Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-19T06:05:29.649Z Has data issue: false hasContentIssue false
  • English
  • Français

The effects of sidewall heat loss on convection in a saturated porous vertical slab

Published online by Cambridge University Press:  20 April 2006

D. R. Kassoy  and
B. Cotte
D. R. Kassoy
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309
B. Cotte
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309
Get access Rights & Permissions [Opens in a new window]

Abstract

The onset of natural convection is considered in a vertically oriented, thin, finite slab of saturated porous media when sidewall heat transfer exists. First, a linear stability analysis is carried out for a system with impermeable boundaries. The sidewall temperature increases linearly with depth while the smaller-area endwalls are insulated. Convection occurs when the Rayleigh number R is asymptotically large relative to the inverse square of the horizontal aspect ratio, H2 [Lt ] 1. The convection pattern is composed of an integer number of vertically oriented three-dimensional, finger-like cells. The wavelength of each cell, relative to the larger horizontal dimension of the slab, is proportional to H2½. This somewhat surprising type of modal configuration is also found when there is a specified vertical mass flux through the slab. In this second example one considers the characteristics of the 3-dimensional fully developed solution for the thin vertical-slab problem which is compatible with a linear temperature increase on the vertical walls. When R is like that found in the first problem, closely spaced finger-like cells are found superimposed on the generally upward fluid flow. It is concluded that sidewall heat loss has a very strong stabilizing effect on the initiation of buoyancy-induced convection relative to the more traditional situation where side- and endwalls are insulated. Furthermore the appearance of slender, finger-like convection cells is characteristic of motion in a narrow vertical-slab configuration. Finally it is noted that the precise modal configuration selected by a system is extremely sensitive to the value of the Rayleigh number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 152 , March 1985 , pp. 361 - 378
Copyright
© 1985 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, 1377–1383.Google Scholar
Catton, I. 1970 Convection in a closed rectangular region: The onset of motion. Trans. ASME C: J. Heat Transfer 92, 186–188Google Scholar
Cole, J. D. 1968 Perturbation Methods in Applied Mathematics, p. 10. Blaisdell.
Davis, S. H. 1967 Convection in a box: linear theory. J. Fluid Mech. 30, 465–478.Google Scholar
Donaldson, I. G. 1970 The simulation of geothermal systems with a simple convective model. Geothermics 2, 649–654.Google Scholar
Kassoy, D. R. & Zebib, A. 1978 Convection fluid dynamics in a model of a fault zone in the Earth's crust. J. Fluid Mech. 88, 769–792.Google Scholar
Lowell, R. P. 1977 Convection and thermoelastic effects in narrow, vertical fracture spaces. Tech. Lett. Rep. 1, School Geophys. Sci., Georgia Inst. Tech. Atlanta.Google Scholar
Lowell, R. P. & Hernandez, H. 1982 Finite amplitude convection in a porous container with fault-like geometry: effect of initial and boundary conditions. Intl J. Heat Mass Transfer 25, 631–641.Google Scholar
Lowell, R. P. & Shyu, C.-T. 1978 On the onset of convection in a water-saturated porous box: effect of conducting walls. Lett. Heat Mass Transfer 5, 371–378.Google Scholar
Murphy, H. D. 1979 Convective instabilities in vertical fractures and faults. J. Geophys. Res. 84, 6121–6130.Google Scholar
Robinson, J. L. & O'Sullivan, M. J.1976 A boundary-layer model of flow in a porous medium at high Rayleigh number. J. Fluid Mech. 75, 459–467.Google Scholar
Straus, J. M. & Schubert, G. 1978 On the existence of three-dimensional convection in a rectangular box containing fluid-saturated porous material. J. Fluid Mech. 87, 385–394.Google Scholar
Turcotte, D. L., Ribando, R. J. & Torrance, K. E. 1977 Numerical calculation of two temperature thermal convection in a porous layer with application to Steamboat Springs thermal system, Nevada. In The Earth's Crust (ed. J. G. Heacock). American Geophysical Union Geophysical Monograph 20.
Zebib, A. & Kassoy, D. R. 1977 Onset of natural convection in a box of water-saturated porous media with large temperature variation. Phys. Fluids 20, 4–9.Google Scholar