Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T21:09:51.888Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional and multicellular steady and unsteady convection in fluid-saturated porous media at high Rayleigh numbers

Published online by Cambridge University Press:  19 April 2006

Gerald Schubert  and
Joe M. Straus
Gerald Schubert
Affiliation:
Space Sciences Laboratory, The Ivan A. Getting Laboratories, The Aerospace Corporation, P.O. Box 92957, Los Angeles, California 90009 Permanent Address: Department of Earth and Space Sciences, UCLA, Los Angeles, California 90024.
Joe M. Straus
Affiliation:
Space Sciences Laboratory, The Ivan A. Getting Laboratories, The Aerospace Corporation, P.O. Box 92957, Los Angeles, California 90009
Get access Rights & Permissions [Opens in a new window]

Abstract

In an effort to determine the characteristics of the various types of convection that can occur in a fluid-saturated porous medium heated from below, a Galerkin approach is used to investigate three-dimensional convection in a cube and two-dimensional convection in a square cross-section. Strictly two-dimensional, single-cell flow in a square cross-section is steady for Rayleigh numbers R between 4π2 and a critical value which lies between 300 and 320; it is unsteady at higher values of R. Double-cell, two-dimensional flow in a square cross-section becomes unsteady when R exceeds a value between 650 and 700, and triple-cell motion is unsteady for R larger than a value between 800 and 1000. Considerable caution must be exercised in attributing physical reality to these flows. Strictly two-dimensional, steady, multicellular convection may not be realizable in a three-dimensional geometry because of instability to perturbations in the orthogonal dimension. For example, even though single-cell, two-dimensional convection in a square cross-section is steady at R = 200, it cannot exist in either an infinitely long square cylinder or in a cube. It could exist, however, in a cylinder whose length is smaller than 0.38 times the dimension of its square cross-section. Three-dimensional convection in a cube becomes unsteady when R exceeds a value between 300 and 320, similar to the unicellular two-dimensional flow in a square cross-section. Nusselt numbers Nu, generally accurate to 1%, are given for the strictly two-dimensional flows up to R = 1000 and for three-dimensional convection in cubes up to R = 500. Single-cell, two-dimensional, steady convection in a square cross-section transports the most heat for R < 97; this mode of convection is also stable in square cylinders of arbitrary length including the cube for R < 97. Steady three-dimensional convection in cubes transports more heat for 97 [lsim ] R [lsim ] 300 than do any of the realizable two-dimensional modes. At R [gsim ] 300 the unsteady modes of convection in both square cylinders and cubes involve wide variations in Nu.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 94 , Issue 1 , 11 September 1979 , pp. 25 - 38
Copyright
© 1979 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

Beck, J. L. 1972 Convection in a box of porous material saturated with fluid. Phys. Fluids 15, 1377.Google Scholar
Caltagirone, J. P. 1975 Thermoconvective instabilities in a horizontal porous layer. J. Fluid Mech. 72, 169.Google Scholar
Caltagirone, J. P., Cloupeau, M. & Combarnous, M. 1971 Convection naturelle fluctuante dans une couche poreuse horizontale. C. R. Acad. Sci. Paris 273 B, 833.Google Scholar
Combarnous, M. & LeFur, B. 1969 Transfert de chaleur par convection naturelle dans une couche poreuse horizontale. C.R. Acad. Sci. Paris 269 B, 1009.Google Scholar
Elder, J. W. 1967 Steady free convection in a porous medium heated from below. J. Fluid Mech. 27, 29.Google Scholar
Holst, P. H. & Aziz, K. 1972 Transient three-dimensional natural convection in confined porous media. Int. J. Heat Mass Transfer 15, 73.Google Scholar
Horne, R. N. 1975 Transient effects in geothermal convective systems. Ph.D. thesis, Auckland University.
Horne, R. N. 1979 Three dimensional natural convection in a confined porous medium heated from below. J. Fluid Mech. 92, 751.Google Scholar
Horne, R. N. & O'Sullivan, M. J. 1974 Oscillatory convection in a porous medium heated from below. J. Fluid Mech. 66, 339.Google Scholar
Horne, R. N. & O'Sullivan, M. J. 1978 Origin of oscillatory convection in a porous medium heated from below. Phys. Fluids 8, 1260.Google Scholar
Horton, C. W. & Rogers, F. T. 1945 Convection currents in a porous medium. J. Appl. Phys. 16, 367.Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508.Google Scholar
Palm, E., Weber, J. E. & Kvernvold, O. 1972 On steady convection in a porous medium. J. Fluid Mech. 54, 153.Google Scholar
Straus, J. M. 1974 Large amplitude convection in porous media. J. Fluid Mech. 64, 51.Google Scholar
Straus, J. M. & Schubert, G. 1978 On the existence of three-dimensional convection in a rectangular box of fluid-saturated porous material. J. Fluid Mech. 87, 385.Google Scholar
Straus, J. M. & Schubert, G. 1979 Three-dimensional convection in a cubic box of fluid-saturated porous material. J. Fluid Mech. 91, 155.Google Scholar
Zebib, A. & Kassoy, D. R. 1978 Three-dimensional natural convection motion in a confined porous medium. Phys. Fluids 21, 1.Google Scholar