Hostname: page-component-7dd5485656-npwhs Total loading time: 0 Render date: 2025-10-30T21:13:34.274Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of initial conditions and lateral boundaries on convection

Published online by Cambridge University Press:  29 March 2006

Theodore D. Foster
Theodore D. Foster
Affiliation:
Yale University, New Haven, Connecticut
Get access Rights & Permissions [Opens in a new window]

Abstract

A theoretical analysis of two-dimensional, finite-amplitude, thermal convection is made for a fluid which has an infinite Prandtl number. The vertical velocity disturbance is expanded in a double Fourier series which satisfies the horizontal and lateral boundary conditions. The resulting coupled sets of non-linear differential equations are solved numerically. It is found that for a particular Rayleigh number the number and size of the convection cells that form depend upon the ratio of the distance between the lateral boundaries to the depth of the fluid layer and on the initial conditions. The steady-state solutions are not unique and the solution for which the heat transport is a maximum is not necessarily the solution that results. Where there are no lateral boundaries, the lateral edges of the cells tend to tilt and the Nusselt number increases slightly.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 37 , Issue 1 , 5 June 1969 , pp. 81 - 94
Copyright
© 1969 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.)

Article purchase

Temporarily unavailable

References

Busse, F. H. 1967 On the stability of two-dimensional convection in a layer heated from below. J. Math. Phys. 46, 140.Google Scholar
Chen, M. M. & WHITEHEAD, J. A. 1968 Evolution of two-dimensional periodic Rayleigh convection cells of arbitrary wave-numbers. J. Fluid Mech. 31, 1.Google Scholar
Foster, T. D. 1965a Stability of a homogeneous fluid cooled uniformly from above. Phys. Fluids 8, 1249.Google Scholar
Foster, T. D. 1965b Onset of convection in a layer of fluid cooled from above. Phys. Fluids 8, 1770.Google Scholar
Herring, J. R. 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20, 325.Google Scholar
Herring, J. R. 1964 Investigation of problems in thermal convection: rigid boundaries. J. Atmos. Sci. 21, 277.Google Scholar
Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. Roy. Soc. A 225, 196.Google Scholar
Meyer, K. A. 1967 Time-dependent numerical study of Taylor vortex flow. Phys. Fluids 10, 1874.Google Scholar
Romanelli, M. J. 1960 Runge-Kutta methods for the solution of ordinary differential equations, in Mathematical Methods for Digital Computers, edited by A. Ralston and H. S. Wilf. New York: Wiley.
Veronis, G. 1966 Large-amplitude Bénard convection. J. Fluid Mech. 26, 49.Google Scholar