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Thermal instability and convection of a thin fluid layer bounded by a stably stratified region

Published online by Cambridge University Press:  29 March 2006

J. A. Whitehead  and
M. M. Chen
J. A. Whitehead
Affiliation:
Yale University, New Haven, Conn. Present address: Institute of Geophysics and Planetary Physics, University of California, Los Angeles.
M. M. Chen
Affiliation:
Yale University, New Haven, Conn. Present address: Department of Mechanical Engineering, New York University, New York.
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Abstract

Results of linear stability calculations and post-stability experimental observations are reported for horizontal fluid layers with upward heat flux bounded below by a stably stratified fluid. Stability calculations were done for several families of continuous and discontinuous temperature distributions, and it was found that as a rule the flow originating in the unstable layer penetrates into the stably stratified region, resulting in increased critical cell size and correspondingly decreased critical Rayleigh number. A notable exception to this occurs for an unstable layer with a linear temperature distribution adjacent to a stable layer of very high stable density gradient. In this case energy pumped from the unstable to the stable region is sufficient to raise the critical Rayleigh number above that of a solid boundary. It is also found that, for density distributions with a more gradual transition between the stable and the unstable regions, the effect of increased cell size upon the critical Rayleigh number is sometimes masked by effects of curvature in the density profile of the unstable region, which tends to increase the critical Rayleigh number. The inadequacy of the usual definition of Rayleigh number to characterize the stability of such complex systems is discussed. Experimentally, such a temperature distribution was produced by radiant energy from above as it was absorbed by the top few centimetres of the fluid. Within an uncertainty of ± 20%, it was found that the critical experimental Rayleigh number agreed with neutral stability calculations. The supercritical convective motion consisted of vertical jets of cool surface fluid which plunged downward into the interior of the fluid. The jets were not arranged in an orderly lattice but were in a constant state of change, each jet having a tendency to merge with a close neighbour. The net loss of jets due to merging was balanced by new jets spontaneously appearing. As Rayleigh number was increased, the mean number of jets and the intermittancy increased proportionally. Temperature scans taken with a movable probe showed that cool surface fluid plunging downward in the jets was confined to a fairly restricted region, the surrounding fluid being quite isothermal.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 40 , Issue 3 , 18 February 1970 , pp. 549 - 576
Copyright
© 1970 Cambridge University Press

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References

Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
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
Deardorff, J. W., Willis, G. E. & Lilly, D. K. 1969 Laboratory investigation of non-steady penetrative convection. J. Fluid Mech. 35, 7.Google Scholar
Foster, T. D. 1965a Stability of 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
Gribov, V. N. & Gurevich, L. E. 1957 On the theory of the stability of a layer located at a superadiabatic temperature gradient in a gravitational fluid. J.E.T.P. 4, 720.Google Scholar
Kraus, E. B. & Rooth, C. 1961 Temperature and steady state vertical flux in the ocean surface layers. Tellus, 13, 231.Google Scholar
Krishnamurti, R. E. 1968a Finite amplitude convection with changing mean temperature. Part 1. Theory. J. Fluid Mech. 33, 445.Google Scholar
Krishnamurti, R. E. 1968b Finite amplitude convection with changing mean temperature. Part 2. An experimental test of theory. J. Fluid Mech. 33, 457.Google Scholar
Krishnamurti, R. E. 1968c On the transition to turbulent convection. Part I. Transition from two to three dimensional flow. Geophysical Fluid Dynamics Institute Technical Report 14. Florida State University.Google Scholar
Krishnamurti, R. E. 1968d On the transition to turbulent convection. Part II. Transition to time-dependent flow. Geophysical Fluid Dynamics Institute Technical Report 16. Florida State University.Google Scholar
Lehnert, B. 1967 Experimental evidence of plasma instabilities. Plasma Physics, 9, 301.Google Scholar
Malkus, W. V. R. 1954 Discrete transitions in turbulent convection. Proc. Roy. Soc. Lond. A 225, 185.Google Scholar
Morton, B. R. 1957 On the equilibrium of a stratified layer of fluid. Quart. J. Mech. Appl. Math. 10, 433.Google Scholar
Rintell, L. 1967 Penetrative convective instabilities. Phys. Fluids, 4, 848.Google Scholar
Roberts, P. H. 1967 On non-linear Bénard convection. J. Fluid Mech. 30, 33.Google Scholar
Saunders, P. M. 1962 Penetrative convection in stably stratified fluids. Tellus, 14, 177.Google Scholar
Segel, L. A. 1967 Non-linear hydrodynamic stability theory and its application to thermal convection and curved flows. Non-Equilibrium Thermodynamics: Variational Techniques and Stability (ed. R. J. Donnelly, L. Prigogine & R. Herman). University of Chicago Press.
Snyder, H. A. 1969 Wave-number selection at finite amplitude in rotating Couette flow. J. Fluid Mech. 35, 273.Google Scholar
Spangenberg, W. C. & Rowland, W. R. 1961 Convective circulation in water induced by evaporative cooling. Phys. Fluids, 4, 743.Google Scholar
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1963 Thermal instability in a horizontal fluid layer, the effect of boundary conditions and non-linear temperature profile. J. Fluid Mech. 18, 513.Google Scholar
Thompson, J. J. 1882 On a changing tesselated structure in certain liquids. Proc. Glasgow Phil. Soc. 13, 464.Google Scholar
Townsend, A. A. 1966 Natural convection in water over an ice surface. Quart. J. Roy. Met. Soc. 90, 248.Google Scholar
Tritton, D. J. & Zarraga, M. N. 1967 Convection in horizontal layers with interna heat generation. J. Fluid Mech. 30, 21.Google Scholar
Veronis, G. 1963 Penetrative convection. Astrophysical J. 137, 641.Google Scholar
Whitehead, J. A. 1968 Stability and convection of a thermally destabilized layer of fluid above a stabilized region. Ph.D. Thesis, Yale University.