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Thermoconvective instabilities in a porous medium bounded by two concentric horizontal cylinders

Published online by Cambridge University Press:  11 April 2006

Jean-Paul Caltagirone
Jean-Paul Caltagirone
Affiliation:
Laboratoire d'Aérothermique du C.N.R.S., 4 ter, Route des Gardes, 92190 Meudon, France
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Abstract

The study of natural convection in a saturated porous medium bounded by two concentric, horizontal, isothermal cylinders reveals different types of evolution according to the experimental conditions and the geometrical configuration of the model. At small Rayleigh numbers the state of the system corresponds to a regime of pseudo-conduction. The isotherms are coaxial with the cylinders. At larger Rayleigh numbers a regime of steady two-dimensional convection sets in between the two cylinders. Finally, for Rayleigh numbers above the critical Rayleigh number Ra*c the phenomena become three-dimensional and fluctuating. The appearance of these different regimes depends, moreover, on the geometry considered and, in particular, on two numbers: R, the ratio of the radii of the cylinders, and A, the ratio of the length of the cylinders to the radius of the inner one. In order to approach these experimental observations and to obtain realistic theoretical models, several methods of solving the equations have been used.

The perturbation method yields information about the thermal field and the heat transfer between the cylinders under conditions close to the equilibrium state.

A numerical two-dimensional model enables us to extend the range of investigation and to represent properly the phenomena when steady convection appreciably modifies the temperature distribution and the velocities within the porous layer.

Neither of these models allows account to be taken of the instabilities observed experimentally above a critical Rayleigh number Ra*c. For this reason, a study of stability has been carried out using a Galerkin method based on equations corresponding to an initial state of steady convection. The results obtained show the importance of three-dimensional effects for the onset of fluctuating convection. The critical transition Rayleigh number Ra*c is thus determined in terms of the ratio of the radii R by solving an eigenvalue problem.

A numerical three-dimensional model based on the method of finite elements has thus been developed in order to point out the different types of evolution with time. Steady two-dimensional convection and fluctuating three-dimensional convection have been actually found by calculation. The solution of the system of equations by the method of finite elements is briefly described.

The experimental and theoretical results are then compared and a general physical interpretation is given.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 76 , Issue 2 , 28 July 1976 , pp. 337 - 362
Copyright
© 1976 Cambridge University Press

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References

Abbott, M. R. 1964 Quart. J. Mech. Appl. Math. 17, 471.
Bishop, E. H. & Carley, C. T. 1966 Proc. Heat Transfer Fluid Mech. Inst., p. 63. Stanford University Press.
Bishop, E. H., Carley, C. T. & Powe, R. E. 1968 Int. J. Heat Mass Transfer. 11, 1741.
Bories, S. A. & Combarnous, M. A. 1973 J. Fluid Mech. 57, 63.
Caltagirone, J. P. 1971 Mém. CNAM, Paris.
Caltagirone, J. P. 1972 C.R. Acad. Sci. Paris. 275, 307.
Caltagirone, J. P. 1975a 6éme Cong. Int. Fondation FranÇaise d'Etudes Nordiques, Le Havre (to be published).
Caltagirone, J. P. 1975b J. Fluid Mech 72, 269.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Cloupeau, M. & Klarsfeld, S. 1973 Appl. Optics. 12, 198.
Combarnous, M. A. & Bories, S. A. 1975 Adv. in Hydrosci 10, 231–307.
Crawford, L. & Lemlich, R. 1962 Indust. Engng Chem. Fund. 1, 260.
Finlayson, B. A. 1968 J. Fluid Mech. 33, 201.
Finlayson, B. A. 1972 The Method of Weighted Residuals and Variational Principles. Academic.
Finlayson, B. A. & Scriven, L. E. 1967 Int. J. Heat Mass Transfer. 10, 799.
Glansdorff, P. & Prigogine, I. 1971 Structure, Stabilité et Fluctuations. Masson.
Grigull, U. & Hauf, W. 1966 Proc. 3rd Int. Heat Transfer Conf., paper 60, vol. 2, p. 182.
Hirasaki, G. J. & Hellums, J. D. 1968 Quart. Appl. Math. 16, 331.
Holst, P. H. & Aziz, K. 1971 Int. J. Heat Mass Transfer. 15, 73.
Kantorovich, L. V. & Krylov, V. I. 1958 Approximate Methods in Higher Analysis. Noordhoff.
LIS, J. 1966 Proc. 3rd Int. Heat Transfer Conf., paper 61, vol. 2, p. 196.
Liu, C. Y., Mueller, W. K. & Landis, F. 1961 Int. Dev. Heat Transfer, paper 117, pt 5, p. 976. A.S.M.E.
Mack, L. R. & Bishop, E. H. 1968 Quart. J. Mech. Appl. Math. 21, 223.
Powe, R. E., Carley, C. T. & Carruth, S. L. 1971 J. Heat Transfer, Trans. A.S.M.E. 93, 210.
Rotem, Z. 1971 Int. J. Heat Mass Transfer. 15, 1679.
Wilson, E. L. & Nickell, R. E. 1966 Nucl. Engng Design. 4, 276.
Zienkiewicz, O. C. 1971 The Finite Element Method in Engineering Science. McGraw-Hill.