Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T16:00:34.545Z Has data issue: false hasContentIssue false
  • English
  • Français

Time-dependent solutions of multimode convection equations

Published online by Cambridge University Press:  20 April 2006

Juri Toomre ,
D. O. Gough  and
E. A. Spiegel
Juri Toomre
Affiliation:
Joint Institute for Laboratory Astrophysics, and Department of Astrophysical, Planetary and Atmospheric Sciences, University of Colorado, Boulder, Colorado 80309, U.S.A.
D. O. Gough
Affiliation:
Institute of Astronomy, and Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England
E. A. Spiegel
Affiliation:
Department of Astronomy, Columbia University, New York, New York 10027, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Truncated modal equations are used to study the time evolution of thermal convection. In the Boussinesq approximation these nonlinear equations are obtained by expanding the fluctuating velocity and temperature fields in a finite set of planforms of the horizontal coordinates. Here we report on numerical studies dealing with two or three modes with triad interactions. We have found rich time dependence in these cases: periodic and aperiodic solutions can be obtained, along with various steady solutions. Three-mode solutions reproduce the qualitative appearance of spoke-pattern convection as observed in experiments at high Prandtl numbers. Though the values of the periods of the time-dependent solutions do not agree with those of the experiments, their variation with Rayleigh number compares favourably. Except at the highest Rayleigh number we have considered (107), the theoretical Nusselt numbers agree well with experiment.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 125 , December 1982 , pp. 99 - 122
Copyright
© 1982 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

Ahlers, G. & Behringer, R. P. 1978 Evolution of turbulence from the Rayleigh-Bénard instability. Phys. Rev. Lett. 40, 712–716.Google Scholar
Bergé, P. & Dubois, M. 1976 Time dependent velocity in Rayleigh-Bénard convection: a transition to turbulence. Optics Commun. 19, 129–133.Google Scholar
Busse, F. H. 1969 On Howard's upper bound for heat transport by turbulent convection. J. Fluid Mech. 37, 457–477.Google Scholar
Busse, F. H. 1972 The oscillatory instability of convection rolls in a low Prandtl number fluid. J. Fluid Mech. 52, 97–112.Google Scholar
Busse, F. H. & Clever, R. M. 1979 Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319–335.Google Scholar
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305–320.Google Scholar
Busse, F. H. & Whitehead, J. A. 1974 Oscillatory and collective instabilities in large Prandtl number convection. J. Fluid Mech. 66, 67–80.Google Scholar
Chan, S. K. 1971 Infinite Prandtl number turbulent convection. Stud. Appl. Math. 50, 13–49.Google Scholar
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625–645.Google Scholar
Curry, J. H. 1978 A generalized Lorenz system. Commun. Math. Phys. 60, 193–204.Google Scholar
Deardorff, J. W. & Willis, G. E. 1967 Investigation of turbulent thermal convection between horizontal plates. J. Fluid Mech. 28, 675–704.Google Scholar
Gollub, J. P. & Benson, S. V. 1978 Chaotic response to periodic perturbation of a convecting fluid. Phys. Rev. Lett. 41, 948–951.Google Scholar
Gollub, J. P. & Benson, S. V. 1980 Many routes to turbulent convection. J. Fluid Mech. 100, 449–470.Google Scholar
Gough, D. O., Spiegel, E. A. & Toomre, J. 1975a Modal equations for cellular convection. J. Fluid Mech. 68, 695–719 (Paper I).Google Scholar
Gough, D. O., Spiegel, E. A. & Toomre, J. 1975b Highly stretched meshes as functional of solutions. In Proc. 4th Int. Conf. Numer. Meth. Fluid Mech. (ed. R. D. Richtmyer). Lecture Notes in Physics, vol. 35, pp. 191–196.
Howard, L. N. 1966 Convection at high Rayleigh number. In Proc. 11th Int. Congr. Appl. Mech. (ed. H. Görtler), pp. 1109–1115. Springer.
Jones, C. A. & Moore, D. R. 1979 The stability of axisymmetric convection. Geophys. Astrophys. Fluid Dyn. 11, 245–270.Google Scholar
Jones, C. A., Moore, D. R. & Weiss, N. O. 1976 Axisymmetric convection in a cylinder. J. Fluid Mech. 73, 353–388.Google Scholar
Keller, J. B. 1966 Periodic oscillations in a model of thermal convection. J. Fluid Mech. 26, 599–606.Google Scholar
Krishnamurti, R. 1970 On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42, 309–320.Google Scholar
Krishnamurti, R. 1973 Some further studies on the transition to turbulent convection. J. Fluid Mech. 60, 285–303.Google Scholar
Ledoux, P., Schwarzschild, M. & Spiegel, E. A. 1961 On the spectrum of turbulent convection. Astrophys. J. 133, 184–197.Google Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Mclaughlin, J. B & Martin, P. C. 1975 Transition to turbulence in a statically stressed fluid system. Phys. Rev. A 12, 186–203.Google Scholar
Maschke, E. K. & Saramito, B. 1979 On the possible importance of generalized Lorenz equations in the theory of magnetohydrodynamic stability. In Intrinsic Stochasticity in Plasmas (ed. G. Laval & D. Gresillon), pp. 383–392. Orsay: Editions de Physique.
Moore, D. R. & Weiss, N. O. 1973 Two-dimensional Rayleigh-Bénard convection. J. Fluid Mech. 58, 289–312.Google Scholar
Roberts, P. H. 1966 On non-linear Bénard convection. In Non-Equilibrium Thermodynamics, Variational Techniques, and Stability (ed. R. Donnelly, R. Hermann & I. Prigogine), pp. 125–162. University of Chicago Press.
Rossby, H. T. 1966 An experimental study of Bénard convection with and without rotation. Ph.D. thesis, Massachusetts Institute of Technology.
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Saltzman, B. 1962 Finite amplitude free convection as an initial value problem. J. Atmos. Sci. 19, 329–341.Google Scholar
Spiegel, E. A. 1971 Convection in stars. I. Basic Boussinesq convection. Ann. Rev. Astron. Astrophys. 9, 323–352.Google Scholar
Toomre, J., Gough, D. O. & Spiegel, E. A. 1977 Numerical solutions of single-mode convection equations. J. Fluid Mech. 79, 1–31 (Paper II).Google Scholar
Welander, P. 1967 On the oscillatory instability of a differentially heated fluid loop. J. Fluid Mech. 29, 17–30.Google Scholar
Whitehead, J. A. & Chan, G. L. 1976 Stability of Rayleigh-Bénard convection rolls and bimodal flow at moderate Prandtl numbers. Dyn. Atmos. Oceans 1, 33–49.Google Scholar
Whitehead, J. A. & Parsons, B. 1978 Observations of convection at Rayleigh numbers up to 760,000 in a fluid with large Prandtl number. Geophys. Astrophys. Fluid Dyn. 9, 201–217.Google Scholar
Willis, G. E. & Deardorff, J. W. 1967 Development of short-period temperature fluctuations in thermal convection. Phys. Fluids 10, 931–937.Google Scholar
Willis, G. E. & Deardorff, J. W. 1970 The oscillatory motions of Rayleigh convection. J. Fluid Mech. 44, 661–672.Google Scholar
Yorke, J. A. & Yorke, E. D. 1981 Chaotic behavior and fluid dynamics. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), pp. 77–95. Springer.