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The spectral dynamics of laminar convection

Published online by Cambridge University Press:  28 March 2006

George W. Platzman
George W. Platzman
Affiliation:
Department of the Geophysical Sciences, The University of Chicago
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Abstract

The non-linear equations of the Bénard convection problem are transformed to the spectral domain. The spectral basis consists of the supercritical normal modes of the characteristic-value problem in which the exponential growth rates are characteristic values. The norm of the spectrum is the variance of an arbitrary finite-amplitude state of convection. The equations that govern the spectrum are solvable by linear methods when the spectrum is truncated by exclusion of all convective modes except those of lowest-order symmetric vertical structure. Numerical computations of heat flux for a spectrum that contains only one convective mode are in good agreement with experimental data for water in the laminar régime.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 23 , Issue 3 , November 1965 , pp. 481 - 510
Copyright
© 1965 Cambridge University Press

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References

Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Christopherson, D. G. 1940 Note on the vibration of membranes. Quart. J. Math., 11, 63–5.Google Scholar
Davey, A. 1962 The growth of the Taylor vortices in flow between rotating cylinders. J. Fluid Mech. 14, 336–68.Google Scholar
Eckart, Carl 1960 Hydrodynamics of Oceans and Atmospheres. London: Pergamon Press.
Herring, J. R. 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20 325–38.Google Scholar
Howard, Louis N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–32.Google Scholar
Kuo, Hsiao-Lan & Platzman, George W. 1961 A normal-mode non-linear solution of the Rayleigh convection problem. Beit. Phys. Atmos. 33, 137–68.Google Scholar
Kuo, H. L. 1961 Solution of the non-linear equations of cellular convection and heat transport. J. Fluid Mech., 10, 611–34.Google Scholar
Ledoux, P., Schwarzschild, M. & Spiegel, E. A. 1961 On the spectrum of turbulent convection. Astrophys. J. 133, 184–97.Google Scholar
Lorenz, E. N. 1955 Available potential energy in the atmosphere. Tellus, 7, 157–67.Google Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225–60.Google Scholar
Nakagawa, Y. 1960 Heat transport by convection. Phys.Fluids, 3, 82–6.Google Scholar
Palm, Enok 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–92.Google Scholar
Pellew, Anne & Southwell, R. V. 1940 On maintained convective motion in a fluid heated from below. Proc. Roy. Soc.A, 176, 312–43.Google Scholar
Platzman, George W. 1964 The spectral dynamics of laminar convection. Department of the Geophysical Sciences, The University of Chicago.
Reid, W. H. & Harris, D. L. 1958 Some further results on the Benard problem. Phys. Fluids, 1, 102–10.Google Scholar
Saltzman, Barry 1962 Finite amplitude free convection as an initial value problem. I. J. Atmos. Sci. 19, 329–41.Google Scholar
Sani, R. 1964 On the non-existence of subcritical instabilities in fluid layers heated from below. J. fluid Mech., 20, 315–19.Google Scholar
Segel, Lee A. & Stuart, J. T. 1962 On the question of the preferred mode in cellular thermal convection. J. Fluid Mech. 13, 289–306.Google Scholar
Segel, Lee A. 1962 The non-linear interaction of two distrubances in the thermal convection problem. J. Fluid Mech. 14, 97–114.Google Scholar
Silveston, P. L. 1958 Wärmedurchgang in waagerechten Flüssigketschichten. Forsch. Ing. Wes. 24, 29–32. 59–69.Google Scholar
Spiegel, Edward A. 1962 On the Malkus theory of turbulence. In Mécanique de la Turbulence (A. Favre, Ed.) Èditions du Centre National de la Recherche Scientifique, Paris. (Also:The Mechanics of Turbulence. New York: Gordon and Breach, 1964.)
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech., 4, 1–21.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech., 9, 353–370.Google Scholar
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layers (L. Rosenhead, Ed.), pp. 492–579. Oxford University Press.
Stuart, J. T. 1964 On the cellular patterns in thermal convection. J. Fluid Mech., 18, 481–98.Google Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane poiseuille flow and for plane Couette flow. J. Fluid Mech., 9, 371–89.Google Scholar