Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-08T04:38:21.582Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear transition in three-dimensional convection

Published online by Cambridge University Press:  21 April 2006

R. Kessler
R. Kessler
Affiliation:
Institute for Theoretical Fluid Mechanics, DFVLR, Bunsenstrasse 10, D-3400 Göttingen, West Germany Present address: Lehrstuhl für Strömungsmechanik, Universität Erlangen, Egerlandstrasse 13, D-8520 Erlangen, West Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

Steady and oscillatory convection in a rectangular box heated from below are studied by means of a numerical solution of the three-dimensional, time-dependent Boussinesq equations. The effect of the rigid sidewalls of the box on the spatial structure and the dynamical behaviour of the flow is analysed. Both conducting and adiabatic sidewalls are considered. Calculated streamlines illustrate the three-dimensional structure of the steady flow with Prandtl numbers 0.71 and 7. The onset and the frequency of the oscillatory instability are calculated and compared with available experimental and theoretical data. With increasing Rayleigh number a subharmonic bifurcation and the onset of a quasi-periodic flow can be observed. A comparison of the different time-dependent solutions shows some interesting relations between the spatial structure and the dynamical behaviour of the confined flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 174 , January 1987 , pp. 357 - 379
Copyright
© 1987 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

Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous fluid. 1. Theory. Proc. R. Soc. Lond. A 359, 1–26.Google Scholar
Busse, F. 1981 Transition to turbulence in Rayleigh-Bénard convection. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), pp. 97–137. Springer.
Curry, J. H., Herring, J. R., Loncaric, J. & Orszag, S. A. 1984 Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 1–38.Google Scholar
Dahlquist, G. 1976 Error analysis for a class of methods for stiff non-linear initial value problems. Lecture Notes in Mathematics, vol. 506 (ed. A. Dold & B. Echmann). Springer.
Davies-Jones, R. P. 1970 Thermal convection in an infinite channel with no-slip sidewalls. J. Fluid Mech. 44, 695–704.Google Scholar
Davis, S. H. 1967 Convection in a box: linear theory. J. Fluid Mech. 30, 465–478.Google Scholar
De Vahl Davis, G. & Jones, I. P. 1983 Natural convection in a square cavity: a comparison exercise. Intl J. Numer. Method. Fluid. 3, 227–248.Google Scholar
Feigenbaum, M. J. 1979 The onset spectrum of turbulence. Phys. Lett. A 74, 375–378.Google Scholar
Frick, H., Busse, F. H. & Clever, R. M. 1983 Steady three-dimensional convection at high Prandtl numbers. J. Fluid Mech. 127, 141–153.Google Scholar
Frick, H. & Clever, R. M. 1980 EinflusZ der Seitenwände auf das Einsetzen der Konvektion in einer horizontalen Flüssigkeitsschicht. Z. angew. Math. Phys. 31, 502–513.Google Scholar
Gollub, J. P. & Benson, S. V. 1980 Many routes to turbulent convection. J. Fluid Mech. 100, 449–470.Google Scholar
Gollub, J. P., Benson, S. V. & Steinman, J. 1980 A subharmonic route to turbulent convection. Ann. N.Y. Acad. Sci. 357, 22–27.Google Scholar
Harris, D. L. & Reid, W. H. 1958 On orthogonal functions which satisfy four boundary conditions. Astrophys. J.S. 3, 429–453.Google Scholar
Jäger, W. 1982 Oszillatorische und turbulente Konvektion. dissertation, Universität Karlsruhe, West Germany.
Jones, I. P. 1979 Natural convection in an enclosed cavity: a comparison problem. Comp. Fluids 7, 315–316.Google Scholar
Kessler, R. 1983 Oszillatorische Konvektion. dissertation, Universität Karlsruhe, West Germany (DFVLR-Rep. FB-84-14).
Kessler, R. 1984 Vectorization of the Galerkin method. In Vectorization of Computer Programs with Applications to Computational Fluid Dynamics (ed. W. Gentzsch), pp. 217–234. Vieweg.
Kirchartz, K.-R. 1980 Zeitabhängige Zellularkonvektion in horizontalen und geneigten Behältern. dissertation, Universität Karlsruhe, West Germany.
Koschmieder, E. L. 1981 Experimental aspects of hydrodynamic instabilities. In Order and Fluctuations in Equilibrium and Non-Equilibrium Statistical Mechanics (ed. E. Nicolis, G. Dewel & J. W. Turner), pp. 159–188. Wiley.
Libchaber, A. & Maurer, J. 1980 Une expérience de Rayleigh-Bénard de géométrie reduite: multiplication, accrochage et démultiplication de fréquences. J. Phys. Paris 41, 51–56.Google Scholar
Libchaber, A., Laroche, C. & Fauve, S. 1982 Period doubling in mercury, a quantitative measurement. J. Phys. Lett. Paris 43, L211–L216.Google Scholar
McLaughlin, J. B. & Orszag, S. A. 1982 Transition from periodic to chaotic thermal convection. J. Fluid Mech. 122, 123–142.Google Scholar
Mallinson, G. D. & De Vahl Davis, G. 1977 Three-dimensional natural convection in a box: a numerical study. J. Fluid Mech. 83, 1–31.Google Scholar
Maurer, J. & Libchaber, A. 1979 Rayleigh-Bénard experiment in liquid helium; frequency locking and the onset of turbulence. J. Phys. Lett. Paris 40, L419–L423.Google Scholar
Maurer, J. & Libchaber, A. 1980 Effect of the Prandtl number on the onset of turbulence in liquid 4He. J. Phys. Lett. Paris 41, L515–L518.Google Scholar
Oertel, H. 1980 Three-dimensional convection within rectangular boxes. In Natural Convection in Enclosures (ed. I. Catton & K. E. Torrance), pp. 11–16. ASME HTD, vol. 8.
Stork, K. & Müller, U. 1972 Convection in boxes: experiments. J. Fluid Mech. 54, 559–611.Google Scholar
Upson, C. D., Gresho, P. M., Sani, R. L., Chan, S. T. & Lee, R. L. 1983 A thermal convection simulation in three dimensions by a modified finite element method. In Numerical Properties and Methodologies in Heat Transfer (ed. T. H. Shih), pp. 245–259. Hemisphere.
Zierep, J. & Oertel Jr. H. 1981 Convective Transport and Instability Phenomena. Braun.