Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-09T16:57:35.739Z Has data issue: false hasContentIssue false
  • English
  • Français

Large-scale flow in turbulent convection: a mathematical model

Published online by Cambridge University Press:  21 April 2006

L. N. Howard  and
R. Krishnamurti
L. N. Howard
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
R. Krishnamurti
Affiliation:
Department of Oceanography, Florida State University, Tallahasse, Florida 32306, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A mathematical model of convection, obtained by truncation from the two-dimensional Boussinesq equations, is shown to exhibit a bifurcation from symmetrical cells to tilted non-symmetrical ones. A subsequent bifurcation leads to time-dependent flow with similarly tilted transient plumes and a large-scale Lagrangian mean flow. This change of symmetry is similar to that occurring with the advent of a large-scale flow and transient tilted plumes seen in laboratory experiments on turbulent convection at high Rayleigh number. Though not intended as a description of turbulent convection, the model does bring out in a theoretically tractable context the possibility of the spontaneous change of symmetry suggested by the experiments.

Further bifurcations of the model lead to stable chaotic phenomena as well. These are numerically found to occur in association with heteroclinic orbits. Some mathematical results clarifying this association are also presented.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 170 , September 1986 , pp. 385 - 410
Copyright
© 1986 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. & Whitehead J. A.1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305–320.Google Scholar
Coddington, E. A. & Levinson N.1955 Theory of Ordinary Differential Equations. McGraw-Hill.
Curry J. H.1978 A generalized Lorenz System. Communs Math. Phys. 60, 193–204.Google Scholar
Feigenbaum M. J.1978 Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19, 25–52.Google Scholar
Krishnamurti R.1970a On the transition to turbulent convection. Part 1. The transition from two- to three-dimensional flow. J. Fluid Mech. 42, 295–307.Google Scholar
Krishnamurti R.1970b On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42, 309–320.Google Scholar
Krishnamurti, R. & Howard L. N.1981 Large-scale flow generation in turbulent convection. Proc. Nat. Acad. Sci. 78, 1981–1985.Google Scholar
Lorenz E. N.1963 Deterministic nonperiodic flow J. Atmos. Sci. 20, 130–141.Google Scholar
Malkus W. V. R.1954 Discrete transitions in turbulent convection Proc. R. Soc. Lond. A 225, 185–195.Google Scholar
Moore D. R., Toomre J., Knobloch, E. & Weiss N. O.1983 Period doubling and chaos in partial differential equations for thermosolutal convection. Nature 303, 663–667.Google Scholar
Silnikov L. P.1965 A case of the existence of a denumerable set of periodic motions. Dokl. Akad. Nauk SSSR 160, 558–561 (Sov. Math. Dokl. 6, 163–166).Google Scholar
Silnikov L. P.1970 A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle-focus type. Mat. Sbornik 81, 123 (Math. USSR Sbornik 10, 91–102).Google Scholar
Sparrow C.1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer.
Tresser C.1984 About some theorems by L. P. Silnikov. Ann. Inst. Henri Poincaré 40, 441–461.Google Scholar
Willis, G. E. & Deardorff J. W.1967a Development of short-period temperature fluctuations in thermal convection. Phys. Fluids 10, 931–937.Google Scholar
Willis, G. E. & Deardorff J. W.1967b Confirmation and renumbering of the discrete heat flux transitions of Malkus. Phys. Fluids 10, 1861–1866.Google Scholar