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On the cellular patterns in thermal convection

Published online by Cambridge University Press:  28 March 2006

J. T. Stuart
J. T. Stuart
Affiliation:
National Physical Laboratory, Teddington, Middlesex
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Abstract

In the theory of thermal convective instability between two horizontal planes there are many solutions that are periodic in the horizontal co-ordinates, while in experiment convection is observed to take place in cellular patterns. It is often assumed, or decided after insufficient argument, that the periodic solutions of the mathematical model ‘explain’ or correspond to these patterns, but a completely satisfactory discussion of this correspondence has not been given. Indeed, with certain mathematical solutions ambiguities arise as to what cell centres and cellular boundaries are. A detailed discussion has recently become especially necessary because attempts are being made to predict which particular cellular pattern will occur in given experimental conditions.

In this paper the topic is studied afresh and the question is asked: what features, in the mathematical model, correspond to what an experimentalist observes in cellular convective motion? In answer a definition of a cell is formulated which relates certain surfaces in the flow field of the mathematical model to steady vertical cellular boundaries that are observed in experiment, and which shows where the cell centres lie. In particular the classical hexagonal cellular pattern, of the mathematical model, is shown to be the prototype pattern of what is experimentally observed. On the other hand the square and so called ‘rectangular’ cases of linearized theory are shown not to correspond truly to square and rectangular cells at all. The new formulation is especially relevant to theoretical work on the prediction of cell shape and direction of flow in cells, since precise knowledge of the shapes of the cellular boundaries and locations of cell centres is essential if predictions are to be compared with observation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 18 , Issue 4 , April 1964 , pp. 481 - 498
Copyright
© 1964 Cambridge University Press

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References

Aris, R. 1962 Vectors, Tensors and the Basic Equations of Fluid Mechanics. New York: Prentice-Hall.
Avsec, D. 1939 Tourbillons thermoconvectifs dans l'air. Application a la météorologie. Pub. Sci. Tech. Min. Air (Paris), no. 155.
Bénard, H. 1901 Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent. Ann. Chim. Phys., 23, 62–144.Google Scholar
Bisshopp, F. E. 1960 On two-dimensional cell patterns. J. Math. Anal. Appl., 1, 373–85.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
De Graaf, J. G. A. & Van Der Held, E. F. M. 1953 The relation between the heat transfer and convection phenomena in enclosed plane air layers. Appl. Sci. Res. A, 3, 393–409.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech., 8, 183–92.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech., 4, 489–500.Google Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Scientific Papers, 6, 432–46.Google Scholar
Reid, W. H. & Harris, D. L. 1959 Streamlines in Bénard convection cells. Phys. Fluids, 2, 716–17.Google Scholar
Schmidt, E. & Silveston, P. L. 1959 Natural convection in horizontal liquid layers. Chem. Engng Progr. 55, 163–9.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, I, II. ForschArb. IngWes. 24, 29–32, 59–69.Google Scholar
Stoker, J. J. 1950 Non-linear Vibrations. New York: Interscience.
Stuart, J. T. 1960 Non-linear effects in hydrodynamic stability. Proc. 10th Int. Congr. Appl. Mech., Stresa, pp. 63–97.
Stuart, J. T., Pankhurst, R. C. & Bryer, D. W. 1963 Particle paths, filament lines and streamlines. NPL Aero Rep. 1057.Google Scholar
Tippelskirch, H. Von 1956 über Konvektionszellen, inbesondere im flüssigen Schwefel. Beitr. Phys. Atmos. 29, 37–54.Google Scholar
Veronis, G. 1959 Cellular convection with finite amplitude in a rotating fluid. J. Fluid Mech., 5, 401–35.Google Scholar
Wehausen, J. V. & Laitone, E. V. 1960 Surface Waves. Handbuch der Physik IX, Fluid Dynamics III, pp. 495–8.