Hostname: page-component-6bb9c88b65-vpjdr Total loading time: 0 Render date: 2025-07-23T12:10:47.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear oscillatory convection

Published online by Cambridge University Press:  21 April 2006

R. M. Clever  and
F. H. Busse
R. M. Clever
Affiliation:
Institut für Physik, Universität Bayreuth, 8580 Bayreuth, West-Germany
F. H. Busse
Affiliation:
Institut für Physik, Universität Bayreuth, 8580 Bayreuth, West-Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

A numerical analysis has been performed of three-dimensional time-dependent solutions which bifurcate supercritically from two-dimensional convection-roll solutions at the onset of the oscillatory instability. The bifurcating solutions describe a periodic shifting forward and backward of the convection rolls and lead to a strong deformation of the rolls as the Rayleigh number increases. Since the bifurcating solution is stable in the form of a travelling wave, the computational expense can be reduced by assuming a moving coordinate. Travelling-wave solutions have been computed in the case of rigid boundaries as a function of the Prandtl number and of the two basic wavenumbers αx, αy of the problem. The onset of oscillations reduces the heat transport in comparison with that of two-dimensional rolls because the occupation of a new degree of freedom of motion by the oscillation reduces the energy of the heat-transporting component of convection. A limited stability analysis of finite-amplitude travelling waves has been performed and the onset of an asymmetric mode of oscillations is determined as a function of the parameters of the problem. This mode appears to be identical with a mode that was observed in the numerical simulations of Lipps (1976) and McLaughlin & Orszag (1982).

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 176 , March 1987 , pp. 403 - 417
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.)

Article purchase

Temporarily unavailable

References

Brown, W. 1973 Heat-flux transitions at low Rayleigh number. J. Fluid Mech. 60, 539–559.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. & Frick, H. 1985 Square-pattern convection in fluids with strongly temperaturedependent viscosity. J. Fluid Mech. 150, 451–465.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625–645.Google Scholar
Clever, R. M. & Busse, F. H. 1978 Large wavelength convection rolls in low Prandtl number fluids. Z. angew. Math. Phys. 29, 711–714.Google Scholar
Clever, R. M. & Busse, F. H. 1981 Low-Prandtl-number convection in a layer heated from below. J. Fluid Mech. 102, 61–74.Google Scholar
Fauve, S. & Libchaber, A. 1981 Rayleigh–Bénard experiment in a low Prandtl number fluid. In Chaos and Order in Nature (ed. H. Haken), pp. 25–35. Springer.
Grötzbach, G. 1982 Direct numerical simulation of laminar and turbulent Bénard convection. J. Fluid Mech. 119, 27–53.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
Krishnamurti, R. 1973 Some further studies on the transition to turbulent convection. J. Fluid Mech. 60, 285–303.Google Scholar
Lipps, F. B. 1976 Numerical simulation of three-dimensional Bénard convection in air. J. Fluid Mech. 75, 113–148.Google Scholar
McLaughlin, J. B. & Orszag, S. A. 1982 Transition from periodic to chaotic thermal convection. J. Fluid Mech. 122, 123–142.Google Scholar
Maeno, Y., Haucke, H. & Wheatley, J. C. 1985 Transition to oscillatory convection in a 3He–superfluid–4He mixture. Phys. Rev. Lett. 54, 340–342.Google Scholar
Meneguzzi, M., Sulem, C., Sulem, P. L. & Thual, O. 1985 Three-dimensional numerical simulation of convection in low-Prandtl-number fluids. J. Fluid Mech. (submitted).Google Scholar
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Willis, G. E. & Deardorff, J. W. 1970 The oscillatory motions of Rayleigh convection. J. Fluid Mech. 44, 661–672.Google Scholar