Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T12:22:14.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Onset of convection in a moderate aspect-ratio rotating cylinder: Eckhaus–Benjamin–Feir instability

Published online by Cambridge University Press:  15 October 2007

J. M. LOPEZ ,
F. MARQUES ,
I. MERCADER  and
O. BATISTE
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
I. MERCADER
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
O. BATISTE
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

A numerical study of the onset of thermal convection in a rotating circular cylinder of radius-to-depth ratio equal to four is considered in a regime dominated by the Coriolis force where the onset is to so-called wall modes. The wall modes consist of hot and cold pairs of thermal plumes rising and descending in the cylinder wall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot/cold plume pairs, m. In the limit of zero centrifugal force, this onset of convection at a critical temperature difference across the depth of the cylinder is via a symmetry-breaking supercritical Hopf bifurcation which leads to retrograde precession of the pattern with respect to the rotation of the cylinder. For temperature differences greater than critical, a number of distinct wall modes, distinguished by m, coexist and are stable. Their dynamics are controlled by an Eckhaus–Benjamin–Feir instability, the most basic features of which had been captured by a complex Ginzburg–Landau equation model. Here, we analyse this instability in rotating convection using direct numerical simulations of the Navier–Stokes equations in the Boussinesq approximation. Several properties of the wall modes are computed, extending the results to far beyond the onset of convection. Extensive favourable comparisons between our numerical results and previous experimental observations and complex Ginzburg–Landau model results are made.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 590 , 10 November 2007 , pp. 187 - 208
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

Ahlers, G., Cannell, D. S., Dominguez-Lerma, M. A. & Heinrichs, R. 1986 Wavenumber selection and Eckhaus instability in Couette–Taylor flow. Physica D 23, 202–219.Google Scholar
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
Benjamin, T. B. & Feir, J. E. 1967 Disintegration of wave trains on deep water. 1. Theory. J. Fluid Mech. 27, 417–430.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Buell, J. C. & Catton, I. 1983 Effect of rotation on the stability of a bounded cylindrical layer of fluid heated from below. Phys. Fluids 26, 892–896.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Choi, W., Prasad, D., Camasa, R. & Ecke, R. E. 2004 Traveling waves in rotating Rayleigh–Bénard convection. Phys. Rev. E 69, 056301.Google ScholarPubMed
Ecke, R. E., Zhong, F. & Knobloch, E. 1992 Hopf-bifurcation with broken reflection symmetry in rotating Rayleigh–Bénard convection. Europhys. Lett. 19, 177–182.CrossRefGoogle Scholar
Eckhaus, W. 1965 Studies in Nonlinear Stability Theory. Springer.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
Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M. 1994 Convection in a rotating cylinder. Part 2. Linear theory for low Prandtl numbers. J. Fluid Mech. 262, 293–324.CrossRefGoogle Scholar
Herrmann, J. & Busse, F. H. 1993 Asymptotic theory of wall-attached convection in a rotating fluid layer. J. Fluid Mech. 255, 183–194.CrossRefGoogle Scholar
Hu, Y., Ecke, R. E. & Ahlers, G. 1997 Convection under rotation for Prandtl numbers near 1. Linear stability, wave-number selection, and pattern dynamics. Phys. Rev. E 55, 6928–6949.Google 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
Janiaud, B., Pumir, A., Benisom, D., Croquette, V., Richter, H. & Kramer, L. 1992 The Eckhaus instability for traveling waves. Physica D 55, 269–286.Google Scholar
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
Krupa, M. 1990 Bifurcations of relative equilibria. SIAM J. Math. Anal. 21, 1453–1486.CrossRefGoogle Scholar
Kuo, E. Y. & Cross, M. C. 1993 Traveling-wave wall states in rotating Rayleigh–Bénard convection. Phys. Rev. E 47, R2245–R2248.Google ScholarPubMed
Kuznetsov, Y. A. 1998 Elements of Applied Bifurcation Theory, 2nd edn. Springer.Google Scholar
Liu, Y. & Ecke, R. E. 1997 Eckhaus–Benjamin–Feir instability in rotating convection. Phys. Rev. Lett. 78, 4391–4394.CrossRefGoogle Scholar
Liu, Y. & Ecke, R. E. 1999 Nonlinear traveling waves in rotating Rayleigh–Bénard convection: stability boundaries and phase diffusion. Phys. Rev. E 59, 4091–4105.Google Scholar
Lopez, J. M., Hart, J. E., Marques, F., Kittelman, S. & Shen, J. 2002 Instability and mode interactions in a differentially-driven rotating cylinder. J. Fluid Mech. 462, 383–409.CrossRefGoogle Scholar
Marques, F. & Lopez, J. M. 2001 Precessing vortex breakdown mode in an enclosed cylinder flow. Phys. Fluids 13, 1679–1682.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
Mercader, I., Net, M. & Falqués, A. 1991 Spectral methods for high order equations. Comput. Meth. Appl. Mech. Engng 91, 1245–1251.CrossRefGoogle Scholar
Mizushima, J. & Fujimura, K. 1992 Higher harmonic resonance of two-dimensional disturbances in Rayleigh–Bénard convection. J. Fluid Mech. 234, 651–667.CrossRefGoogle Scholar
Nagata, M. 1995 Bifurcations at the Eckhaus points in two-dimensional Rayleigh–Bénard convection. Phys. Rev. E 52, 6141–6145.Google ScholarPubMed
Ning, L. & Ecke, R. E. 1993 Rotating Rayleigh–Bénard convection: aspect-ratio dependence of the initial bifurcations. Phys. Rev. E 47, 3326–3333.Google ScholarPubMed
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.CrossRefGoogle Scholar
Pfotenhauer, J. M., Niemela, J. J. & Donnelly, R. J. 1987 Stability and heat transfer of rotating cryogens. Part 3. Effects of finite cylindrical geometry and rotation on the onset of convection. J. Fluid Mech. 175, 85–96.CrossRefGoogle Scholar
Plaut, E. 2003 Nonlinear dynamics of traveling waves in rotating Rayleigh–Bénard convection: effects of the boundary conditions and of the topology. Phys. Rev. E 67, 046303.Google ScholarPubMed
Porter, J. & Knobloch, E. 2000 Complex dynamics in the 1:3 spatial resonance. Physica D 143 (1–4), 138–168.Google Scholar
Prat, J., Mercader, I. & Knobloch, E. 1998 Resonant mode interactions in Rayleigh–Bénard convection. Phys. Rev. E 58, 3145–3156.Google Scholar
Rand, D. 1982 Dynamics and symmetry. Predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79, 1–38.CrossRefGoogle Scholar
Rossby, H. T. 1967 A study of Bénard convection with and without rotation. J. Fluid Mech. 29, 673–690.Google Scholar
Rüdiger, S. & Feudel, F. 2000 Pattern formation in Rayleigh–Bénard convection in a cylindrical container. Phys. Rev. E 62, 4927–4931.Google Scholar
Rüdiger, S. & Knobloch, E. 2003 Mode interaction in rotating Rayleigh–Bénard convection. Fluid Dyn. Res. 33, 477–492.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.Google ScholarPubMed
Scheel, J. D., Paul, M. R., Cross, M. C. & Fischer, P. F. 2003 Traveling waves in rotating Rayleigh–Bénard convection: analysis of modes and mean flow. Phys. Rev. E 68, 066216.Google ScholarPubMed
Tuckerman, L. S. & Barkley, D. 1990 Bifurcation analysis of the Eckhaus instability. Physica D 46, 57–86.Google Scholar
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
Zhong, F., Ecke, R. & Steinberg, V. 1993 Rotating Rayleigh–Bénard convection: asymmetric modes and vortex states. J. Fluid Mech. 249, 135–159.CrossRefGoogle Scholar