Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-18T19:12:08.015Z Has data issue: false hasContentIssue false
  • English
  • Français

Onset of Rayleigh–Bénard convection in a rigid channel

Published online by Cambridge University Press:  26 April 2006

M. S. Chana  and
P. G. Daniels
M. S. Chana
Affiliation:
Department of Mathematics, The City University, Northampton Square, London, EC1V 0HB, UK
P. G. Daniels
Affiliation:
Department of Mathematics, The City University, Northampton Square, London, EC1V 0HB, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

A two-dimensional Galerkin formulation of the three-dimensional Oberbeck-Boussinesq equations is used to describe the onset of convection in an infinite rigid horizontal channel uniformly heated from below. The dependence of the critical Rayleigh number on the channel aspect ratio is determined and results are compared with those of an idealized model studied by Davies-Jones (1970). Asymptotic results are derived for both narrow and wide channels, corresponding to limits of small and large aspect ratios respectively. In the latter case the main core flow, consisting of two-dimensional rolls with axes perpendicular to the vertical walls of the channel, can be represented by the solution of an amplitude equation. Close to the walls, however, the motion remains fully three-dimensional and a reversal of the vertical flow is associated with a local subdivision of each main roll into a pair of co-rotating rolls.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 199 , February 1989 , pp. 257 - 279
Copyright
© 1989 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

Brown, S. N. & Stewartson, K., 1977 Stud. Appl. Math. 57, 187.
Buhler, K., Kirchartz, K. R. & Oertel, H., 1979 Acta Mech. 31, 155.
Catton, I.: 1970 Trans. ASME C: J. Heat Transfer 92, 186.
Chana, M. S.: 1986 Thermal convection in channels and long boxes. Ph.D. thesis, The City University, London.
Cross, M. C., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D., 1983 J. Fluid Mech. 127, 155.
Daniels, P. G.: 1977 Proc. R. Soc. Lond. A A358, 173.
Daniels, P. G.: 1978 Mathematika 25, 216.
Daniels, P. G. & Ong, C. F., 1988 Intl J. Heat Mass Transfer (submitted).
Davies-Jones, R. P.: 1970 J. Fluid Mech. 44, 695.
Davis, S. H.: 1967 J. Fluid Mech. 30, 465.
Drazin, P. G.: 1975 Z. angew Math. Phys. 26, 239.
Harris, D. L. & Reid, W. H., 1958 Astrophys. J. Suppl. Series 3, 429.
Kelly, R. E. & Pal, D., 1978 J. Fluid Mech. 86, 433.
Kessler, R.: 1987 J. Fluid Mech. 174, 357.
Koschmieder, E. L.: 1966 Beitr. Phys. Atmos. 39, 1.
Koschmieder, E. L. & Pallas, S. G., 1974 Intl J. Heat Mass Transfer 17, 991.
Luijkx, J. M. & Platten, J. K., 1981 J. Non-Equilib. Thermodyn. 6, 141.
Newell, A. C. & Whitehead, J. A., 1969 J. Fluid Mech. 38, 279.
Oertel, H.: 1980 In Natural Convection in Enclosures (ed. K. E. Torrance & I. Catton), vol. 8, p. 11. ASME Heat Transf. Div.
Reid, W. H. & Harris, D. L., 1958 Astrophys. J. Suppl. Series 3, 448.
Segel, L. A.: 1969 J. Fluid Mech. 38, 203.