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Transitions to three-dimensional flows in a cylinder driven by oscillations of the sidewall

Published online by Cambridge University Press:  24 June 2011

C. PANADES ,
F. MARQUES  and
J. M. LOPEZ
C. PANADES
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
F. MARQUES
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
J. M. LOPEZ*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
*
†Email address for correspondence: [email protected]
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Abstract

The transition from two-dimensional to three-dimensional flows in a finite circular cylinder driven by an axially oscillating sidewall is explored in detail. The complete symmetry group of this flow, including a spatio-temporal symmetry related to the oscillating sidewall, is Z2 × O(2). Previous studies in flows with the same symmetries, such as symmetric bluff-body wakes and periodically forced rectangular cavities, were unable to obtain the theoretically predicted bifurcation to modulated travelling waves. In the simpler cylindrical geometry, where the azimuthal direction is physically periodic, we have found these predicted modulated travelling waves as stable fully saturated nonlinear solutions for the first time. A careful analysis of the base states and their linear stability identifies different parameter regimes where three-dimensional states are either synchronous with the forcing or quasi-periodic, corresponding to different symmetry-breaking processes. These results are in good agreement with theoretical predictions and previous results in similar flows. These different regimes are separated by three codimension-two bifurcation points that are yet to be fully analysed theoretically. Finally, the saturated nonlinear states and their properties in different parameter regimes are analysed.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Instability: Nonlinear instability Instability: Parametric instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 681 , 25 August 2011 , pp. 515 - 536
Copyright
Copyright © Cambridge University Press 2011

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References

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Panades et al. supplementary movie

Movie 1a: Animation of the streamfunction of the basic state at $St=10$ and $Re=340$. The left boundary is the cylinder axis and the right boundary is the axially-oscillating cylinder sidewall. Solid (dashed) contours are positive (negative); yellow/red colors correspond to negative/positive values. Movies 1a-h correspond to figure 2 in the paper.

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Video 1.1 MB

Panades et al. supplementary movie

Movie 1a: Animation of the streamfunction of the basic state at $St=10$ and $Re=340$. The left boundary is the cylinder axis and the right boundary is the axially-oscillating cylinder sidewall. Solid (dashed) contours are positive (negative); yellow/red colors correspond to negative/positive values. Movies 1a-h correspond to figure 2 in the paper.

Download Panades et al. supplementary movie(Video)
Video 1.2 MB

Panades et al. supplementary movie

Movie 2a: Animation of isosurfaces of radial vorticity (solid) and of azimuthal vorticity (translucent) for the synchronous state $B_1$ at $St=10$ and $Re=340$. Movies 2a-c correspond to figure 6 in the paper.

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Video 705.3 KB

Panades et al. supplementary movie

Movie 2a: Animation of isosurfaces of radial vorticity (solid) and of azimuthal vorticity (translucent) for the synchronous state $B_1$ at $St=10$ and $Re=340$. Movies 2a-c correspond to figure 6 in the paper.

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Video 717.9 KB

Panades et al. supplementary movie

Movie 3a: Animation of isosurfaces of radial vorticity (solid) and of azimuthal vorticity (translucent) for the quasi-periodic state $MRW_1$ at $St=50$ and $Re=615$. Movies 3a and b correspond to figure 10 in the paper.

Download Panades et al. supplementary movie(Video)
Video 755 KB

Panades et al. supplementary movie

Movie 3a: Animation of isosurfaces of radial vorticity (solid) and of azimuthal vorticity (translucent) for the quasi-periodic state $MRW_1$ at $St=50$ and $Re=615$. Movies 3a and b correspond to figure 10 in the paper.

Download Panades et al. supplementary movie(Video)
Video 766.8 KB

Panades et al. supplementary movie

Movie 4a: Same as in movie 3a, but strobed once per period. Movies 4a and b correspond to figure 11 in the paper.

Download Panades et al. supplementary movie(Video)
Video 1.2 MB

Panades et al. supplementary movie

Movie 4a: Same as in movie 3a, but strobed once per period. Movies 4a and b correspond to figure 11 in the paper.

Download Panades et al. supplementary movie(Video)
Video 1.3 MB

Panades et al. supplementary movie

Movie 2b: Same as in movie 2a, but for the synchronous state $B_2$ at $St=32$ and $Re=525$.

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Video 823 KB

Panades et al. supplementary movie

Movie 2b: Same as in movie 2a, but for the synchronous state $B_2$ at $St=32$ and $Re=525$.

Download Panades et al. supplementary movie(Video)
Video 835 KB

Panades et al. supplementary movie

Movie 3b: Same as in movie 3a, but for the quasi-periodic state $MSW_1$ at the same point in parameter space. Movies 3a and b correspond to figure 10 in the paper.

Download Panades et al. supplementary movie(Video)
Video 724.9 KB

Panades et al. supplementary movie

Movie 3b: Same as in movie 3a, but for the quasi-periodic state $MSW_1$ at the same point in parameter space. Movies 3a and b correspond to figure 10 in the paper.

Download Panades et al. supplementary movie(Video)
Video 738 KB

Panades et al. supplementary movie

Movie 4b: Same as in movie 3b, but strobed once per period.

Download Panades et al. supplementary movie(Video)
Video 907.2 KB

Panades et al. supplementary movie

Movie 4b: Same as in movie 3b, but strobed once per period.

Download Panades et al. supplementary movie(Video)
Video 997.1 KB

Panades et al. supplementary movie

Movie 1b: Same as in movie 1a, but for St=32 and Re=525.

Download Panades et al. supplementary movie(Video)
Video 1.2 MB

Panades et al. supplementary movie

Movie 1b: Same as in movie 1a, but for St=32 and Re=525.

Download Panades et al. supplementary movie(Video)
Video 1.4 MB

Panades et al. supplementary movie

Movie 1c: Same as in movie 1a, but for St=50 and Re=615.

Download Panades et al. supplementary movie(Video)
Video 1.3 MB

Panades et al. supplementary movie

Movie 1c: Same as in movie 1a, but for St=50 and Re=615.

Download Panades et al. supplementary movie(Video)
Video 1.4 MB

Panades et al. supplementary movie

Movie 2c: Same as in movie 2a, but for the synchronous state $A_2$ at $St=100$ and $Re=700$.

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Video 683.7 KB

Panades et al. supplementary movie

Movie 2c: Same as in movie 2a, but for the synchronous state $A_2$ at $St=100$ and $Re=700$.

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Video 715.6 KB

Panades et al. supplementary movie

Movie 1d: Same as in movie 1a, but for St=100 and Re=700.

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Video 1.3 MB

Panades et al. supplementary movie

Movie 1d: Same as in movie 1a, but for St=100 and Re=700.

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Video 1.4 MB

Panades et al. supplementary movie

Movie 1e: Same as in movie 1a, but showing the azimuthal component of vorticity.

Download Panades et al. supplementary movie(Video)
Video 618.9 KB

Panades et al. supplementary movie

Movie 1e: Same as in movie 1a, but showing the azimuthal component of vorticity.

Download Panades et al. supplementary movie(Video)
Video 663.6 KB

Panades et al. supplementary movie

Movie 1f: Same as in movie 1b, but showing the azimuthal component of vorticity.

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Video 692.6 KB

Panades et al. supplementary movie

Movie 1f: Same as in movie 1b, but showing the azimuthal component of vorticity.

Download Panades et al. supplementary movie(Video)
Video 744.1 KB

Panades et al. supplementary movie

Movie 1g: Same as in movie 1c, but showing the azimuthal component of vorticity.

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Video 707.2 KB

Panades et al. supplementary movie

Movie 1g: Same as in movie 1c, but showing the azimuthal component of vorticity.

Download Panades et al. supplementary movie(Video)
Video 768.3 KB

Panades et al. supplementary movie

Movie 1h: Same as in movie 1d, but showing the azimuthal component of vorticity.

Download Panades et al. supplementary movie(Video)
Video 645.5 KB

Panades et al. supplementary movie

Movie 1h: Same as in movie 1d, but showing the azimuthal component of vorticity.

Download Panades et al. supplementary movie(Video)
Video 722 KB