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Heat transport by turbulent convection

Published online by Cambridge University Press:  28 March 2006

Louis N. Howard
Louis N. Howard
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology
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Abstract

Upper bounds for the heat flux through a horizontally infinite layer of fluid heated from below are obtained by maximizing the heat flux subject to (a) two integral constraints, the ‘power integrals’, derived from the equations of motion, and (b) the continuity equations. This variational problem is solved completely, for all values of the Rayleigh number R, when only the constrains (a) are imposed, and it is thus shown that the Nusselt number N for any statistically steady convective motion cannot exceed a certain value N1(R), which for large R is approximately (3R/64)½. When (b) is included as a constraint, the variational problem is solved for large R, under the additional hypothesis that the solution has a single horizontal wave number; the associated upper bound on the Nusselt number is (R–248)⅜. The mean properties of this maximizing ‘flow’, in particular the mean temperature and mean square temperature deviation fields, are found to resemble the mean properties of the real flow observed by Townsend; the results thus tend to support Malkus's hypothesis that turbulent convection maximizes heat flux.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 17 , Issue 3 , November 1963 , pp. 405 - 432
Copyright
© 1963 Cambridge University Press

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References

Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Globe, S. & Dropkin, D. 1959 Trans. Amer. Soc. Mech. Engrs, J. Heat Transf, 81, 24.
Gor'kov, L. P. 1958 Sov. Phys. JETP, 6, 311.
Kraichnan, R. 1962 Phys. Fluids, 5, 1374.
Malkus, W. V. R. 1954a Proc. Roy. Soc. A, 225, 185.
Malkus, W. V. R. 1954b Proc. Roy. Soc. A, 225, 196.
Malkus, W. V. R. & Veronis, G. 1958 J. Fluid Mech. 4, 225.
Mihaljan, J. 1960 Tech. Rep. no. 1 on Thermal Convection in Rotating Systems. Nat. Sci. Foundation.
Pellew, A. & Southwell, R. V. 1940 Proc. Roy. Soc. A, 176, 312.
Spiegel, E. & Veronis, G. 1960 Astrophys. J. 131, 442.
Townsend, A. A. 1959 J. Fluid Mech. 5, 209.
Townsend, A. A. 1962 Mecanique de la turbulence, p. 167. Centre National de la Recherche Scientifique, Paris.