Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-20T07:44:20.108Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Avila, M., Marques, F., Lopez, J. M. & Meseguer, A. 2007 Stability control and catastrophic transition in a forced Taylor–Couette system. J. Fluid Mech. 590, 471–496.CrossRefGoogle Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57, 1437–1458.CrossRefGoogle Scholar
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Barkley, D., Tuckerman, L. S. & Golubitsky, M. 2000 Bifurcation theory for three-dimensional flow in the wake of a circular cylinder. Phys. Rev. E 61, 5247–5252.CrossRefGoogle ScholarPubMed
Blackburn, H. M. 2002 Three-dimensional instability and state selection in an oscillatory axisymmetric swirling flow. Phys. Fluids 14 (11), 3983–3996.CrossRefGoogle Scholar
Blackburn, H. M. & Lopez, J. M. 2002 Modulated rotating waves in an enclosed swirling flow. J. Fluid Mech. 465, 33–58.CrossRefGoogle Scholar
Blackburn, H. M. & Lopez, J. M. 2003 a On three-dimensional quasi-periodic Floquet instabilities of two-dimensional bluff body wakes. Phys. Fluids 15 (8), L57–L60.CrossRefGoogle Scholar
Blackburn, H. M. & Lopez, J. M. 2003 b The onset of three-dimensional standing and modulated travelling waves in a periodically driven cavity flow. J. Fluid Mech. 497, 289–317.CrossRefGoogle Scholar
Blackburn, H. M., Marques, F. & Lopez, J. M. 2005 Symmetry breaking of two-dimensional time-periodic wakes. J. Fluid Mech. 522, 395–411.CrossRefGoogle Scholar
Blackburn, H. M. & Sheard, G. J. 2010 On quasiperiodic and subharmonic Floquet wake instabilities. Phys. Fluids 22, 031701–1–4.CrossRefGoogle Scholar
Blackburn, H. M. & Sherwin, S. J. 2004 Formulation of a Galerkin spectral element-Fourier method for three-dimensional incompressible flows in cylindrical geometries. J. Comput. Phys. 197 (2), 759–778.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Carmo, B. S., Sherwin, S. J., Bearman, P. W. & Willden, R. H. J. 2008 Wake transition in flow around two circular cylinders in staggered arrangements. J. Fluid Mech. 597, 1–29.CrossRefGoogle Scholar
Crawford, J. D. & Knobloch, E. 1991 Symmetry and symmetry-breaking bifurcations in fluid dynamics. Annu. Rev. Fluid Mech. 23, 341–387.CrossRefGoogle Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
Henderson, R. D. 1997 Nonlinear dynamics and pattern formation in three-dimensional wakes. J. Fluid Mech. 352, 65–112.CrossRefGoogle Scholar
Henderson, R. D. & Barkley, D. 1996 Secondary instability in the wake of a circular cylinder. Phys. Fluids 8 (6), 1683–1685.CrossRefGoogle Scholar
Julien, S., Lasheras, J. & Chomaz, J.-M. 2003 Three-dimensional instability and vorticity patterns in the wake of a flat plate. J. Fluid Mech. 479, 155–189.CrossRefGoogle Scholar
Kuznetsov, Y. A. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.CrossRefGoogle Scholar
Lasheras, J. C. & Meiburg, E. 1990 Three-dimensional vorticity modes in the wake of a flat plate. Phys. Fluids A 5, 371–380.CrossRefGoogle Scholar
Leung, J. J. F., Hirsa, A., Blackburn, H. M., Lopez, J. M. & Marques, F. 2005 Three-dimensional modes in a periodically driven elongated cavity. Phys. Rev. E 71, 026305–1–7.CrossRefGoogle Scholar
Leweke, T. & Provansal, M. 1995 The flow behind rings: bluff body wakes without end effects. J. Fluid Mech. 288, 265–310.CrossRefGoogle Scholar
Marques, F. & Lopez, J. M. 1997 Taylor–Couette flow with axial oscillations of the inner cylinder: Floquet analysis of the basic flow. J. Fluid Mech. 348, 153–175.CrossRefGoogle Scholar
Marques, F. & Lopez, J. M. 2000 Spatial and temporal resonances in a periodically forced extended system. Physica D 136, 340–352.CrossRefGoogle Scholar
Marques, F., Lopez, J. M. & Blackburn, H. M. 2004 Bifurcations in systems with Z 2 spatio-temporal and O(2) spatial symmetry. Physica D 189 (3–4), 247–276.CrossRefGoogle Scholar
Meiburg, E. & Lasheras, J. C. 1988 Experimental and numerical investigation of the three-dimensional transition in plane wakes. J. Fluid Mech. 190, 1–37.CrossRefGoogle Scholar
Rubio, A., Lopez, J. M. & Marques, F. 2010 Onset of Küppers–Lortz-like dynamics in finite rotating thermal convection. J. Fluid Mech. 644, 337–357.CrossRefGoogle Scholar
Sheard, G. J., Fitzgerald, M. J. & Ryan, K. 2009 Cylinders with square cross-section: wake instabilities with incidence angle variation. J. Fluid Mech. 630, 43–69.CrossRefGoogle Scholar
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2003 From spheres to circular cylinders: the stability and flow structures of bluff ring wakes. J. Fluid Mech. 492, 147–180.CrossRefGoogle Scholar
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2004 From spheres to circular cylinders: non-axisymmetric transitions in the flow past rings. J. Fluid Mech. 506, 45–78.CrossRefGoogle Scholar
Sinha, M., Kevrekidis, I. G. & Smits, A. J. 2006 Experimental study of a Neimark–Sacker bifurcation in axially forced Taylor–Couette flow. J. Fluid Mech. 558, 1–32.CrossRefGoogle Scholar
Swift, J. W. & Wiesenfeld, K. 1984 Suppression of period doubling in symmetric systems. Phys. Rev. Lett. 52 (9), 705–708.CrossRefGoogle Scholar
Tuckerman, L. S. & Barkley, D. 2000 Bifurcation analysis for time steppers. In Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems (ed. Doedel, E. & Tuckerman, L. S.), pp. 453–566. Springer.CrossRefGoogle Scholar
Vogel, M. J., Hirsa, A. H. & Lopez, J. M. 2003 Spatio-temporal dynamics of a periodically driven cavity flow. J. Fluid Mech. 478, 197–226.CrossRefGoogle Scholar
Weisberg, A. Y., Kevrekidis, I. G. & Smits, A. J. 1997 Delaying transition in Taylor–Couette flow with axial motion of the inner cylinder. J. Fluid Mech. 348, 141–151.CrossRefGoogle Scholar
Williamson, C. H. K. 1989 Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 206, 579–627.CrossRefGoogle Scholar
Williamson, C. H. K. 1996 Three-dimensional wake transition. J. Fluid Mech. 328, 345–407.CrossRefGoogle Scholar

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.

Download Blackburn supplementary material(Video)
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.

Download Blackburn supplementary material(Video)
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.

Download Blackburn supplementary material(Video)
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).

Download Blackburn supplementary material(Video)
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.

Download Blackburn supplementary material(Video)
Video 6 MB

Blackburn supplementary material

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

Download Blackburn supplementary material(Video)
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.

Download Blackburn supplementary material(Video)
Video 5.6 MB

Blackburn supplementary material

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

Download Blackburn supplementary material(Video)
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.

Download Blackburn supplementary material(Video)
Video 3.3 MB

Blackburn supplementary material

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

Download Blackburn supplementary material(Video)
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.

Download Blackburn supplementary material(Video)
Video 6.5 MB

Blackburn supplementary material

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

Download Blackburn supplementary material(Video)
Video 2.8 MB