Hostname: page-component-5c6d5d7d68-tdptf Total loading time: 0 Render date: 2024-08-09T04:15:06.494Z Has data issue: false hasContentIssue false
  • English
  • Français

The planforms and onset of convection with a temperature-dependent viscosity

Published online by Cambridge University Press:  21 April 2006

David B. White
David B. White
Affiliation:
Schlumberger Cambridge Research, P.O. Box 153, Cambridge CB3 0HG, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

An experimental investigation was made of convection in a fluid with a strongly temperature-dependent viscosity. The determination of the critical Rayleigh number, Rc, using the appearance of convection to define onset, was complicated by the occurrence of subcritical instabilities initiated by horizontal temperature gradients at the side boundaries. The increase in Rc over the expected value was less than predicted by linear theory, probably owing to the effect of finite conductivity boundaries and the temperature dependence of other fluid properties.

The stability of various convective planforms was studied as a function of Rayleigh number, wavenumber and viscosity variation using controlled initial conditions to specify the wavenumber and pattern, Rayleigh numbers of up to 63000 and viscosity variations of up to 1000. In addition to the rolls and hexagons seen in constant- and weakly temperature-dependent-viscosity fluid, a new planform of squares was observed at large viscosity variations.

Experiments with viscosity variations of 50 and 1000 showed that hexagons and squares were stable at Rayleigh numbers less than 25000 over a limited range of wavenumbers, which was shifted to higher values with increasing viscosity variation. Temperature profiles through the layer revealed that this shift in wavenumber was associated with the development of a thick, stagnant, cold boundary layer which reduced the effective depth of the layer.

Experiments with a fixed wavenumber showed that rolls were unstable at all Rayleigh numbers for a viscosity contrast greater than 40, whereas squares did not become stable until the viscosity contrast exceeded 6. At low viscosity variations and high Rayleigh numbers rolls became unstable to a bimodal pattern, but at high viscosity variations and a Rayleigh number of 25000 squares broke down into the spoke pattern, a convective flow not observed until Rayleigh numbers of around 100000 in a constant-viscosity fluid.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 191 , June 1988 , pp. 247 - 286
Copyright
© 1988 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

Booker, J. R. 1976 Thermal convection with strongly temperature-dependent viscosity. J. Fluid Mech. 76, 741–754.Google Scholar
Busse, F. H. 1967a On the stability of two-dimensional convection in a layer heated from below. J. Math. Phys. 46, 140–149.Google Scholar
Busse, F. H. 1967b The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.Google Scholar
Busse, F. H. & Frick, H. 1985 Square-pattern convection in fluids with strongly temperature-dependent viscosity. J. Fluid Mech. 150, 415–465.Google Scholar
Busse, F. H. & Riahi, N. 1980 Non-linear convection in a layer with nearly insulating boundaries. J. Fluid Mech. 96, 243–256.Google Scholar
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305–320.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Chen, M. M. & Whitehead, J. A. 1968 Evolution of two-dimensional periodic Rayleigh convection cells of arbitrary wavenumbers. J. Fluid Mech. 31, 1–15.Google Scholar
Hoard, C. O., Robertson, C. R. & Acrivos, A. 1970 Experiments on cellular structure in Bénard convection. Intl. J. Heat Mass Transfer 13, 849.Google Scholar
Krishnamurti, R. 1968 Finite amplitude convection with changing mean temperature. Part 1. Theory. J. Fluid Mech. 33, 445–455.Google Scholar
Oliver, D. S. & Booker, J. R. 1983 Planform and heat transport of convection with strongly temperature dependent viscosity. Geophys. Astrophys. Fluid Dyn. 27, 73–850.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.Google Scholar
Palm, E., Ellingsen, T. & Gjevik, B. 1967 On the occurrence of cellular motion in Bénard convection. J. Fluid Mech. 30, 651–661.Google Scholar
Proctor, M. R. E. 1981 Planform selection by finite amplitude thermal convection between poorly conducting slabs. J. Fluid Mech. 113, 469–485.Google Scholar
Rayleigh, Lord. 1916 On convection currents in a horizontal layer of fluid when the higher temperature is on the underside. Phil. Mag. (6) 32, 529.Google Scholar
Richter, F. M. 1978 Experiments on the stability of convection rolls in fluids whose viscosity depends on temperature. J. Fluid Mech. 89, 553–560.Google Scholar
Richter, F. M., Nataf, H.-C. & Daley, S. F. 1983 Heat transfer and horizontally averaged temperature of convection with large viscosity variations. J. Fluid Mech. 129, 173–192.Google Scholar
Somerscales, E. F. C. & Dougherty, T. S. 1970 Observed flow patterns at the initiation of convection in a horizontal liquid heated from below. J. Fluid Mech. 42, 755–768.Google Scholar
Stengel, K. C., Oliver, D. S. & Booker, J. R. 1982 Onset of convection in a variable viscosity fluid. J. Fluid Mech. 120, 411–431.Google Scholar
White, D. B. 1981 Experiments with convection in a variable viscosity fluid. Ph.D. thesis, University of Cambridge.
Whitehead, J. A. & Chan, G. L. 1978 Stability of Rayleigh-Bénard convection rolls and bimodal flow at moderate Prandtl numbers. Dyn. Atmos. Oceans 1, 33–49.Google Scholar
Whitehead, J. A. & Parsons, B. 1978 Observations of convection at Rayleigh numbers up to 760,000 in a fluid with a large Prandtl number. Geophys. Astrophys. Fluid Dyn. 19, 201–217.Google Scholar
Wray, F. 1978 Some convective flows of geophysical interest. Ph.D. thesis, University of Cambridge.