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Stability of the flow in a differentially heated inclined box

Published online by Cambridge University Press:  29 March 2006

John E. Hart
John E. Hart
Affiliation:
Department of Meteorology, M.I.T., Cambridge, Massachusetts Present address: D.A.M.T.P., Cambridge University.
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Abstract

The effect of sloping boundaries on thermal convection is studied theoretically and in the laboratory in the context of a model in which fluid is contained in a differentially heated rectangular box of small aspect ratio (depth/length), inclined at an angle δ to the vertical. Like its two limiting cases, Bénard convection and convection in the vertical slot, a basic state which exists for low Rayleigh numbers becomes unstable as this parameter is increased. The types of instability and indeed the manner in which the motions become turbulent depend crucially on δ. In our work with water the following general picture of the primary instabilities applies:

  1. For 90° > δ > 10° with the bottom plate hotter, the instabilities are stationary longitudinal convectively driven rolls with axes oriented up the slope. Near δ = 10° there is an upper and lower Rayleigh number cut off. If the Rayleigh number is too small diffusion damps the instabilities, but if it is too large they are damped by the development of a stable upslope temperature gradient in the mean flow.

  2. For 10° > δ > −10° (negative angles imply a hotter upper plate), transverse travelling waves oriented across the slope are the first instabilities of the mean flow. They obtain their kinetic energy via the working of the upslope buoyancy force.

  3. For - 10° > δ > −85° longitudinal modes are again observed. These are rather curious in that they may exist when the stratification $-\hat{g}\cdot\nabla T $ is everywhere positive. The necessary energy for these modes comes out of the mean velocity field and out of the mean available potential energy.

Agreement between the stability theory and the experiments is generally quite good over the whole range of δ, considering the approximations involved in finding a suitable basic flow solution.

For Rayleigh numbers less than ∼ 106 turbulence is only possible for positive angles. For 85° > δ > 20° the development of unsteadiness involves the occurrence and the breaking of wavy longitudinal vortices in a manner reminiscent of the development of turbulence in cylindrical Couette flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 47 , Issue 3 , 14 June 1971 , pp. 547 - 576
Copyright
© 1971 Cambridge University Press

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References

Baker, D. J. 1966 J. Fluid Mech. 26, 573.
Batchelor, G. K. 1954 Quart. Appl. Math. 12, 209–233.
Birikh, R. V., Gershuni, G. Z., Zhukhovitskii, E. M. & Rudakov, R. N. 1968 Prikl. Mat. i Mekh. 32, 256–263.
Birikh, R. V., Gershuni, G. Z., Zhukhovitskii, E. M. & Rudakov, R. N. 1969 Prikl. Mat. i Mekh. 33, 958.
Davis, S. H. 1967 J. Fluid Mech. 30, 465–478.
Degraff, J. G. A. & Van der Held, E. F. M. 1953 Appl. Sci. Res. A, 3, 393.
Eckert, E. R. G. & Carlson, W. O. 1961 Int. J. Heat & Mass Transfer, 2, 106–120.
Elder, J. W. 1965a J. Fluid Mech. 23, 77–98.
Elder, J. W. 1965b J. Fluid Mech. 23, 99–111.
Gallagher, A. & McDmercer, A. 1965 Proc. Roy. Soc A 286, 117–128.
Gershuni, G. Z. & Zhukhovitskii, E. M. 1969 Prikl. Mat. i Mekh. 33, 855.
Gill, A. E. 1966 J. Fluid Mech. 26, 515–536.
Gill, A. E. & Davey, A. 1969 J. Fluid Mech. 35, 775–778.
Gill, A. E. & Kirkham, C. C. 1970 J. Fluid Mech. 42, 125–127.
Hart, J. E. 1971 in preparation.
Ingersoll, A. P. 1966 Phys. Fluids 9, 682–689.
Koschmieder, E. 1966 Beitr. Phys. Atmos. 39, 1–11.
Krishnamurti, R. 1970 J. Fluid Mech. 42, 295–321.
Kurtweg, V. H. 1970 J. Heat Transfer, 14, 190.
Liang, S. F. & Acrivos, A. 1970 Int. J. Heat & Mass Transfer, 13, 449.
Mikhlin, S. G. 1964 Variational Methods in Mathematical Physics. New York: Pergammon.
Rayleigh, L. 1916 Phil. Mag. 32, series 6.
Rudakov, R. N. 1967 Prikl. Mat. i Mekh. 31, 349–355.
Segel, L. 1969 J. Fluid Mech. 38, 203–224.
Vest, C. M. & Arpaci, V. S. 1969 J. Fluid Mech. 36, 1.