Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-03T20:31:57.399Z Has data issue: false hasContentIssue false
  • English
  • Français

Onset of Küppers–Lortz-like dynamics in finite rotating thermal convection

Published online by Cambridge University Press:  28 January 2010

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, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The onset of thermal convection in a finite rotating cylinder is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime in which spatio-temporal complexity is observed directly after onset. The system is examined in the non-physical limit of zero centrifugal force as well as with an experimentally realizable centrifugal force, leading to two different paths to Küppers–Lortz-like spatio-temporal chaos. In the idealized case, neglecting centrifugal force, the onset of convection occurs directly from a conduction state, resulting in square patterns with slow roll switching, followed at higher thermal driving by straight roll patterns with faster roll switching. The case with a centrifugal force typical of laboratory experiments exhibits target patterns near the theoretically predicted onset of convection, followed by a rotating wave that emerges via a Hopf bifurcation. A subsequent Hopf bifurcation leads to ratcheting states with sixfold symmetry near the axis. With increasing thermal driving, roll switching is observed within the ratcheting lattice before Küppers–Lortz-like spatio-temporal chaos is observed with the dissolution of the lattice at a slightly stronger thermal driving. For both cases, all of these states are observed within a 2% variation in the thermal driving.

JFM classification

Convection: Convection Geophysical and Geological Flows: Rotating flows Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 644 , 10 February 2010 , pp. 337 - 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

Ahlers, G. 2006 Experiments with Rayleigh–Bénard convection. In Dynamics of Spatio-Temporal Cellular Structures: Henri Bénard Centenary Review (ed. Mutabazi, I., Wesfreid, J. E. & Guyon, E.), vol. 207, pp. 67–94. Springer.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
Bajaj, K. M. S., Ahlers, G. & Pesch, W. 2002 Rayleigh–Bénard convection with rotation at small Prandtl numbers. Phys. Rev. E 65, 056309.CrossRefGoogle ScholarPubMed
Bajaj, K. M. S., Liu, J., Naberhuis, B. & Ahlers, G. 1998 Square patterns in Rayleigh–Bénard convection with rotation about a vertical axis. Phys. Rev. Lett. 81, 806–809.CrossRefGoogle Scholar
Barcilon, V. & Pedlosky, J. 1967 On the steady motions produced by a stable stratification in a rapidly rotating fluid. J. Fluid Mech. 29, 673–690.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
Becker, N. & Ahlers, G. 2006 a Domain chaos puzzle and the calculation of the structure factor and its half-width. Phys. Rev. E 73, 046209.CrossRefGoogle ScholarPubMed
Becker, N. & Ahlers, G. 2006 b Local wave director analysis of domain chaos in Rayleigh–Bénard convection. J. Stat. Mech. 2006, P12002.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Bodenschatz, E., Cannell, D. S., de Bruyn, J. R., Ecke, R., Hu, Y.-C., Lerman, K. & Ahlers, G. 1992 Experiments on three systems with non-variational aspects. Physica D 61, 77–93.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Brummell, N., Hart, J. E. & Lopez, J. M. 2000 On the flow induced by centrifugal buoyancy in a differentially-heated rotating cylinder. Theoret. Comput. Fluid Dyn. 14, 39–54.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Cox, S. M. & Matthews, P. C. 2000 Instability of rotating convection. J. Fluid Mech. 403, 153–172.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., Meiron, D. & Tu, Y. 1994 Chaotic domains: a numerical investigation. Chaos 4, 607–619.CrossRefGoogle ScholarPubMed
Daniels, P. G. 1980 The effect of centrifugal acceleration on axisymmetric convection in a shallow rotating cylinder or annulus. J. Fluid Mech. 99, 65–84.CrossRefGoogle Scholar
Fantz, M., Friedrich, R., Bestehorn, M. & Haken, H. 1992 Pattern formation in rotating Bénard convection. Physica D 61, 147–154.CrossRefGoogle Scholar
Fornberg, B. 1998 A Practical Guide to Pseudospectral Methods. Cambridge University Press.Google Scholar
Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M. 1993 Convection in a rotating cylinder. Part 1. Linear theory for moderate Prandtl numbers. J. Fluid Mech. 248, 583–604.CrossRefGoogle Scholar
Golubitsky, M., LeBlanc, V. G. & Melbourne, I. 2000 Hopf bifurcation from rotating waves and patterns in physical space. J. Nonlin. Sci. 10, 69–101.CrossRefGoogle Scholar
Gorman, M., el Hamdi, M., Pearson, B. & Robbins, K. A. 1996 Ratcheting motion of concentric rings in cellular flames. Phys. Rev. Lett. 76, 228–231.CrossRefGoogle ScholarPubMed
Hart, J. E. 2000 On the influence of centrifugal buoyancy on rotating convection. J. Fluid Mech. 403, 133–151.CrossRefGoogle Scholar
Heikes, K. E. & Busse, F. H. 1980 a Convection in a rotating layer: a simple case of turbulence. Science 208, 173–175.Google Scholar
Heikes, K. E. & Busse, F. H. 1980 b Weakly nonlinear turbulence in a rotating convection layer. Ann. NY Acad. Sci. 357, 28–36.CrossRefGoogle Scholar
Homsy, G. M. & Hudson, J. L. 1969 Centrifugally driven thermal convection in a rotating cylinder. J. Fluid Mech. 35, 33–52.CrossRefGoogle Scholar
Homsy, G. M. & Hudson, J. L. 1971 Centrifugal convection and its effect on the asymptotic stability of a bounded rotating fluid heated from below. J. Fluid Mech. 48, 605–624.CrossRefGoogle Scholar
Hu, Y., Ecke, R. E. & Ahlers, G. 1997 Convection under rotation for Prandtl numbers near 1: linear stability, wavenumber selection, and pattern dynamics. Phys. Rev. E 55, 6928–6949.CrossRefGoogle Scholar
Hu, Y., Pesch, W., Ahlers, G. & Ecke, R. E. 1998 Convection under rotation for Prandtl numbers near 1: Küppers–Lortz instability. Phys. Rev. E 58, 5821–5833.CrossRefGoogle Scholar
Hughes, 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
Jayaraman, A., Scheel, J. D., Greenside, H. S. & Fisher, P. F. 2006 Characterization of the domain chaos convection state by the largest Lyapunov exponent. Phys. Rev. E 74, 016209.CrossRefGoogle ScholarPubMed
Knobloch, E. 1994 Bifurcations in rotating systems. In Lectures on Solar and Planetary Dynamos (ed. Proctor, M. R. E. & Gilbert, A. D.), pp. 331–372. Cambridge University Press.CrossRefGoogle Scholar
Knobloch, E. 1996 Symmetry and instability in rotating hydrodynamic and magnetohydrodynamic flows. Phys. Fluids 8, 1446–1454.CrossRefGoogle Scholar
Knobloch, E. 1998 Rotating convection: recent developments. Intl J. Engng Sci. 36, 1421–1450.CrossRefGoogle Scholar
Koschmieder, E. L. 1967 On convection on a uniformly heated rotating plane. Beitr. Phys. Atmos. 40, 216–225.Google Scholar
Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.Google Scholar
Küppers, G. 1970 The stability of steady finite amplitude convection in a rotating fluid layer. Phys. Lett. A 32, 7–8.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
Lewis, G. M. & Nagata, W. 2003 Double Hopf bifurcations in the differentially heated rotating annulus. SIAM J. Appl. Math. 63, 1029–1055.CrossRefGoogle Scholar
Lopez, J. M., Cui, Y. D. & Lim, T. T. 2006 a An experimental and numerical investigation of the competition between axisymmetric time-periodic modes in an enclosed swirling flow. Phys. Fluids 18, 104106.CrossRefGoogle 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.CrossRefGoogle Scholar
Lopez, J. M., Rubio, A. & Marques, F. 2006 b travelling circular waves in axisymmetric rotating convection. J. Fluid Mech. 569, 331–348.CrossRefGoogle Scholar
Marques, F., Gelfgat, A. Y. & Lopez, J. M. 2003 A tangent double Hopf bifurcation in a differentially rotating cylinder flow. Phys. Rev. E 68, 016310.CrossRefGoogle Scholar
Marques, F. & Lopez, J. M. 2008 Influence of wall modes on the onset of bulk convection in a rotating cylinder. Phys. Fluids 20, 024109.CrossRefGoogle Scholar
Marques, F., Lopez, J. M. & Shen, J. 2002 Mode interactions in an enclosed swirling flow: a double Hopf bifurcation between azimuthal wavenumbers 0 and 2. J. Fluid Mech. 455, 263–281.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. Comp. Meth. Appl. Mech. Engng 91, 1245–1251.CrossRefGoogle Scholar
Niemela, J. J. & Donnelly, R. J. 1986 Direct transition to turbulence in rotating Bénard convection. Phys. Rev. Lett. 57, 2524–2527.CrossRefGoogle ScholarPubMed
Ning, L., Hu, Y., Ecke, R. & Ahlers, G. 1993 Spatial and temporal averages in chaotic patterns. Phys. Rev. Lett. 71, 2216–2219.CrossRefGoogle ScholarPubMed
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.CrossRefGoogle Scholar
Ponty, Y., Passot, T. & Sulem, P. L. 1997 Pattern dynamics in rotating convection at finite Prandtl number. Phys. Rev. E 56, 4162–4178.CrossRefGoogle Scholar
Rodríguez, J. M., Pérez-García, C., Bestehorn, M., Fantz, M. & Friedrich, R. 1992 Pattern formation in convection of rotating fluids with broken vertical symmetry. Phys. Rev. A 46, 4729–4735.CrossRefGoogle ScholarPubMed
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.CrossRefGoogle Scholar
Rubio, A., Lopez, J. M. & Marques, F. 2008 Modulated rotating convection: radially travelling concentric rolls. J. Fluid Mech. 608, 357–378.CrossRefGoogle Scholar
Rubio, A., Lopez, J. M. & Marques, F. 2009 Interacting oscillatory boundary layers and wall modes in modulated rotating convection. J. Fluid Mech. 625, 75–96.CrossRefGoogle Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Comm. Math. Phys. 20, 167.CrossRefGoogle 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 72, 036307.CrossRefGoogle ScholarPubMed
Scheel, J. D. 2007 The amplitude equation for rotating Rayleigh–Bénard convection. Phys. Fluids 19, 104105.CrossRefGoogle 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, 04618.CrossRefGoogle ScholarPubMed
Torrest, M. A. & Hudson, J. L. 1974 The effect of centrifugal convection on the stability of a rotating fluid heated from below. Appl. Sci. Res. 29, 273–289.CrossRefGoogle Scholar
Tu, Y. & Cross, M. C. 1992 Chaotic domain structure in rotating convection. Phys. Rev. Lett. 69, 2515–2518.CrossRefGoogle ScholarPubMed
Zhong, F., Ecke, R. & Steinberg, V. 1991 Asymmetric modes and the transition to vortex structures in rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 67, 2473–2476.CrossRefGoogle ScholarPubMed

Rubio et al. supplementary movie

Movie 1. This movie corresponds to figure 5(a) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2372, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 76,500 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 858.6 KB

Rubio et al. supplementary movie

Movie 1. This movie corresponds to figure 5(a) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2372, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 76,500 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 667.4 KB

Rubio et al. supplementary movie

Movie 2. This movie corresponds to figure 5(b) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2373, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 95,400 viscous times are shown at a rate of 4,500 viscous times per second.

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

Rubio et al. supplementary movie

Movie 2. This movie corresponds to figure 5(b) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2373, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 95,400 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 940.8 KB

Rubio et al. supplementary movie

Movie 3. This movie corresponds to figure 5(c) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2374, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 130,050 viscous times are shown at a rate of 4,500 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 2.1 MB

Rubio et al. supplementary movie

Movie 3. This movie corresponds to figure 5(c) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 0.005 for a solution with Ra=2374, Ω0=19.7, σ=4.5, γ=11.8 & Fr=0. Here 130,050 viscous times are shown at a rate of 4,500 viscous times per second.

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

Rubio et al. supplementary movie

Movie 4. This movie corresponds to figure 13(b) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2390, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 47250 viscous times are shown at a rate of 6750 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 2.1 MB

Rubio et al. supplementary movie

Movie 4. This movie corresponds to figure 13(b) in the paper. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2390, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 47250 viscous times are shown at a rate of 6750 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 2.1 MB

Rubio et al. supplementary movie

Movie 5. This movie corresponds to figure 13(c) in the paper over a time span indicated by box C in figure 14. Shown are contour levels of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2400, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 47250 viscous times are shown at a rate of 6750 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 3.9 MB

Rubio et al. supplementary movie

Movie 5. This movie corresponds to figure 13(c) in the paper over a time span indicated by box C in figure 14. Shown are contour levels of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2400, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 47250 viscous times are shown at a rate of 6750 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 2.6 MB

Rubio et al. supplementary movie

Movie 6. This movie corresponds to figure 13(d) in the paper over a time span indicated by box A in figure 14. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 7.5 MB

Rubio et al. supplementary movie

Movie 6. This movie corresponds to figure 13(d) in the paper over a time span indicated by box A in figure 14. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 4 MB

Rubio et al. supplementary movie

Movie 7. This movie corresponds to figure 13(d) in the paper over a time span indicated by box B in figure 14. Shown are contour levels of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 7.6 MB

Rubio et al. supplementary movie

Movie 7. This movie corresponds to figure 13(d) in the paper over a time span indicated by box B in figure 14. Shown are contour levels of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=8.82 x 10-3. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 4 MB

Rubio et al. supplementary movie

Movie 8. This movie corresponds to figure 5(f) in the paper over a time span indicated by box A in figure 7. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=0. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 7.2 MB

Rubio et al. supplementary movie

Movie 8. This movie corresponds to figure 5(f) in the paper over a time span indicated by box A in figure 7. Shown are isosurfaces of the temperature perturbation at Θ=+/- 5ε for a solution with Ra=2420, Ω0=19.7, σ=4.5 & γ=11.8 and Fr=0. Here 4050 viscous times are shown at a rate of 135 viscous times per second.

Download Rubio et al. supplementary movie(Video)
Video 4 MB