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Bursting dynamics due to a homoclinic cascade in Taylor–Couette flow

Published online by Cambridge University Press:  01 October 2008

J. ABSHAGEN ,
J. M. LOPEZ ,
F. MARQUES  and
G. PFISTER
J. ABSHAGEN
Affiliation:
Institute of Experimental and Applied Physics, University of Kiel, 24105 Kiel, Germany
J. M. LOPEZ
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
F. MARQUES
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, 08034, Barcelona, Spain
G. PFISTER
Affiliation:
Institute of Experimental and Applied Physics, University of Kiel, 24105 Kiel, Germany
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Abstract

Transitions between regular oscillations and bursting oscillations that involve a bifurcational process which culminates in the creation of a relative periodic orbit of infinite period and infinite length are investigated both experimentally and numerically in a short-aspect-ratio Taylor–Couette flow. This bifurcational process is novel in that it is the accumulation point of a period-adding cascade at which the mid-height reflection symmetry is broken. It is very rich and complex, involving very-low-frequency states arising via homoclinic and heteroclinic dynamics, providing the required patching between states with very different dynamics in neighbouring regions of parameter space. The use of nonlinear dynamical systems theory together with symmetry considerations has been crucial in interpreting the laboratory experimental data as well as the results from the direct numerical simulations. The phenomenon corresponds to dynamics well beyond the first few bifurcations from the basic state and so is beyond the reach of traditional hydrodynamic stability analysis, but it is not fully developed turbulence where a statistical or asymptotic approach could be employed. It is a transitional phenomenon, where the phase dynamics of the large-scale structures (jets of angular momentum emanating from the boundary layer on the rotating inner cylinder) becomes complicated. Yet the complicated phase dynamics remains accessible to an analysis of its space–time characteristics and a comprehensive mechanical characterization emerges. The excellent agreement between the experiments and the numerical simulations demonstrates the robustness of this complex bifurcation phenomenon in a physically realized system with its inherent imperfections and noise. Movies are available with the online version of the paper.

JFM classification

Instability: Taylor-Couette flow Nonlinear Dynamical Systems: Bifurcation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 613 , 25 October 2008 , pp. 357 - 384
Copyright
Copyright © Cambridge University Press 2008

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References

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Abshagen et sl. supplementary movie

Movie 1. An animation of isosurfaces of the axial component of the angular momentum, rv, for the symmetric modulated rotating wave MRWs at Re=830 and Γ=3.14. MRWs is a quasi-periodic solution with two frequencies, but one of these corresponds to the precession frequency of the underlying symmetric rotating wave and plays no dynamic role. The other (modulation) frequency corresponds to the period over which the tilted jets oscillate in and out of phase relative to each other, as is evident in the movie, which is shown in the precessing frame of reference. (This movie corresponds to figure 6 in the paper.)

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Video 804 KB

Abshagen et al. supplementary movie

Movie 2. The same rv isosurfaces, but for the asymmetric MRWa at the same Re=830, but larger Γ=3.17. As with MRWs, MRWa is also periodic in a frame of reference rotating at its precession frequency, but for MRWa both jets undergo modulated oscillations in the azimuthal direction, with the phase difference oscillating about a mean value. Moreover, the lower jet is thicker during the first half of the modulation period than in the second half.

Download Abshagen et al. supplementary movie(Video)
Video 685.2 KB

Abshagen et al. supplementary movie

Movie 3 shows the same MRWa as in movie 2, but in the stationary frame of reference. (These movies correspond to figure 10 in the paper.)

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Video 819.6 KB