Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2025-01-04T05:19:47.369Z Has data issue: false hasContentIssue false
  • English
  • Français

Modal equations for cellular convection

Published online by Cambridge University Press:  29 March 2006

D. O. Gough ,
E. A. Spiegel  and
Juri Toomre
D. O. Gough
Affiliation:
Institute of Astronomy and Department of Applied Mathematics and Theoretical Physics, University of Cambridge
E. A. Spiegel
Affiliation:
Astronomy Department, Columbia University, New York 10027
Juri Toomre
Affiliation:
Joint Institute for Laboratory Astrophysics and Department of Astro-Geophysics, University of Colorado, Boulder
Get access Rights & Permissions [Opens in a new window]

Abstract

We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possibly useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of equations which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of ‘modal equations’ is analysed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 68 , Issue 4 , 29 April 1975 , pp. 695 - 719
Copyright
© 1975 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Busse, F. H. 1969 On Howard's upper bound for heat transport by turbulent convection. J. Fluid Mech. 37, 457–477.Google Scholar
Chan, S. K. 1971 Infinite Prandtl number turbulent convection. Studies in Appl. Math. 50, 13–49.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Chu, T. Y. & Goldstein, R. J. 1973 Turbulent convection in a horizontal layer of water. J. Fluid Mech. 60, 141–159.Google Scholar
Elder, J. W. 1969 The temporal development of a model of high Rayleigh number convection. J. Fluid Mech. 35, 417–437.Google Scholar
Glansdorff, P. & Prigogine, I. 1964 On a general evolution criterion in macroscopic physics. Physica, 30, 351–374.Google Scholar
Herring, J. R. 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20, 325–338.Google Scholar
Herring, J. R. 1964 Investigation of problems in thermal convection: rigid boundaries. J. Atmos. Sci. 21, 277–290.Google Scholar
Herring, J. R. 1966 Some analytic results in the theory of thermal convection. J. Atmos. Sci. 23, 672–677.Google Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.Google Scholar
Howard, L. N. 1965 Boundary layer treatment of the mean field equations. Woods Hole Oceanographic Inst. GFD Notes, no. 65–51, I, pp. 124–126.Google Scholar
Kraichnan, R. H. 1961 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids, 5, 1374–1389.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. Roy. Soc. A 225, 196–212.Google Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225–260.Google Scholar
Murphy, J. O. 1971a Nonlinear thermal convection with free boundaries. Austr. J. Phys. 24, 587–592.Google Scholar
Murphy, J. O. 1971b Non-linear convection with hexagonal planform. Proc. Astron. Soc. Austr. 2, 53–54.Google Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32, 529–546.Google Scholar
Reid, W. H. & Harris, D. L. 1958 Some further results on the Bénard problem. Phys. Fluids, 1, 102–110.Google Scholar
Roberts, P. H. 1966 On non-linear Bénard convection. In Non-Equilibrium Thermo-dynamics, Variational Techniques, and Stability (ed. R. Donnelly, R. Hermann & I. Prigogine), pp. 125–162. University of Chicago Press.
Rossby, H. T. 1969 A study of BBnard convection with or without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Schlüter, A., Lortz, D. & Busse, F. H. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Segel, L. A. & Stuart, J. T. 1962 On the question of the preferred mode in cellulm thermal convection. J. Fluid Mech. 13, 289–306.Google Scholar
Spiegiel, E. A. 1962 On the Malkus theory of turbulence. In Mécanique de la Turbulence, pp. 181–201. Paris: C.N.R.S.
Spiegel, E. A. 1967 The theory of turbulent convection. 28th I.A.U. Symp., pp. 348–366. Academic.
Spiegel, E. A. 1971 Convection in stars. I. Basic Boussinesq convection. Ann. Rev. Astron. Astrophys. 9, 323–352.Google Scholar
Stewartson, K. 1966 Asymptotic theory in limit R →. ∞ In Non-Equilibrium Thermodynamics, Variational Techniques, and Stability (ed. R. Donnelly, R. Hermann & I. Prigogine), pp. 158–162. University of Chicago Press.
Van Der Borget, R. 1971 Asymptotic solutions in nonlinear convection with free boundaries. Austr. J. Phys. 24, 579–585.Google Scholar