Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-19T15:16:16.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Modulated rotating convection: radially travelling concentric rolls

Published online by Cambridge University Press:  11 July 2008

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

Abstract

Recent experiments in rotating convection have shown that the spatio-temporal bulk convective state with Küppers–Lortz dynamics can be suppressed by small-amplitude modulations of the rotation rate. The resultant axisymmetric pulsed target patterns were observed to develop into axisymmetric travelling target patterns as the modulation amplitude and Rayleigh number were increased. Using the Navier–Stokes–Boussinesq equations with physical boundary conditions, we are able to numerically reproduce the experimental results and gain physical insight into the responsible mechanism, relating the onset of the travelling target patterns to a symmetry-restoring saddle-node on an invariant circle bifurcation. Movies are available with the online version of the paper.

JFM classification

Nonlinear Dynamical Systems: Pattern formation Nonlinear Dynamical Systems: Chaos
Type
Papers
Information
Journal of Fluid Mechanics , Volume 608 , 10 August 2008 , pp. 357 - 378
Copyright
Copyright © Cambridge University Press 2008

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

Abshagen, J., Lopez, J. M., Marques, F. & Pfister, G. 2005 a Mode competition of rotating waves in reflection-symmetric Taylor–Couette flow. J. Fluid Mech. 540, 269–299.CrossRefGoogle Scholar
Abshagen, J., Lopez, J. M., Marques, F. & Pfister, G. 2005 b Symmetry breaking via global bifurcations of modulated rotating waves in hydrodynamics. Phys. Rev. Lett. 94, 074101.CrossRefGoogle ScholarPubMed
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
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Becker, N., Scheel, J. D., Cross, M. C. & Ahlers, G. 2006 Effect of the centrifugal force on domain chaos in Rayleigh–Bénard convection. Phys. Rev. E 73, 066309.Google ScholarPubMed
Bhattacharjee, J. K. 1990 Convective instability in a rotating fluid layer under modulation of the rotating rate. Phys. Rev. A 41, 5491–5494.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Davis, S. H. 1976 The stability of time-periodic flows. Annu. Rev. Fluid Mech. 8, 57–74.CrossRefGoogle Scholar
Fornberg, B. 1998 A Practical Guide to Pseudospectral Methods. Cambridge University Press.Google Scholar
Hugues, S. & Randriamampianina, A. 1998 An improved projection scheme applied to pseudospectral methods for the incompressible Navier–Stokes equations. Intl J. Numer. Meth. Fluids 28, 501–521.3.0.CO;2-S>CrossRefGoogle Scholar
yon Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.CrossRefGoogle Scholar
Knobloch, E. 1998 Rotating convection: recent developments. Intl J.Engng Sci. 36, 1421–1450.CrossRefGoogle Scholar
Küppers, G. & Lortz, D. 1969 Transition from laminar convection to thermal turbulence in a rotating fluid layer. J. Fluid Mech. 35, 609–620.CrossRefGoogle Scholar
Kuznetsov, Y. A. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.CrossRefGoogle Scholar
Lopez, J. M., Rubio, A. & Marques, F. 2006 Travelling circular waves in axisymmetric rotating convection. J. Fluid Mech. 569, 331–348.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.CrossRefGoogle Scholar
Lopez, J. M., Cui, Y. D., Marques, F. & Lim, T. T. 2008 Quenching of vortex breakdown oscillations via harmonic modulation. J. Fluid Mech. 599, 441–464.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., 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.CrossRefGoogle Scholar
Mercader, I., Net, M. & Falqués, A. 1991 Spectral methods for high order equations. Comput. Meth. Appl. Mech. Engng 91, 1245–1251.CrossRefGoogle Scholar
Niemela, J. J., Smith, M. R. & Donnelly, R. J. 1991 Convective instability with time-varying rotation. Phys. Rev. A 44, 8406–8409.CrossRefGoogle ScholarPubMed
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.CrossRefGoogle Scholar
Roppo, M. N., Davis, S. H. & Rosenblat, S. 1984 Bénard convection with time-periodic heating. Phys. Fluids 27, 796–803.CrossRefGoogle Scholar
Roxin, A. & Riecke, H. 2002 Rotating convection in an anisotropic system. Phys. Rev. E 65, 046219.Google Scholar
Sánchez-Álvarez, J. J., Serre, E., Crespo del Arco, E. & Busse, F. H. 2005 Square patterns in rotating Rayleigh–Bénard convection. Phys. Rev. E 036307.CrossRefGoogle Scholar
Scheel, J. D. & Cross, M. C. 2005 Scaling laws for rotating Rayleigh–Bénard convection. Phys. Rev. E 72, 056315.Google ScholarPubMed
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
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8.Google Scholar
Thompson, K. L., Bajaj, K. M. S. & Ahlers, G. 2002 Travelling concentric-roll patterns in Rayleigh–Bénard convection with modulated rotation. Phys. Rev. E 65, 046218.Google ScholarPubMed
Tuckerman, L. S. & Barkley, D. 1988 Global bifurcation to travelling waves in axisymmetric convection. Phys. Rev. Lett. 61, 408–411.CrossRefGoogle ScholarPubMed
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
Yih, C.-S. 1977 Fluid Mechanics. West River Press, Ann Arbor, Michigan.Google Scholar

Rubio et al. supplementary movie

Movie 1. Numerical simulation of Küpper--Lortz spatio-temporal chaos in Rayleigh--Bénard convection with constant rotation, showing contours of the temperature perturbation field, Θ, at mid-height (z=0) for Rayleigh number Ra=2868.8, mean angular velocity Ω0=23.6 and aspect ratio γ=11.8 over 450 viscous times at 108 viscous times per second, repeated 10 times. This movie corresponds to figure 1 in the paper.

Download Rubio et al. supplementary movie(Video)
Video 4.9 MB

Rubio et al. supplementary movie

Movie 2. Numerical simulation of a pulsed target pattern in Rayleigh--Bénard convection with modulated rotation. At each forcing period, Stokes layers form at the top and bottom boundaries before being deflected into the sidewall to create a strong radial jet that displaces the target pattern over the course of the forcing period. Shown are contours of the temperature field for Ra=2700, Ω0=Ωm=23.6 (where Ωm is the modulation frequency),  relative modulation amplitude A=0.05 and γ=11.8. Ten forcing periods are shown at a rate of 0.266 viscous times per second. This movie corresponds to figure 7(b) in the paper.

Download Rubio et al. supplementary movie(Video)
Video 2.9 MB

Rubio et al. supplementary movie

Movie 3. Numerical simulation of a travelling target state in Rayleigh--Bénard convection with modulated rotation. By raising the Rayleigh number from Ra=2700 to Ra=2705 the target pattern loses stability to a travelling target pattern in which the axisymmetric plumes slowly recede into the centre. Shown are contours of the temperature field for Ra=2705, Ω0=Ωm=23.6, A=0.05 and γ=11.8 over 2500 viscous times at a rate of 102 viscous times per second. This movie corresponds to the space-time diagram shown in figure 12(a) in the paper.

Download Rubio et al. supplementary movie(Video)
Video 489.5 KB

Rubio et al. supplementary movie

Movie 4. Numerical simulation of a travelling target state. Beyond the onset of the travelling target patterns, the solution behaves more like a harmonic oscillator, although the slow-fast nature of the solution is still evident. Shown are contours of the temperature field at Ra=2844, Ω0=Ωm=23.6, A=0.18 and γ=11.8 over 2500 viscous times at a rate of 102 viscous times per second. This movie corresponds to the space--time diagram shown in figure 12(b) in the paper.

Download Rubio et al. supplementary movie(Video)
Video 1.7 MB

Rubio et al. supplementary movie

Movie 5. Numerical simulation of a travelling target state. Far beyond the onset of the travelling target patterns, the solution behaves like a harmonic oscillator. Shown are contours of the temperature field at Ra=2840, Ω0=Ωm=23.6, A=0.06 and γ=11.8 over 2500 viscous times at a rate of 102 viscous times per second. This movie corresponds to the space--time diagram shown in figure 12(c) in the paper.

Download Rubio et al. supplementary movie(Video)
Video 5.2 MB