From Periodic Solutions with Spatiotemporal Symmetry , 1999 .

[10]  Eckart Meiburg,et al.  Experimental and numerical investigation of the three-dimensional transition in plane wakes , 1988, Journal of Fluid Mechanics.

[11]  BIFURCATIONS OF PERIODIC ORBITS WITH SPATIO-TEMPORAL SYMMETRIES , 1997, patt-sol/9704002.

[12]  Pascal Chossat,et al.  The Couette-Taylor Problem , 1992 .

[13]  C. Williamson.  The Existence of Two Stages in the Transition to Three-Dimensionality of a Cylinder Wake , 1988 .

[14]  M. Thompson,et al.  From spheres to circular cylinders: non-axisymmetric transitions in the flow past rings , 2004, Journal of Fluid Mechanics.

[15]  J L S T E V E N S,et al.  Oscillatory flow states in an enclosed cylinder with a rotating endwall , 1998 .

[16]  Symmetry–breaking instabilities of convection in squares , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  Juan Lopez,et al.  Spatio-temporal dynamics of a periodically driven cavity flow , 2002, Journal of Fluid Mechanics.

[18]  P. Chossat,et al.  Methods in Equivariant Bifurcations and Dynamical Systems , 2000 .

[19]  H. Blackburn,et al.  On three-dimensional quasiperiodic Floquet instabilities of two-dimensional bluff body wakes , 2003 .

[20]  C. Williamson.  Three-dimensional wake transition , 1996, Journal of Fluid Mechanics.

[21]  L. Tuckerman,et al.  Bifurcation Analysis for Timesteppers , 2000 .

[22]  Juan M. Lopez,et al.  Small aspect ratio Taylor-Couette flow: onset of a very-low-frequency three-torus state. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  S. Vanka,et al.  THREE-DIMENSIONAL FLOQUET INSTABILITY OF THE WAKE OF SQUARE CYLINDER , 1999 .

[24]  R. Henderson,et al.  Three-dimensional Floquet stability analysis of the wake of a circular cylinder , 1996, Journal of Fluid Mechanics.

[25]  Eckart Meiburg,et al.  Three‐dimensional vorticity modes in the wake of a flat plate , 1990 .

[26]  F. M A R Q U E S,et al.  Mode interactions in an enclosed swirling flow : a double Hopf bifurcation between azimuthal wavenumbers 0 and 2 , 2001 .

[27]  Martin Golubitsky,et al.  Pattern Formation in Continuous and Coupled Systems , 1999 .

[28]  M. Golubitsky,et al.  The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space , 2002 .

[29]  Bifurcations in systems with Z 2 spatio-temporal and O ( 2 ) spatial symmetry , 2003 .

[30]  M. Thompson,et al.  From spheres to circular cylinders: the stability and flow structures of bluff ring wakes , 2003, Journal of Fluid Mechanics.

[31]  Hendrik C. Kuhlmann,et al.  The two-sided lid-driven cavity: experiments on stationary and time-dependent flows , 2002, Journal of Fluid Mechanics.

[32]  H. Blackburn,et al.  Modulated rotating waves in an enclosed swirling flow , 2002, Journal of Fluid Mechanics.

[33]  Claudia Wulff,et al.  Bifurcation from relative periodic solutions , 2001, Ergodic Theory and Dynamical Systems.

[34]  T. Benjamin,et al.  Bifurcation phenomena in steady flows of a viscous fluid II. Experiments , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[35]  Three-dimensional instability and state selection in an oscillatory axisymmetric swirling flow , 2002 .

[36]  G. Iooss,et al.  Topics in bifurcation theory and applications , 1999 .

[37]  Daniel D. Joseph,et al.  Stability of fluid motions , 1976 .

[38]  J. Lamb,et al.  Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions , 2003 .

[39]  F Marques,et al.  Tangent double Hopf bifurcation in a differentially rotating cylinder flow. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  E. Knobloch,et al.  Symmetry and Symmetry-Breaking Bifurcations in Fluid Dynamics , 1991 .

[41]  Kurt Wiesenfeld,et al.  Suppression of period doubling in symmetric systems , 1984 .

[42]  T. Benjamin.  Bifurcation phenomena in steady flows of a viscous fluid. I. Theory , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[43]  Hugh Maurice Blackburn,et al.  Bifurcations in systems with Z2 spatio-temporal and O(2) spatial symmetry , 2004 .