Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-19T00:50:04.650Z Has data issue: false hasContentIssue false
  • English
  • Français

Preferred pattern of convection in a porous layer with a spatially non-uniform boundary temperature

Published online by Cambridge University Press:  26 April 2006

D. N. Riahi
D. N. Riahi
Affiliation:
Department of Theoretical and Applied Mechanics, 216 Talbot Laboratory, University of Illinois, Urbana, Illinois 61801, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of finite-amplitude thermal convection in a porous layer between two horizontal walls with different mean temperatures is considered when spatially non-uniform temperature with amplitude L* is prescribed at the lower wall. The nonlinear problem of three-dimensional convection for values of the Rayleigh number close to the classical critical value is solved by using a perturbation technique. Two cases are considered: the wavelength γ(b)n of the nth mode of the modulation is equal to or not equal to the critical wavelength γc for the onset of classical convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. The most surprising results for the case γ(b)n = γc for all n are that regular or non-regular solutions in the form of multi-modal pattern convection can become preferred in some range of L*, provided the wave vectors of such pattern are contained in the set of wave vectors representing the spatially non-uniform boundary temperature. There can be critical value(s) L*c of L* below which the preferred flow pattern is different from the one for L* > L*c. The most surprising result for the case γ(b)n ≠ γc and γ(b)n ≡ γ(b) for all n is that some three-dimensional solution in the form of multi-modal convection can be preferred, even if the boundary modulation is one-dimensional, provided that the wavelength of the modulation is not too small. Here γ(b) is a constant independent of n.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 246 , January 1993 , pp. 529 - 543
Copyright
© 1993 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

Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid. Mech. 30, 625– 649.Google Scholar
Busse, F. H. 1972 On the mean flow induced by a thermal wave. J. Atmos. Sci. 29, 1423– 1429.Google Scholar
Ernbux, T. & Cohen, D. S. 1983 Imperfect bifurcation near a double eigenvalue: transition between nonsymmetric and symmetric patterns. SIAM J. Appl. Maths 43, 1042– 1060.Google Scholar
Hall, P. & Walton, I. C. 1978 The smooth transition to a convective regime in a two-dimensional box. Proc. R. Land. Soc. A 358, 199– 221.Google Scholar
Jospeh, D. D. 1976 Stability of Fluid Motions, vols I and II. Springer.
Keller, J. B. 1966 Periodic oscillations in a model of thermal convection. J. Fluid. Mech. 26, 599– 606.Google Scholar
Kelly, R. E. & Pal, D. 1976 Thermal convection induced between non-uniformity heated horizontal surfaces. Proc. 1976 Heat Transfer and Fluid Mech. Inst., pp. 1– 17. Stanford University Press.
Kelly, R. E. & Pal, D. 1978 Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation. J. Fluid. Mech. 86, 433– 456.Google Scholar
Krettenauer, K. & Schumann, U. 1989 Direct numerical simulation of thermal convection over a wavy surface. Met. Atmos. Phys. 41, 165– 179.Google Scholar
Krettenauer, K. & Schumann, U. 1992 Numerical simulation of turbulent convection over wavy terrain. J. Fluid. Mech. 237, 261– 299.Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil Soc. 44, 508– 521.Google Scholar
Matkowsky, B. J. & Reiss, E. L. 1977 Singular perturbations of bifurcations. SIAM J Appl Maths 33, 230– 255.Google Scholar
Pal, D. & Kelly, R. E. 1978 Thermal convection with spatially periodic nonuniform heating nonresonant wavelength excitation. Proc. 6th Intl Heat Transfer Conf. Toronto, pp. 235– 238.
Palm, E., Weber, J. E. & Kvernvold, O. 1972 ON steady convection in a porous medium. J. Fluid. Mech. 54, 153– 161.Google Scholar
Reiss, E. L. 1976 Imperfect bifurcation. Adv. Seminar on Appl. of Bifurcation Theory pp 37– 71. Academic.
Riahi, N. 1983 Nonlinear convection in a porous layer with finite conducting boundaries J Fluid Mech. 129, 153– 171.Google Scholar
Riahi, N. 1985 Nonlinear thermal convection with finite conducting boundaries. J. Fluid. Mech. 152, 113– 123.Google Scholar
Straus, J. M. 1974 Large amplitude convection in porous media. J. Fluid. Mech. 64, 51– 63.Google Scholar
Tavantzis, J., Reiss, E. L. & Matkowsky, B. J. 1978 On the smooth transition to convection. SIAM J. Appl. Maths 34, 322– 337.Google Scholar
Walton, I. C. 1982 On the onset of Rayleigh–Bénard convection in a fluid layer of slowly increasing depth. Stud. Appl. Maths 67, 199– 216.Google Scholar
Walton, I. C. 1983 The onset of cellular convection in a shallow two-dimensional container of fluid heated non-uniformity from below. J. Fluid. Mech. 131, 455– 470.Google Scholar
Yoo, J.-S. & Kim, M.-U. 1991 Two-dimensional convection in a horizontal fluid layer with spatially periodic boundary temperatures. Fluid Dyn. Res. 7, 181– 200.Google Scholar
Young, R. E., Schubert, G. & Torrance, K. E. 1972 Nonlinear motions induced by moving thermal waves. J. Fluid. Mech. 54, 163– 187.Google Scholar