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Natural convection about a heated horizontal cylinder in a porous medium

Published online by Cambridge University Press:  21 April 2006

D. B. Ingham  and
I. Pop
D. B. Ingham
Affiliation:
Department of Applied Mathematical Studies, The University of Leeds, Leeds LS2 9JT, West Yorkshire, UK
I. Pop
Affiliation:
Faculty of Mathematics, University of Cluj, Cluj, Romania
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Abstract

The natural convection from a heated circular cylinder in an unbounded region of porous medium is investigated for the full range of Rayleigh numbers. At small Rayleigh numbers a qualitative solution is obtained and at large Rayleigh numbers the second-order boundary-layer solution is found that takes into account the first-order plume solution. In order to find the solution at finite Rayleigh numbers the two governing coupled, nonlinear, elliptic partial differential equations are expressed in finite-difference form using a specialized technique which is second-order accurate everywhere. Further, methods are devised which deal with the plume and infinity boundary conditions. Although numerical results are presented for Rayleigh numbers up to 400 solutions of the finite-difference equations can be obtained for higher values of the Rayleigh numbers but in these cases the mesh size used is too large to adequately deal with the developing boundary-layer on the cylinder and the plume.

The numerical results show how the theories at both low and high Rayleigh numbers are approached. The plume solution which develops with increasing Rayleigh number agrees with that predicted by the theory presented using the boundary-layer approximation. No separation of the flow at the top of the cylinder is found and there are no indications that it will appear at higher values of the Rayleigh number. The results presented here give reasonable agreement with the existing experimental results for Rayleigh numbers of order unity. However as the Rayleigh number increases to order 102 there is a large discrepancy between the theoretical and experimental results and this is because at these higher values of the Rayleigh number the Darcy approximation has been violated in the experimental results. This indicates the severe limitations of some of the existing theories which use boundary-layer analyses and the Darcy approximation for flows in a porous medium. The application of Darcy's law requires that the size of the pores be much smaller than the scale of the bulk flow and inertial and thermal lengthscales.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 184 , November 1987 , pp. 157 - 181
Copyright
© 1987 Cambridge University Press

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References

Ackroyd, J. A. D. 1967 Proc. Camb. Phil. Soc. 63, 871.
De Allen, D. N. G. & Southwell, R. V. 1955 Q. J. Mech. Appl. Maths 8, 129.
Banks, W. H. H. 1965 Q. J. Mech. Appl. Maths 18, 443.
Banks, W. H. H. 1976 Acta Mech. 24, 273.
Beavers, G. S. & Sparrow, E. M. 1969 Trans. ASME E: J. Appl. Mech. 26, 711.Google Scholar
Burns, P. J. & Tien, C. L. 1979 Intl J. Heat Mass Transfer 22, 929.
Caltagirone, J. P. 1976 J. Fluid Mech. 76, 337.
Cheng, P. 1985 In Proc. Advanced Study Institute on Natural Convection: Fundamentals and Applications (ed. S. Kakac, W. Aung & R. Viskanta), pp. 312–351. Hemisphere.
Cheng, P. & Hsu, C. T. 1984 Trans. ASME C: J. Heat Transfer 106, 143.Google Scholar
Darcy, H. 1856 Les Fontaines Publiques de la Ville de Dijon. Dalmont.
Dennis, S. C. R. 1960 Q. J. Mech. Appl. Maths 13, 487.
Dennis, S. C. R. 1973 Proc. 3rd Intl Conf. on Numerical Methods in Fluid Mechanics, vol. 2 (ed. H. Cabannes & R. Teman), Lecture Notes in Physics, vol. 19, p. 129. Springer.
Dennis, S. C. R. & Hudson, J. D. 1978 Proc. Intl Conf. on Numerical Methods in Laminar and Turbulent Flow, Swansea, UK, p. 69. London: Pentech.
Dennis, S. C. R., Ingham, D. B. & Cook, R. N. 1979 J. Comp. Phys. 33, 325.
Fand, R. M., Steinberger, T. E. & Cheng, P. 1986 Intl J. Heat Mass Transfer 29, 119.
Fendell, F. E. 1968 J. Fluid Mech. 34, 163.
Forchheimer, P. H. 1901 Zeitschrift des Vereines, Deutscher Ing. 45, 1782.
Fornberg, B. 1980 J. Fluid Mech. 97, 819.
Hardee, H. C. 1976 Sandia Laboratories Rep. SAND 76–0075.
Ingham, D. B. 1987 Intl J. Heat Mass Transfer (to be published).
Kuehn, T. H. & Goldstein, R. J. 1980 Intl J. Heat Mass Transfer 23, 971.
Mahony, J. J. 1957 Proc. R. Soc. Lond. A 238, 412.
Merkin, J. H. 1979 Intl J. Heat Mass Transfer 22, 1461.
Muskat, M. 1946 The Flow of Homogeneous Fluids Through Porous Media. Michigan: J. W. Edwards.
Nield, D. A. 1985 Proc. CSIRO/DSIR Seminar on Convective Flows in Porous Media, Wellington, New Zealand, p. 6.
Nigam, S. D. 1954 Z. angew Math. Phys. 5, 151.
Roscoe, D. F. 1975 J. Inst. Maths Applics 16, 291.
Roscoe, D. F. 1976 Intl J. Numer. Meth. Engng 10, 1299.
Scheidegger, A. E. 1974 The Physics of Flow Through Porous Media. University of Toronto Press.
Smith, F. T. & Duck, P. W. 1977 Q. J. Mech. Appl. Maths 30, 143.
Spalding, D. B. 1972 Intl J. Numer. Meth. Engng 10, 1299.
Stewartson, K. 1958 In Boundary-Layer Research Symp. Freiburg, p. 59. Springer.
Wooding, R. A. 1963 J. Fluid Mech. 15, 527.
Yamamoto, K. 1974 J. Phys. Soc. Japan 37, 1164.
Yih, C.-S. 1965 Dynamics of Non-homogeneous Fluids. Macmillan.