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Steady free convection in a porous medium heated from below

Published online by Cambridge University Press:  28 March 2006

J. W. Elder
J. W. Elder
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Cambridge
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Abstract

This is an experimental and numerical study of steady free convection in a porous medium, a system dominated by a single non-linear process, the advection of heat. The paper presents results on three topics: (1) a system uniformly heated from below, for which the flow is cellular, as in the analogous Bénard-Rayleigh flows, (ii) the role of end-effects, and (iii) the role of mass discharge. Measurements of heat transfer are used to establish further the validity of the numerical scheme proposed by the author (1966a), while the other flows allow a more extensive study of the numerical scheme under various boundary conditions. The results are very satisfactory even though only moderately non-linear problems can be treated at present.

The main new results are as follows. For the Rayleigh-type flow, above a critical Rayleigh number of about 40, the heat transferred across the layer is proportional to the square of the temperature difference across the layer and is independent of the thermal conductivity of the medium or the depth of the layer. This result is modified when the boundary-layer thickness is comparable to the grain size of the medium. The investigation of end-effects reveals variations in horizontal wave-number and a pronounced hysteresis and suggests an alternative explanation of some observations by Malkus (1954).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 27 , Issue 1 , 11 January 1967 , pp. 29 - 48
Copyright
© 1967 Cambridge University Press

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References

Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Oxford: Clarendon Press.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Coles, D. 1965 J. Fluid Mech. 21, 385.
Darcy, H. P. G. 1856 Les Fontaines publiques de la ville de Dijon. Paris: Victor Dalmont.
Donaldson, I. G. 1962 J. Geophys. Res. 67, 3449.
Einarsson, T. 1942 Rit. Visind. Isl. 26, 1.
Elder, J. W. 1958 Ph.D. Thesis, Cambridge.
Elder, J. W. 1965 Terrestrial Heat Flow, chap. 8. (Ed. W. H. K. Lee.) Amer. Geophys. U. Monograph, no. 8.Google Scholar
Elder, J. W. 1966a J. Fluid Mech. 24, 823.
Elder, J. W. 1966b Bull. Volcan. 29 (in the Press).
Foster, T. D. 1965 Personal communication, to be published.
Fox, L. 1962 Numerical Solution of Ordinary and Partial Differential Equations. London: Pergamon Press.
Frank, F. C. 1965 Rev. Geophys. 3, 485.
Garabedian, P. R. 1956 Math. Tables Aids Comput. 10, 183.
HELE-SHAW, H. S. J. 1898 Trans. Inst. Nav. Arch. 40, 21.
Horton, C. W. & Rogers, F. T. 1945 J. Applied Phys. 16, 367. (See also 22, 1476.)
Jakob, M. 1949 Heat Transfer. New York: Wiley.
Lapwood, E. R. 1948 Proc. Camb. Phil. Soc. 44, 508.
Malkus, W. V. R. 1954 Proc. Roy. Soc. A, 225, 185.
Richardson, J. G. 1961 Handbook of Fluid Dynamics, sec. 16 (ed. V. L. Streeter). New York: McGraw-Hill.
Saffman, P. G. & Taylor, G. I. 1958 Proc. Roy. Soc. A, 245, 312.
Schneider, K. J. 1963 11th Int. Cong. of Refrigeration, paper 11-4, Munich.
Silveston, P. L. 1958 Forsch. Ing. 24, 29.
Wooding, R. A. 1957 J. Fluid Mech. 2, 273.
Wooding, R. A. 1958 J. Fluid Mech. 3, 582.
Wooding, R. A. 1959 Proc. Roy. Soc. A, 252, 120.
Wooding, R. A. 1960a J. Fluid Mech. 7, 501.
Wooding, R. A. 1960b J. Fluid Mech. 9, 183.
Wooding, R. A. 1962 J. Fluid Mech. 13, 129.
Wooding, R. A. 1963 J. Fluid Mech. 15, 527.
Zierep, J. 1961 ZAMM. 41, 114.