Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-25T01:31:22.187Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of two-dimensional convection in a fluid-saturated porous medium

Published online by Cambridge University Press:  26 April 2006

M. De La Torre Juárez  and
F. H. Busse
M. De La Torre Juárez
Affiliation:
ETSI Aeronáuticos, Pza. Cardenal Cisneros 3, 28040 Madrid, Spain
F. H. Busse
Affiliation:
Institut für Theoretische Physik IV, Universität Bayreuth, 95440 Bayreuth, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

The range of existence and the stability of spatially periodic solutions has been studied for steady and oscillatory two-dimensional convection in a fluid-saturated porous medium. We have analysed the limit where viscous effects are dominant and Darcy's law can be applied. A Galerkin method has been used to obtain the steady and the centrosymmetric oscillatory solutions that appear in nonlinear convection at Rayleigh numbers up to 20 times the critical value. Their stability boundaries to arbitrary infinitesimal perturbations have been obtained. Above a given Rayleigh number stable oscillatory solutions are possible at wavenumbers close to the critical values. The stability of this oscillatory state with respect to infinitesimal perturbations of any wavenumber has also been studied. The resulting temporal dynamics in the different unstable regimes is briefly discussed. We show the existence of 3:1 spatial resonances of the steady roll solutions and the existence of stable centrosymmetric and non-centrosymmetric oscillatory solutions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 292 , 10 June 1995 , pp. 305 - 323
Copyright
© 1995 Cambridge University Press

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

Ahlers, G. & Behringer, R. P. 1978 Evolution of turbulence from the Rayleigh-Bénard instability. Phys. Rev. Lett. 40, 712–716.Google Scholar
Bergé, P. & Dubois, M. 1979 Study of unsteady convection through simultaneous velocity and interferometric measurements. J. Phys. Lett. 40, 505.Google Scholar
Busse, F. H. 1978 Nonlinear properties of thermal convection. Rep. Prog. Phys. 41, 1929.Google Scholar
Caltagirone, J. P. 1975 Thermoconvective instabilities in a horizontal porous layer. J. Fluid Mech. 72, 269–287.Google Scholar
Caltagirone, J. P. & Fabrie, P. 1989 Natural convection in a porous medium at high Rayleigh numbers. Part I - Darcy's model. Eur. J. Mech. B/ Fluids 8, 207–227.Google Scholar
Daviaud, F., Bonetti, M. & Dubois, M. 1990 Transition to turbulence via spatiotemporal intermittency in one-dimensional Rayleigh Bénard convection. Phys. Rev. A 42, 3388–3399.Google Scholar
Frick, H. & Müller, U. 1983 Oscillatory Hele-Shaw convection. J. Fluid Mech. 126, 521–532.Google Scholar
Gollub, J. P. & Benson, S. V. 1980 Many routes to turbulent convection. J. Fluid Mech. 100, 449–470.Google Scholar
Graham, M. D. & Steen, P. 1992 Strongly interacting traveling waves and quasiperiodic dynamics in porous medium convection. Physica D 54, 231–350.Google Scholar
Graham, M. D. & Steen, P. 1994 Plume formation and resonant bifurcations in porous media-convection. J. Fluid Mech. 212, 67–89.Google Scholar
Graham, M. D., M$uuller, U. & Steen, P. H. 1992 Time-periodic thermal convection in Hele-Shaw slots: The diagonal oscillation. Phys. Fluids A 4, 2382–2391.Google Scholar
Hegseth, J., Vince, J. M., Dubois, M. & Berg$ea, P. 1992 Pattern domains in Rayleigh-Bénard slot convection. Europhys. Lett. 17, 413–418.Google Scholar
Kimura, S., Schubert, G. & Straus, J. M. 1986 Route to chaos in porous-medium thermal convection. J. Fluid Mech. 166, 305–324.Google Scholar
Koster, J.N. & M$uuller, U. 1982 Free convection in vertical gaps. J. Fluid Mech. 125, 429–451.Google Scholar
Koster, J. N. & M$uuller, U. 1984 Oscillatory convection in vertical slots. J. Fluid Mech. 139, 363–390.Google Scholar
Krishnamurti, R. 1970a On the transition to turbulent convection. Part 1. The transition from two- to three-dimensional flow. J. Fluid Mech. 42, 295–308.Google Scholar
Krishnamurti, R. 1970b On the transition to turbulent convection. Part 2. The transition to time dependent flow. J. Fluid Mech. 42, 309 320.Google Scholar
Kvernvold, O. 1979 On the stability of non-linear convection in a Hele-Shaw cell. Intl J. Heat Mass Transfer 22, 395–400.Google Scholar
Maurer, J. & Libchaber, A. 1979 Rayleigh Bénard experiment in liquid helium: frequency locking and onset of turbulence. J. Phys. Paris 40, 419–423.Google Scholar
Mizushima, J. & Fujimura K. 1992 Higher harmonic resonance of two-dimensional disturbances in Rayleigh-Bénard convection. J. Fluid Mech. 234, 651–667.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1988 Numerical Recipes in C, pp. 382–394. Cambridge University Press.
Riley, D. S. & Winters, K. H. 1991 Time-periodic convection in porous media: the evolution of Hopf bifurcations with aspect ratio. J. Fluid Mech. 223, 457–474.Google Scholar
Steen, P. H. & Aidun, C. K. 1988 Time periodic convection in porous media: transition mechanism. J. Fluid Mech. 196, 263–290.Google Scholar
Strauss, J. M. 1974 Large amplitude convection in porous media. J. Fluid Mech. 64, 51–63.Google Scholar
Veronis, G. 1966 Large-amplitude Bénard convection J. Fluid Mech. 26 49–68.Google Scholar
Wooding, R. A. 1960, Instability of a viscous liquid of variable density in a vertical Hele-Shaw cell. J. Fluid Mech. 7, 501–515.Google Scholar