Hostname: page-component-6bb9c88b65-kfd97 Total loading time: 0 Render date: 2025-07-23T09:01:33.266Z Has data issue: false hasContentIssue false
  • English
  • Français

Time-periodic convection in porous media: transition mechanism

Published online by Cambridge University Press:  21 April 2006

Paul H. Steen  and
Cyrus K. Aidun
Paul H. Steen
Affiliation:
School of Chemical Engineering and Mathematical Sciences Institute, Cornell University, Ithaca, NY 14853, USA
Cyrus K. Aidun
Affiliation:
Engineering Division, Institute of Paper Chemistry, P.O. Box 1039, Appleton. WI 54912, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We resolve the disturbance structures that destabilize steady convection rolls in favour of a time-periodic pattern in two-dimensional containers of fluid-saturated porous material. Analysis of these structures shows that instability occurs as a travelling wave propagating in a closed loop outside the nearly motionless core. The travelling wave consists of five pairs of thermal cells and four pairs of vorticity disturbances in the case of a square container. The wave speed of the thermal disturbances is determined by an average base-state velocity and their structure by a balance between convection and thermal diffusion. Interpretation of the ‘exact’ solution is aided by a one-dimensional convection-loop model which correlates (i) point of transition, (ii) disturbance wavenumber, and (iii) oscillation frequency given the base-state temperature and velocity profiles. The resulting modified Mathieu-Hill equation clarifies the role of the vertical pressure gradient, induced by the impenetrable walls, and the role of the base-state thermal layer.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 196 , November 1988 , pp. 263 - 290
Copyright
© 1988 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.)

Article purchase

Temporarily unavailable

References

Aidun, C. K. & Steen, P. H. 1986 AIAA/ASME 4th Joint Thermophysics and Heat Trans. Conf., AIAA paper 86–1264.
Aidun, C. K. & Steen, P. H. 1987 J. Thermophys. 1, 268.
Beck, J. L. 1972 Phys. Fluids 15 1377.
Bolton, E. W., Busse, F. H. & Clever, R. M. 1986 J. Fluid Mech. 164, 469.
Caltagirone, J.-P. 1974 C. R. Acad. Sci. Paris 278, 259.
Caltagirone, J.-P. 1975 J. Fluid Mech. 72, 269.
Caltagirone, J.-P., Cloupeau, M. & Combarnous, M. 1971 C. R. Acad. Sci. Paris 273, 833.
Combarnous, M. & Lefur, B. 1969 C. R. Acad. Sci. Paris 269, 1009.
Doedel, E. J. 1980 AUTO: A Program for the Automatic Bifurcation Analysis of Autonomous Systems. In Proc. 10th Manitoba Conf. on Numerical Math. and Comp., p. 265.
Eriksen, J. L. & Truesdell, C. 1958 Arch. Rat. Mech. Anal. 1, 295.
Frick, H. & Clever, R. M. 1982 J. Fluid Mech. 114, 467.
Frick, H. & Muller, U. 1983 J. Fluid Mech. 126, 521.
Hartline, B. K. & Lister, C. R. B. 1977 J. Fluid Mech. 79, 379.
Horne, R. N. & Caltagirone, J.-P. 1980 J. Fluid Mech. 100, 385.
Horne, R. N. & O'Sullivan, M. J. 1974 J. Fluid Mech. 66, 339.
Horne, R. N. & O'Sullivan, M. J. 1978 Phys. Fluids 21, 1260.
Keller, H. B. 1977 Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems. In Applications of Bifurcation Theory (ed. P. H. Rabinowitz), p. 359. Academic.
Kimura, S., Schubert, G. & Straus, J. M. 1986 J. Fluid Mech. 166, 305.
Kimura, S., Schubert, G. & Straus, J. M. 1987 Trans. ASME J. Heat Transfer 109, 350.
Koster, J. N. & Muller, U. 1980 In Natural Convection in Enclosures. ASME HTD, vol. 8, p. 27.
Koster, J. N. & Muller, U. 1981 Phys. Rev. Lett. 47, 1599.
Koster, J. N. & Muller, U. 1982 J. Fluid Mech. 125, 429.
Koster, J. N. & Muller, U. 1984 J. Fluid Mech. 139, 363.
Kvernvold, O. 1979 Intl. J. Heat Mass Transfer 22, 395.
Lapwood, E. R. 1948 Proc. Camb. Phil. Soc. 44, 508.
Mielke, A. 1987 Mathematical Sciences Institute, Cornell University, Rep. 87–25.
Rosenblat, S. 1979 Stud. Appl. Maths 60, 241.
Schubert, G. & Straus, J. M. 1982 J. Fluid Mech. 121, 301.
Steen, P. H. 1986 Phys. Fluids 29, 925.