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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: lopez@math.la.asu.edu
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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

REFERENCES

Allen, J. J. & Lopez, J. M. 2007 Transition processes for junction vortex flow. J. Fluid Mech. 585, 457–467.CrossRefGoogle Scholar
Avila, M., Belisle, M. J., Lopez, J. M., Marques, F. & Saric, W. S. 2008 Mode competition in modulated Taylor–Couette flow. J. Fluid Mech. 601, 381–406.CrossRefGoogle Scholar
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.Google 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.Google Scholar
Blackburn, H. M. & Lopez, J. M. 2003 a On three-dimensional quasiperiodic Floquet instabilities of two-dimensional bluff body wakes. Phys. Fluids 15, L57–L60.Google 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.Google Scholar
Blackburn, H. M. & Lopez, J. M. 2011 Modulated waves in a periodically driven annular cavity. J. Fluid Mech. 667, 336–357.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.Google Scholar
Chossat, P. & Iooss, G. 1994 The Couette–Taylor Problem. Springer.CrossRefGoogle Scholar
Chossat, P. & Lauterbach, R. 2000 Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific.CrossRefGoogle Scholar
Crawford, J. D. & Knobloch, E. 1991 Symmetry and symmetry-breaking bifurcations in fluid dynamics. Annu. Rev. Fluid Mech. 23, 341–387.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
Do, Y., Lopez, J. M. & Marques, F. 2010 Optimal harmonic response in a confined Bödewadt boundary layer flow. Phys. Rev. E 82, 036301.Google Scholar
Golubitsky, M. & Schaeffer, D. G. 1985 Singularities and Groups in Bifurcation Theory, vol. I. Springer.CrossRefGoogle Scholar
Golubitsky, M. & Stewart, I. 2002 The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. Birkhäuser.CrossRefGoogle Scholar
Golubitsky, M., Stewart, I. & Schaeffer, D. G. 1988 Singularities and Groups in Bifurcation Theory, vol. II. Springer.Google Scholar
Hoyle, R. 2006 Pattern Formation, An Introduction to Methods. Cambridge University Press.Google Scholar
Iooss, G. & Adelmeyer, M. 1998 Topics in Bifurcation Theory and Applications, 2nd edn. World Scientific.Google Scholar
Kuznetsov, Y. A. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.CrossRefGoogle Scholar
Lamb, J. S. W. 1996 Local bifurcations in k-symmetric dynamical systems. Nonlinearity 9, 537–557.CrossRefGoogle Scholar
Lamb, J. S. W. & Melbourne, I. 1999 Bifurcation from discrete rotating waves. Arch. Rat. Mech. Anal. 149, 229–270.CrossRefGoogle Scholar
Leung, J. J. F., Hirsa, A. H., Blackburn, H. M., Marques, F. & Lopez, J. M. 2005 Three-dimensional modes in a periodically driven elongated cavity. Phys. Rev. E 71, 026305.Google Scholar
Lopez, J. M. & Hirsa, A. 2001 Oscillatory driven cavity with an air/water interface and an insoluble monolayer: Surface viscosity effects. J. Colloid Interface Sci. 242, 1–5.CrossRefGoogle Scholar
Lopez, J. M. & Marques, F. 2000 Dynamics of 3-tori in a periodically forced Navier–Stokes flow. Phys. Rev. Lett. 85, 972–975.Google Scholar
Lopez, J. M. & Marques, F. 2009 Centrifugal effects in rotating convection: nonlinear dynamics. J. Fluid Mech. 628, 269–297.CrossRefGoogle Scholar
Lopez, J. M., Marques, F., Mercader, I. & Batiste, O. 2007 Onset of convection in a moderate aspect-ratio rotating cylinder: Eckhaus–Benjamin–Feir instability. J. Fluid Mech. 590, 187–208.Google Scholar
Lopez, J. M., Marques, F., Rubio, A. M. & Avila, M. 2009 Crossflow instability of finite Bödewadt flows: transients and spiral waves. Phys. Fluids 21, 114107.Google Scholar
Lopez, J. M. & Shen, J. 1998 An efficient spectral-projection method for the Navier–Stokes equations in cylindrical geometries I. Axisymmetric cases. J. Comput. Phys. 139, 308–326.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.Google Scholar
Marques, F. & Lopez, J. M. 2000 Spatial and temporal resonances in a periodically forced extended system. Physica D 136, 340–352.Google 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, 247–276.Google Scholar
Marques, F., Mercader, I., Batiste, O. & Lopez, J. M. 2007 Centrifugal effects in rotating convection: axisymmetric states and three-dimensional instabilities. J. Fluid Mech. 580, 303–318.Google Scholar
Mercader, I., Batiste, O. & Alonso, A. 2010 An efficient spectral code for incompressible flows in cylindrical geometries. Comput. Fluids 39, 215–224.Google Scholar
Mercader, I., Net, M. & Falqués, A. 1991 Spectral methods for high order equations. Comput. Meth. Appl. Mech. Engng 91, 1245–1251.Google 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.Google Scholar
Schlichting, H. & Kestin, J. 1979 Boundary-Layer Theory, 7th edn. McGraw-Hill.Google 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

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