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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
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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

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