Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-18T18:38:59.839Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolution of two-dimensional disturbances in the Rayleigh–Bénard problem and their preferred wavenumbers

Published online by Cambridge University Press:  20 April 2006

A. V. Getling
A. V. Getling
Affiliation:
Institute of Nuclear Physics, M. V. Lomonosov Moscow State University, Moscow 117234, U.S.S.R.
Get access Rights & Permissions [Opens in a new window]

Abstract

A nonlinear non-stationary problem of the development of two-dimensional convective motions in a plane horizontal fluid layer bounded by free surfaces and heated from below is studied. Horizontal dependences of the velocity and temperature are not assumed to be periodic or almost periodic; they have continuous wavenumber spectra and are represented by Fourier integrals. Vertical dependence of each variable is represented by several Fourier harmonics. Spectrum evolution is studied by means of the numerical integration of an initial-value problem. Initial disturbances of two qualitatively different classes are considered; viz those localized in horizontal extent within a narrow part of the layer as well as having the form of a roll set throughout a rather wide region. In both cases convective flows tend to evolve towards the arrays of well-established rolls with the same horizontal wavenumber ap, which apparently seems to be the physically optimal (preferred) one for two-dimensional convection at given Rayleigh number R and Prandtl number P. We see no indications that the deviation of the initial roll-disturbance wavenumber from ap must exceed some threshold value for giving rise to the flow readjustment in wavenumber. At sufficiently small P a decrease in ap with increasing R is observed which agrees with experiments. A comparison is made of various theoretical models and various experimental circumstances, whence it is seen that the less stable the flow (i.e. the easier it can readjust), the better the preferred wavenumber manifests itself. In particular, roll flows periodic in a horizontal direction all over the infinite layer are highly stable, and when only such flows are considered, as has most often been the case, the preferred wavenumber is not revealed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 130 , May 1983 , pp. 165 - 186
Copyright
© 1983 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

Berdnikov, V. S. & Kirdyashkin, A. G. 1979 On the spatial form of cellular convection Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 15, 812–819.Google Scholar
Busse, F. H. 1967 On the stability of two-dimensional convection in a layer heated from below J. Maths & Phys. 46, 140–150.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
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
Cross, M. C., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D. 1980 Effect of distant sidewalls on wave-number selection in Rayleigh–Bénard convection Phys. Rev. Lett. 45, 898–901.Google Scholar
Davis, S. H. 1968 Convection in a box: on the dependence of preferred wavenumber upon the Rayleigh number at finite amplitude. J. Fluid Mech. 32, 619–624.
Foster, T. D. 1969 The effect of initial conditions and lateral boundaries on convection J. Fluid Mech. 37, 81–94.Google Scholar
GERTSENSHTEÍN, S. YA., RODICHEV, E. B., SEMIN, V. N. & SHMIDT, V. M. 1981 On the nonlinear convective motions in ‘double-diffusive’ media. Dokl. Akad. Nauk SSSR 257, 570–574.Google Scholar
Hamming, R. W. 1962 Numerical Methods for Scientists and Engineers. McGraw-Hill.
Koschmieder, E. L. 1969 On the wavelength of convective motions J. Fluid Mech. 35, 527–530.Google Scholar
Krishnamurti, R. 1970 On the transition to turbulent convection. Part 1. The transition from two- to three-dimensional flow J. Fluid Mech. 42, 295–307.Google Scholar
Kutateladze, S. S., Kirdyashkin, A. G. & Berdnikov, V. S. 1974 The velocity field in a convection cell of a horizontal fluid layer at thermal gravitational convection Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 10, 137–145.Google Scholar
Lipps, F. B. & Somerville, R. C. J. 1971 Dynamics of variable wavelength in finite-amplitude Bénard convection Phys. Fluids 14, 759–765.Google Scholar
Nield, D. A. 1968 The Rayleigh–Jeffreys problem with boundary slab of finite conductivity J. Fluid Mech. 32, 393–398.Google Scholar
Ogura, Y. 1971 A numerical study of wavenumber selection in finite-amplitude Rayleigh convection J. Atmos. Sci. 28, 709–717.Google Scholar
Vasin, V. G. & Vlasyuk, M. P. 1974 On the wavelengths of two-dimensional convective motions in a horizontal fluid layer heated from below. Inst. Appl. Math. Acad. Sci. USSR, Preprint no. 84 (in Russian).
Willis, G. E., Deardorff, J. W. & Somerville, R. C. J. 1972 Roll-diameter dependence in Rayleigh convection and its effect upon the heat flux J. Fluid Mech. 54, 351–367.Google Scholar