Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-08T16:46:12.197Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability control and catastrophic transition in a forced Taylor–Couette system

Published online by Cambridge University Press:  15 October 2007

M. AVILA ,
F. MARQUES ,
J. M. LOPEZ  and
A. MESEGUER
M. AVILA
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
F. MARQUES
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
J. M. LOPEZ
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
A. MESEGUER
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

Harmonic axial motion of the inner cylinder in the Taylor–Couette system can efficiently shift the onset of instability to larger inner cylinder rotation rates. However, once instability has set in, a rapid sequence of symmetry-breaking bifurcations results in complex spatio-temporal dynamics even for very low post-critical values of the rotation rate. Using spectral computations, we present a detailed study of this sudden transition, shedding light on the nature of the complex flows observed in recent laboratory experiments. In particular, it is shown that these bifurcations are responsible for some of the experimentally observed frequencies which had been attributed to background noise. Movies are available with the online version of the paper.

JFM classification

Turbulent Flows: Turbulent transition Nonlinear Dynamical Systems: Bifurcation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 590 , 10 November 2007 , pp. 471 - 496
Copyright
Copyright © Cambridge University Press 2007

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

Ali, M. E. & Weidman, P. D. 1993 On the linear stability of cellular spiral Couette flow. Phys. Fluids A 5, 1188–1200.CrossRefGoogle Scholar
Andereck, C. D., Liu, S. S. & Swinney, H. L. 1986 Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183.CrossRefGoogle Scholar
Arnold, V. I., Afrajmovich, V. S., Il'yashenko, Y. S. & Shil'nikov, L. P. 1999 Bifurcation Theory and Catastrophe Theory. Springer.Google Scholar
Avila, M., Meseguer, A. & Marques, F. 2006 Double Hopf bifurcation in corotating spiral Poiseuille flow. Phys. Fluids 18, 064101.CrossRefGoogle Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.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
Hu, H. C. & Kelly, R. E. 1995 Effect of a time-periodic axial shear flow upon the onset of Taylor vortices. Phys. Rev. E 51, 3242–3251.Google ScholarPubMed
Iooss, G. & Adelmeyer, M. 1998 Topics in Bifurcation Theory and Applications, 2nd edn. Advanced Series in Nonlinear Dynamics, vol. 3. World Scientific.Google Scholar
Jones, C. A. 1985 Numerical methods for the transition to wavy Taylor vortices. J. Comput. Phys. 61, 321–344.CrossRefGoogle Scholar
Krupa, M. 1990 Bifurcations of relative equilibria. SIAM J. Math. Anal. 21, 1453–1486.CrossRefGoogle Scholar
Kuznetsov, Y. A. 1998 Elements of Applied Bifurcation Theory, 2nd edn. Springer.Google Scholar
Lamb, J. S. W. & Melbourne, I. 1999 Bifurcation from discrete rotating waves. Arch. Rat. Mech. Anal. 149, 229–270.Google Scholar
Lamb, J. S. W., Melbourne, I. & Wulff, C. 2003 Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions. J. Diffl Equat. 191, 377–407.CrossRefGoogle Scholar
Ludwieg, H. 1964 Experimentelle Nachprufung des stabilitatstheorien fur reibungsfreie Stromungen mit schraubenlinienformigen stromlinien. Z. Flugwiss 12, 304–309.Google Scholar
Lueptow, R. M., Docter, A. & Min, K. 1992 Stability of axial flow in an annulus with a rotating inner cylinder. Phys. Fluids A 4, 2446–2455.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 hydrodynamic 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
Meseguer, A. & Marques, F. 2000 On the competition between centrifugal and shear instability in spiral Couette flow. J. Fluid Mech. 402, 33–56.Google Scholar
Meseguer, A. & Marques, F. 2002 On the competition between centrifugal and shear instability in spiral Poiseuille flow. J. Fluid Mech. 455, 129–148.Google Scholar
Rand, D. 1982 Dynamics and symmetry. Predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79, 1–37.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
Snyder, H. A. 1962 Experiments on the stability of spiral flow at low axial Reynolds numbers. Proc. R. Soc. Lond. A 265, 198–214.Google Scholar
Swift, J. W. & Wiesenfeld, K. 1984 Suppression of period doubling in symmetric systems. Phys. Rev. Lett. 52, 705–708.Google Scholar
Tagg, R. 1994 The Couette-Taylor problem. Nonlinear Sci. Today 4, 1–25.Google Scholar
Takeuchi, D. I. & Jankowski, D. F. 1981 A numerical and experimental investigation of the stability of spiral Poiseuille flow. J. Fluid Mech. 102, 101–126.CrossRefGoogle Scholar
Tsameret, A. & Steinberg, V. 1994 Competing states in a Couette–Taylor system with an axial flow. Phys. Rev. E 49, 4077–4086.Google 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
Wereley, S. T. & Lueptow, R. M. 1999 Velocity field for Taylor-Couette flow with an axial flow. Phys. Fluids 11, 3637–3649.CrossRefGoogle Scholar
Wulff, C., Lamb, J. S. W. & Melbourne, I. 2001 Bifurcation from relative periodic solutions. Ergod. Theor. Dyna. Sysst. 21, 605–635.Google Scholar

Avila et al, supplementary movie

Movie 1. (a) Planar rendering of a gray-scale colormap of the azimuthal vorticity of the perturbation u at the outer cylinder r=ro, where black (white) corresponds to positive (negative) values. (b) Distribution of the kinetic modal energy decomposition, where black corresponds to the energy of the leading mode and white to a level seven orders of magnitude lower. The flow is a left-handed M1 spiral wave at A=79.6 and Re=246.32. Note that M1 is a rotating and traveling wave, precessing in the positive azimuthal direction while traveling in the positive axial direction. The spiral angle is about -23 degrees.

Download Avila et al, supplementary movie(Video)
Video 961.1 KB

Avila et al, supplementary movie

Movie 1. (a) Planar rendering of a gray-scale colormap of the azimuthal vorticity of the perturbation u at the outer cylinder r=ro, where black (white) corresponds to positive (negative) values. (b) Distribution of the kinetic modal energy decomposition, where black corresponds to the energy of the leading mode and white to a level seven orders of magnitude lower. The flow is a left-handed M1 spiral wave at A=79.6 and Re=246.32. Note that M1 is a rotating and traveling wave, precessing in the positive azimuthal direction while traveling in the positive axial direction. The spiral angle is about -23 degrees.

Download Avila et al, supplementary movie(Video)
Video 779.9 KB

Avila et al. supplementary movie

Movie 2. Perspective view (parallel projection of the cylinder). The gray-scale colormap shows the intensity of azimuthal vorticity of the perturbation u for left and right handed M1 spiral waves respectively. The left-handed M1 is the same as in Movie 1. The right-handed M1 also precesses in the positive azimuthal direction, but travels in the negative axial direction. Its spiral angle is about 23 degrees. Note that the motion of the spiral patterns is modulated by the harmonic axial forcing. The parameter values are A=79.6 and Re=246.32.

Download Avila et al. supplementary movie(Video)
Video 905.7 KB

Avila et al. supplementary movie

Movie 2. Perspective view (parallel projection of the cylinder). The gray-scale colormap shows the intensity of azimuthal vorticity of the perturbation u for left and right handed M1 spiral waves respectively. The left-handed M1 is the same as in Movie 1. The right-handed M1 also precesses in the positive azimuthal direction, but travels in the negative axial direction. Its spiral angle is about 23 degrees. Note that the motion of the spiral patterns is modulated by the harmonic axial forcing. The parameter values are A=79.6 and Re=246.32.

Download Avila et al. supplementary movie(Video)
Video 804.2 KB

Avila et al. supplementary movie

Movie 3. Same as Movie 2 but for M2 wavy spirals at A=79.6 and Re=247. The time-dependent inclination of the spirals and the waviness, associated to the fast wavy frequency, can be clearly cobserved in contrast to Movie 2.

Download Avila et al. supplementary movie(Video)
Video 1.1 MB

Avila et al. supplementary movie

Movie 3. Same as Movie 2 but for M2 wavy spirals at A=79.6 and Re=247. The time-dependent inclination of the spirals and the waviness, associated to the fast wavy frequency, can be clearly cobserved in contrast to Movie 2.

Download Avila et al. supplementary movie(Video)
Video 777.2 KB

Avila et al. supplementary movie

Movie 4. Same as Movie 1. The flow is wavy spirals (M2) at A=79.6 and Re=247. Waviness is superposed now to the spirals which also feature a time-dependent angle. The modal kinetic energy is changed by the presence of the wavy mode (l,n)=(0,5).

Download Avila et al. supplementary movie(Video)
Video 986.4 KB

Avila et al. supplementary movie

Movie 4. Same as Movie 1. The flow is wavy spirals (M2) at A=79.6 and Re=247. Waviness is superposed now to the spirals which also feature a time-dependent angle. The modal kinetic energy is changed by the presence of the wavy mode (l,n)=(0,5).

Download Avila et al. supplementary movie(Video)
Video 842.8 KB

Avila et al. supplementary movie

Movie 5. Sames as Movie 3 for secondary wavy spirals (M3) at A=79.6 and Re=249. The flow is very similar to the wavy spirals. However, the defects on the wavy azimuthal wavenumber are visible in contrast to Movie 3.

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

Avila et al. supplementary movie

Movie 5. Sames as Movie 3 for secondary wavy spirals (M3) at A=79.6 and Re=249. The flow is very similar to the wavy spirals. However, the defects on the wavy azimuthal wavenumber are visible in contrast to Movie 3.

Download Avila et al. supplementary movie(Video)
Video 861 KB

Avila et al. supplementary movie

Movie 6. Sames as Movie 4. The flow is M3 at A=79.6 and Re=249. Deffects on the wavy azimuthal wave number can be clearly observed in M3 as opposite to M2 (compare with Movie 4). This can also be seen in the modal kinetic energies distribution featuring a broad-band of azimuthal wavenumbers.

Download Avila et al. supplementary movie(Video)
Video 1.1 MB

Avila et al. supplementary movie

Movie 6. Sames as Movie 4. The flow is M3 at A=79.6 and Re=249. Deffects on the wavy azimuthal wave number can be clearly observed in M3 as opposite to M2 (compare with Movie 4). This can also be seen in the modal kinetic energies distribution featuring a broad-band of azimuthal wavenumbers.

Download Avila et al. supplementary movie(Video)
Video 962.7 KB

Avila et al. supplementary movie

Movie 7. Sames as Movies 4 and 6. The flow is defective wavy spirals (M4) at A=79.6 and Re=250. M4 has lost all the symmetries and deffects on the axial and azimuthal wavenumbers can be clearly observed. The modal kinetic energy features a broad-band in both azimuthal and axial wavenumbers, illustrating the spatio-temporal complexity of the flow (compare to Movies 4 and 6).

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

Avila et al. supplementary movie

Movie 7. Sames as Movies 4 and 6. The flow is defective wavy spirals (M4) at A=79.6 and Re=250. M4 has lost all the symmetries and deffects on the axial and azimuthal wavenumbers can be clearly observed. The modal kinetic energy features a broad-band in both azimuthal and axial wavenumbers, illustrating the spatio-temporal complexity of the flow (compare to Movies 4 and 6).

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

Avila et al. supplementary movie

Movie 8. Same as Movies 3 and 5 for M4 defective wavy spirals at A=79.6 and Re=250. The flow is spatio-temporally complex, featuring defects in both the axial and aziumthal wavenumber (compare to Movies 3 and 5).

Download Avila et al. supplementary movie(Video)
Video 2 MB

Avila et al. supplementary movie

Movie 8. Same as Movies 3 and 5 for M4 defective wavy spirals at A=79.6 and Re=250. The flow is spatio-temporally complex, featuring defects in both the axial and aziumthal wavenumber (compare to Movies 3 and 5).

Download Avila et al. supplementary movie(Video)
Video 1.6 MB