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Oscillatory Hele-Shaw convection

Published online by Cambridge University Press:  20 April 2006

H. Frick  and
U. MÜLler
H. Frick
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, Los Angeles
U. MÜLler
Affiliation:
Kernforschungszentrum Karlsruhe, Institut für Reaktorbauelemente, Karlsruhe, West Germany
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Abstract

Time-dependent convection motions in the form of rolls in a thin vertical fluid layer (Hele-Shaw cell) heated from below are investigated numerically. Perpendicular to the convection-roll axis the fluid is bounded by parallel adiabatic rigid sidewalls. Stress-free top, bottom and end boundaries are assumed. The horizontal extension of the convection rolls is described by the wavenumber α. Solutions for the time-dependent behaviour of the convective motion are presented for a range of wavenumbers between α = ½π and 2π. The onset of the oscillation is shifted to higher Rayleigh numbers with increasing wavenumber. The oscillatory Hele-Shaw convection is caused by an instability of the thermal boundary layer, as is evident from the plotted temperature field and streamlines. From the variation of the Nusselt number with time it is found that the oscillatory motion starts with a sinusoidal time dependence and passes into a periodic state with several frequencies as the Rayleigh number is increased. Quantitative and qualitative agreement with previous experimental and numerical results is found.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 126 , January 1983 , pp. 521 - 532
Copyright
© 1983 Cambridge University Press

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References

Ahlers, G. & Behringer, R. P. 1978 Evolution of turbulence from the Rayleigh-Bénard instability Phys. Rev. Lett. 40, 711.Google Scholar
Berge, P. & Dubois, M. 1979 Study of unsteady convection through simultaneous velocity and interferometric measurements J. Phys. Leti. 40, 505.Google Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection Rep. Prog. Phys. 41, 1929.Google Scholar
Caltagirone, J. P. 1975 Thermoconvective instabilities in a horizontal porous layer J. Fluid Mech. 72, 269.Google Scholar
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection J. Fluid Mech. 65, 625.Google Scholar
Elder, J. W. 1967 Steady free convection in a porous medium heated from below J. Fluid Mech. 72, 29.Google Scholar
Frick, H. 1981 Zellularkonvektion in Fluidschichten mit zwei festen seitlichen Berandungen. Dissertation, Universität Karlsruhe (KfK 3109).
Frick, H. & Clever, R. M. 1980 Einfluss der Seitenwände auf das Einsetzen der Konvektion in einer horizontalen Flüssigkeitsschicht Z. angew. Math. Phys. 31, 502.Google Scholar
Frick, H. & Clever, R. M. 1982 The influence of side walls on finite-amplitude convection in a layer heated from below J. Fluid Mech. 114, 467.Google Scholar
Gollub, P. & Benson, S. V. 1980 Many routes to turbulent convection J. Fluid Mech. 100, 449.Google Scholar
Hartline, B. K. & Lister, C. R. B. 1977 Thermal convection in a Hele-Shaw cell J. Fluid Mech. 79, 379.Google Scholar
HELE-SHAW, H. S. J. 1898 Investigations of the nature of surface resistance of water and a stream motion under certain experimental conditions Trans. Inst. Nav. Arch. 40, 21.Google Scholar
Horne, R. N. & O'SULLIVAN, M. J. 1974 Oscillatory convection in a porous medium heated from below J. Fluid Mech. 66, 339.Google Scholar
Horton, C. W. & Rogers, F. T. 1945 Convection currents in a porous medium J. Appl. Phys. 16, 367.Google Scholar
Howard, L. N. 1964 Convection at high Rayleigh number. In Proc. 11th Int. Congr. on Appl. Mech., Munich (ed. H. Görtler), p. 1109. Springer.
Koster, J. N. 1980 Freie Konvektion in vertikalen Spalten. Dissertation, Universität Karlsruhe (KfK 3066).
Krishnamurti, R. 1970a On the transition to turbulent convection. Part 1. The transition from two- to three-dimensional flow. J. Fluid Mech. 42, 295.Google Scholar
Krishnamurti, R. 1970b On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42, 309.Google Scholar
Krishnamurti, R. 1973 Some further studies on transition to turbulent convection J. Fluid Mech. 60, 285.Google Scholar
Kvernvold, O. 1979 On the stability of non-linear convection in a Hele-Shaw cell Int. J. Heat Mass Transfer 22, 395.Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium Proc. Camb. Phil. Soc. 44, 508.Google Scholar
Maurer, J. & Libchaber, A. 1979 Rayleigh-Bénard experiment in liquid helium: frequency locking and onset of turbulence. J. Phys. (Paris) 40, 419.Google Scholar
PUTIN G. F. & TKACHEVA, E. A. 1979 Experimental investigation of supercritical convective motions in a Hele-Shaw cell. Izv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza 14.Google Scholar
Schubert, G. & Straus, J. M. 1979 Three-dimensional and multicellular steady and unsteady convection in fluid-saturated porous media at high Rayleigh numbers J. Fluid Mech. 94, 25.Google Scholar
Wooding, R. A. 1960 Instability of a viscous liquid of variable density in a vertical Hele-Shaw cell J. Fluid Mech. 7, 501.Google Scholar