Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-16T14:05:10.654Z Has data issue: false hasContentIssue false
  • English
  • Français

Heat source-driven thermal convection at arbitrary Prandtl number

Published online by Cambridge University Press:  19 April 2006

F. B. Cheung
F. B. Cheung
Affiliation:
Reactor Analysis and Safety Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439
Get access Rights & Permissions [Opens in a new window]

Abstract

A theoretical investigation is made of turbulent thermal convection in a horizontally infinite layer of fluid confined between a rigid isothermal upper plate and a rigid adiabatic lower plate, driven by a temperature difference between the plates that is totally induced by volumetric heating of the layer. The dependence of upper surface Nusselt number, Nu, on both Prandtl number, Pr, and internal Rayleigh number, RaI, is obtained from considerations of the Boussinesq equations of motion. Also obtained is the dependence of various turbulence quantities upon distance from the upper plate. At a sufficiently high Rayleigh number, the present theory gives $Nu \sim Ra^{\frac{1}{4}}_I$ for large Pr and Nu ∼ Pr¼Ra¼I for small Pr. At lower Rayleigh numbers, however, the Nusselt number is found to vary according to $Nu \sim (a-b\,Ra^{-\frac{1}{12}}_I)^{-1} Ra^{\frac{1}{4}}_I$, where a and b are coefficients dependent upon Pr. The asymptotic $Ra^{\frac{1}{4}}_I$ law tends to support the boundary layer instability model of Howard (1966), although significant deviation from the model is predicted by the present theory over the range of Rayleigh numbers explored experimentally (Kulacki & Nagle 1975; Kulacki & Emara 1977). Based upon the results of this study the empirical power-law representation of Nu is critically examined and found to be adequate within finite ranges of RaI. Comparison of the present flow situation is made with the corresponding case of turbulent Bénard convection.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 97 , Issue 4 , 29 April 1980 , pp. 743 - 758
Copyright
© 1980 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

Bergholz, R. F., Chen, M. M. & Cheung, F. B. 1979 Generalization of heat transfer results for turbulent free convection adjacent to horizontal surfaces. Int. J. Heat Mass Transfer 22, 763–769.Google Scholar
Cheung, F. B. 1977 Natural convection in a volumetrically heated fluid layer at high Rayleigh numbers. Int. J. Heat Mass Transfer 20, 499–506.Google Scholar
Cheung, F. B. 1978a Turbulent natural convection in a horizontal fluid layer with time dependent volumetric energy sources. A.I.A.A./A.S.M.E. Thermophysics & Heat Transfer Conf., Palo Alto, paper No. 78-HT-6.
Cheung, F. B. 1978b Correlation equations for turbulent thermal convection in a horizontal fluid layer heated internally and from below. J. Heat Transfer 100, 416–422.Google Scholar
Chu, T. Y. & Goldstein, R. J. 1973 Turbulent convection in a horizontal layer of water. J. Fluid Mech. 60, 141–159.Google Scholar
Davenport, F. & King, C. J. 1975 A note on Hoard's model for turbulent natural convection. J. Heat Transfer 97, 476–478.Google Scholar
Fiedler, H. & Wille, R. 1971 Wärmetransport bie frier Konvektion in einer horizontalen Flussigkeitsschicht mit Volumenheizung, Teil 1: Integraler Wärmetransport. Rep. Dtsch Forschungs Varsuchsanstalt Luft-Raumfahrt, Inst. Turbulenzfenschung, Berlin.
Garon, A. M. & Goldstein, R. J. 1973 Velocity and heat transfer measurements in thermal convection. Phys. Fluids 16, 1818–1825.Google Scholar
Herring, J. R. 1964 Investigation of problems in thermal convection: Rigid boundaries. J. Atmos. Sci. 21, 277–290.Google Scholar
Howard, L. N. 1966 Convection at high Rayleigh number. In Proc. 11th Cong. Appl. Mech. (ed. H. Gö;rtler), pp. 1109–1115.
Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389.Google Scholar
Kulacki, F. A. & Emara, A. A. 1977 Steady and transient thermal convection in a fluid layer with uniform volumetric energy sources. J. Fluid Mech. 83, 375–395.Google Scholar
Kulacki, F. A. & Nagle, M. E. 1975 Natural convection in a horizontal fluid layer with volumetric energy sources. J. Heat Transfer 91, 204–211.Google Scholar
Long, R. R. 1976a The relation between Nusselt number and Rayleigh number in turbulent thermal convection. J. Fluid Mech. 73, 445–451.Google Scholar
Long, R. R. 1976b Theories of turbulent thermal convection. Heat Transfer and Turbulent Buoyant Convection (ed. D. B. Spalding & N. Afgan), pp. 79–91. Hemisphere.
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. Roy. Soc. A 225, 196–212.Google Scholar
Manton, M. J. 1975 On the high Rayleigh number heating of a fluid above a horizontal surface. Appl. Sci. Res. 31, 267–277.Google Scholar
Priestley, C. H. B. 1959 Turbulent Transfer in the Lower Atmosphere. University of Chicago Press.
Sparrow, E. M., Husar, R. B. & Goldstein, R. J. 1970 Observations and other characteristics of thermals. J. Fluid Mech. 41, 793–800.Google Scholar
Spiegel, E. A. 1971 Convection in stars, I. Basic Boussinesq convection. Ann. Rev. Astronomy & Astrophys. 9, 323–352.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. Massachusetts Institute of Technology Press.
Threlfall, D. C. 1975 Free convection in low-temperature gaseous helium. J. Fluid Mech. 67, 17–28.Google Scholar