Symmetry breaking and period doubling on a torus in the VLF regime in Taylor-Couette flow

Abstract

We present an extensive experimental study of the very-low-frequency (VLF) mode, a very slow time-periodic oscillation with azimuthal wave number m=0 in axisymmetric Taylor-Couette flow. The VLF mode appears as a secondary or higher time-dependent instability in the entire wavelength range for flow systems with radius ratio 0.5. We focus on measurements which cover a parameter range that reaches from the onset of time dependence to the transition to chaos in the wavelength range \ensuremath{\lambda}1.78d (d is the gap width of the cylinder) appearing in flow systems having 10--50 vortices. It was found that, increasing the Reynolds number, one observes---independently of the number of vortices of the flow system---always the same ``sequence'' of states. This is, first, the transition from Taylor vortex flow to the onset of the time-periodic small-jet mode via a Hopf bifurcation going along with a simultaneous breaking of the axial symmetry of the flow, second, the onset of the VLF mode via a homoclinic bifurcation for smaller cylinders where the underlying wavy Taylor vortex flow is still the small-jet mode (therefore we have a ${\mathit{T}}^{2}$ torus); and finally, the transitions to chaos, which were found to occur as period-doubling routes on ${\mathit{T}}^{2}$ tori. Additionally a quantitative description of this transition to chaos is given, calculating the correlation dimension on the basis of a proper reconstructed phase space. A model of interacting time-dependent Taylor vortex flow is discussed and compared to the appearance of VLF-mode oscillations in the flow. \textcopyright{} 1996 The American Physical Society.