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Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 1. Effect of aspect ratio and Prandtl number

Published online by Cambridge University Press:  10 September 2009

T. P. LYUBIMOVA ,
D. V. LYUBIMOV ,
V. A. MOROZOV ,
R. V. SCURIDIN ,
H. BEN HADID  and
D. HENRY
T. P. LYUBIMOVA
Affiliation:
Institute of Continuous Media Mechanics UB RAS, 1 Koroleva Street, 614013 Perm, Russia
D. V. LYUBIMOV
Affiliation:
Perm State University, 15 Bukirev Street, 614990 Perm, Russia
V. A. MOROZOV
Affiliation:
Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA
R. V. SCURIDIN
Affiliation:
Institute of Continuous Media Mechanics UB RAS, 1 Koroleva Street, 614013 Perm, Russia
H. BEN HADID*
Affiliation:
Université de Lyon, École Centrale de Lyon/Université Lyon 1/INSA de Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR-CNRS 5509, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
D. HENRY
Affiliation:
Université de Lyon, École Centrale de Lyon/Université Lyon 1/INSA de Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR-CNRS 5509, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
*
†Email address for correspondence: [email protected]
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Abstract

The paper deals with the numerical investigation of the steady convective flow in a horizontal channel of rectangular cross-section subjected to a uniform longitudinal temperature gradient imposed at the walls. It is shown that at zero Prandtl number the solution of the problem corresponds to a plane-parallel flow along the channel axis. In this case, the fluid moves in the direction of the imposed temperature gradient in the upper part of the channel and in the opposite direction in the lower part. At non-zero values of the Prandtl number, such solution does not exist. At any small values of Pr all three components of the flow velocity differ from zero and in the channel cross-section four vortices develop. The direction of these vortices is such that the fluid moves from the centre to the periphery in the vertical direction and returns to the centre in the horizontal direction. The stability of these convective flows (uniform along the channel axis) with regard to small three-dimensional perturbations periodical in the direction of the channel axis is studied. It is shown that at low values of the Prandtl number the basic state loses its stability due to the steady hydrodynamic mode related to the development of vortices at the boundary of the counter flows. The growth of the Prandtl number results in the strong stabilization of this instability mode and, beyond a certain value of the Prandtl number depending on the cross-section aspect ratio, a new steady hydrodynamic instability mode becomes the most dangerous. This mode is characterized by the localization of the perturbations near the sidewalls of the channel. At still higher values of the Prandtl number, the spiral perturbations (rolls with axis parallel to the temperature gradient) become the most dangerous modes, at first the oscillatory spiral perturbations and then the Rayleigh-type steady spiral perturbations. The influence of the channel width on these different instabilities is also emphasized.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 635 , 25 September 2009 , pp. 275 - 295
Copyright
Copyright © Cambridge University Press 2009

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