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Modulated waves in a periodically driven annular cavity

Published online by Cambridge University Press:  25 November 2010

H. M. BLACKBURN  and
J. M. LOPEZ
H. M. BLACKBURN*
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, Vic 3800, Australia
J. M. LOPEZ
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
*
†Email address for correspondence: Hugh.Blackburn@monash.edu
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Abstract

Time-periodic flows with spatio-temporal symmetry Z2 × O(2) – invariance in the spanwise direction generating the O(2) symmetry group and a half-period-reflection symmetry in the streamwise direction generating a spatio-temporal Z2 symmetry group – are of interest largely because this is the symmetry group of periodic laminar two-dimensional wakes of symmetric bodies. Such flows are the base states for various three-dimensional instabilities; the periodically shedding two-dimensional circular cylinder wake with three-dimensional modes A and B being the generic example. However, it is not easy to physically realize the ideal flows owing to the presence of end effects and finite spanwise geometries. Flows past rings are sometimes advanced as providing a relevant idealization, but in fact these have symmetry group O(2) and only approach Z2 × O(2) symmetry in the infinite aspect ratio limit. The present work examines physically realizable periodically driven annular cavity flows that possess Z2 × O(2) spatio-temporal symmetry. The flows have three distinct codimension-1 instabilities: two synchronous modes (A and B), and two manifestations of a quasi-periodic (QP) mode, either as modulated standing waves or modulated travelling waves. It is found that the curvature of the system can determine which of these modes is the first to become unstable with increasing Reynolds number, and that even in the nonlinear regime near onset of three-dimensional instabilities the dynamics are dominated by mixed modes with complicated spatio-temporal structure. Supplementary movies illustrating the spatio-temporal dynamics are available at journals.cambridge.org/flm.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Pattern formation Wakes/Jets: Vortex streets
Type
Papers
Information
Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 336 - 357
Copyright
Copyright © Cambridge University Press 2010

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References

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Blackburn supplementary material

Movie 1. Perspective cut-away view of azimuthal vorticity isosurfaces in the base flow for radius ratio (driven/stationary) Ψ=4/3 (i.e. where the outer wall is driven), St=100 and Re=1200; the outer wall translates ±12/2π cavity heights during each motion cycle for this parameter combination. The two isosurfaces (red and yellow) have vorticity of equal magnitude but opposite sign, initially forming as shear layers on the oscillatory outer wall.

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Video 391.7 KB

Blackburn supplementary material

Movie 2. Mode QP modulated travelling wave (TW) solution for Ψ=4/3, Re=1200, in the k=33 subspace. Translucent red/blue isosurfaces represent azimuthal vorticity component, while solid red/yellow isosurfaces represent radial vorticity component of equal magnitude and opposite sign. The animation runs for four wall motion cycles (in a loop): during this time the wave travels almost a complete azimuthal wavelength. Note that the shape of the wave is identical (modulo a constant azimuthal translation) after each motion cycle.

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Video 4.2 MB

Blackburn supplementary material

Movie 3. Mode QP modulated standing wave (SW) solution for Ψ=4/3, Re=1200, in the k=33 subspace. Translucent red/blue isosurfaces represent azimuthal vorticity component, while solid red/yellow isosurfaces represent radial vorticity component of equal magnitude and opposite sign. As for movie 2, the animation runs over four wall cycles, but in this case the shape of the wave is different at the end of each cycle, as a result of quasi-periodicity and spatial symmetry.

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Video 3.3 MB

Blackburn supplementary material

Movie 4. Mode A solution for Ψ=4/3, Re=1320, in the k=7 subspace. Translucent red/blue isosurfaces represent azimuthal vorticity component, while solid red/yellow isosurfaces represent radial vorticity component of equal magnitude and opposite sign. In contrast to the quasi-periodic states shown in movies 2 and 3, this mode is synchronous with the wall motion: no new frequency is introduced with the bifurcation, and the structure stays in a fixed azimuthal location. The animation represents one wall motion cycle (running in a loop).

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Video 1.5 MB

Blackburn supplementary material

Movie 5. Full-annulus simulation for Ψ=4/3, Re=1200 during the mixed-mode phase, unfiltered. View is along axis, and strobed at floor period. The animation runs over 140 wall motion cycles, i.e. approximately two of the long-period modulations of energy seen in figure 5. Isosurfaces represent azimuthal velocity component at levels ±9% of the peak wall speed.

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Video 6 MB

Blackburn supplementary material

Movie 6. As for movie 5, but radial view.

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Video 2.8 MB

Blackburn supplementary material

Movie 7. Full-annulus simulation for Ψ=4/3, Re=1200 during the mixed-mode phase — the same data as for movie 5, but here spatially filtered at k=33 and harmonics to represent TW component of the mixed-mode. View along axis, strobed at floor period. Isosurfaces are of azimuthal velocity component. Note the clear long-period modulation of energy in this state.

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Video 5.6 MB

Blackburn supplementary material

Movie 8. As for movie 7, but using a radial view.

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Video 3.5 MB

Blackburn supplementary material

Movie 9. Full-annulus simulation for Ψ=4/3, Re=1200 during the mixed-mode phase, as for movie 5, but here spatially filtered at k=6 and harmonics to represent the mode A component of the mixed-mode. View along axis, strobed at floor period. Isosurfaces are of azimuthal velocity component. Note the long-period modulation of energy in this state, and also that it rotates retrograde compared to the sense shown in the k=33n-filtered structures seen in movie 7.

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Video 3.3 MB

Blackburn supplementary material

Movie 10. As for movie 9, but radial view.

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Video 2.3 MB

Blackburn supplementary material

Movie 11. Full-annulus simulation for Ψ=4/3, Re=1200 during the fully-saturated phase, no spatial filtering. View is along axis, strobed at floor period, and again the animation represents 140 wall motion cycles.

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Video 6.5 MB

Blackburn supplementary material

Movie 12. As for movie 11, but radial view.

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Video 2.8 MB