Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-18T19:05:41.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolution of two-dimensional periodic Rayleigh convection cells of arbitrary wave-numbers

Published online by Cambridge University Press:  28 March 2006

Michael M. Chen  and
John A. Whitehead
Michael M. Chen
Affiliation:
Yale University, New Haven, Connecticut
John A. Whitehead
Affiliation:
Yale University, New Haven, Connecticut
Get access Rights & Permissions [Opens in a new window]

Abstract

By introducing small controlled perturbations prior to the onset of motion, two-dimensional convection cells of arbitrary width-depth ratios are produced in a horizontal fluid layer heated from below. The conditions employed correspond to the finite-amplitude Rayleigh stability problem for a constant viscosity, large Prandtl number, Boussinesq fluid with rigid, conducting boundaries. It was found that two-dimensional cells with width-depth ratios close to unity are stable at all Rayleigh numbers investigated (Rc [Lt ] R [Lt ] 2·5Rc). Cells whose widthdepth ratios are moderately too large or too small tend to undergo size adjustments toward a preferred value of about 1·1. If the width-depth ratios are much too large or too small, they tend to develop three-dimensional instabilities in the form of cell boundary distortions and transverse secondary cells, respectively. Eventually the flow settles into a new family of rolls with a more preferred widthdepth ratio. It is suggested that these observations may have implications on nonlinear interchange instability problems and geophysical flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 31 , Issue 1 , 8 January 1968 , pp. 1 - 15
Copyright
© 1968 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

Bénard, H. 1901 Les Tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent. Ann. Chim. Phys. 23, 62–144.Google Scholar
Busse, F. 1966 On the stability of two-dimensional convection in a layer heated from below. Unpublished report.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Chen, M. M. 1966 Experimental investigation of artificially initiated convective cells Bull. Am. Phys. Soc. 11, 617.Google Scholar
Fromm, J. E. 1965 Numerical solutions of the non-linear equations for a heated fluid layer Phys. Fluids, 8, 1757.Google Scholar
Koschmieder, E. L. 1966 On convection on a uniformly heated plane Beit. Z. Phys. d. Atmos. 39, 1.Google Scholar
Krishnamurti, R. E. 1967 Thermal convection in a horizontal layer with changing mean temperature. Bull. Am. Phys. Soc. 12, 847, also Ph.D. thesis, University of California, Los Angeles, Calif.Google Scholar
Kuo, H. L. 1961 Solution of the non-linear equations of motion of cellular convection and heat transport J. Fluid Mech. 10, 611–634.Google Scholar
Malkus & Veronis 1958 Finite amplitude cellular convection. J. Fuid Mech. 4, 225.Google Scholar
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of finite amplitude convection J. Fluid Mech. 23, 129–144.Google Scholar
Schmidt, R. J. & Milverton, S. W. 1935 On the stability of a fluid when heated from below. Proc. Roy. Soc. A 152, 586–594.Google Scholar
Segel, L. A. 1965 The non-linear interaction of a finite number of disturbances to a layer of fluid heated from below J. Fluid Mech. 21, 359–384.Google Scholar
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1964 Thermal instability in a horizontal fluid layer; effect of boundary conditions and non-linear temperature profile J. Fluid Mech. 18, 513–528.Google Scholar
Veronis, G. 1966 Large amplitude Bénard convection J. Fluid Mech. 26, 49–68.Google Scholar