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The non-linear interaction of a finite number of disturbances to a layer of fluid heated from below

Published online by Cambridge University Press:  28 March 2006

Lee A. Segel
Lee A. Segel
Affiliation:
Rensselaer Polytechnic Institute, Troy, New York
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Abstract

It is predicted that, at a temperature difference a little less than that at which motion starts according to linear stability theory, a steady hexagonal convective pattern will develop from finite-amplitude instabilities in a horizontal layer of fluid heated from below. This is because the first disturbances to start growing must be the triplet of two-dimensional ‘rolls’ which form angles of 60° with each other and whose amplitudes and phases first fall in certain critical ranges. The growth of these disturbances stabilizes all other disturbances and is such that ultimately the right phases and amplitudes occur to give hexagonal cells. If the temperature difference is increased somewhat beyond its critical value, the hexagonal pattern becomes unstable and a two-dimensional roll pattern is predicted. In an intermediate temperature range, rolls are unstable but transport more heat than hexagons. ‘Free–free’ boundary conditions, a viscosity which varies with temperature, and a fixed disturbance wave-number are assumed in this extension of the work of Palm (1960) and Segel & Stuart (1962). Other theoretical results and some experimental results are compared with the present predictions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 21 , Issue 2 , February 1965 , pp. 359 - 384
Copyright
© 1965 Cambridge University Press

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References

Bisshopp, F. 1960 On two-dimensional cell patterns. J. Math. Anal. Appl. 1, 373–85.Google Scholar
Busse, F. 1962 Das Stabilitätsverhalten der Zellularkonvektion bei endlicher Amplitude. Inaugural Dissertation, Ludwig-Maximilians-Universität, Munich.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Furumoto, A. & Rooth, C. 1961 Observations on convection in water cooled from below. Notes of summer study program in geophysical fluid dynamics, Woods Hole Oceanographic Institute (unpublished).
Herring, J. R. 1964 Investigation of problems in thermal convection: rigid boundaries. Institute for Space Studies, Goddard Space Flight Center.
Howard, L. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–32.Google Scholar
Lortz, D. 1961 Instabilitäten der stationären Konvektionsströmungen endlicher Amplitude. Inaugural Dissertation, Ludwigs-Maximilians-Universität, Munich.
Malkus, W. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225–60.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–92.Google Scholar
Palm, E. & Oiann, H. 1964 Contribution to the theory of cellular thermal convection. J. Fluid Mech. 19, 353–367.Google Scholar
Segel, L. A. 1962 The non-linear interaction of two disturbances in the thermal convection problem. J. Fluid Mech. 14, 97–114.Google Scholar
Segel, L. A. 1965 The structure of non-linear cellular solutions to the Boussinesq equations. J. Fluid Mech. 21, 345–358.Google Scholar
Segel, L. A. & Stuart, J. T. 1962 On the question of the preferred mode in cellular thermal convection. J. Fluid Mech. 13, 289–306.Google Scholar
Silveston, P. L. 1958 Wärmedurchgang in waagerechten Flüssigkeitsschichten. Forsch. Ing. Wes. 24, 29–32, 59–69.Google Scholar
Veronis, G. 1959 Cellular convection with finite amplitude in a rotating fluid. J. Fluid Mech. 5, 401–35.Google Scholar
Veronis, G. 1963 Penetrative convection. Astrophys. J. 137, 641–63.Google Scholar
Whitney, L. F. 1961 Another view from Tiros I of a severe weather situation. Mon. Weath. Rev. 89, 447–60.Google Scholar