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Envelope equations for the Rayleigh–Bénard–Poiseuille system. Part 1. Spatially homogeneous case

Published online by Cambridge University Press:  01 March 2004

PHILIPPE CARRIÈRE ,
PETER A. MONKEWITZ  and
DENIS MARTINAND
PHILIPPE CARRIÈRE
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon-Université Claude Bernard Lyon I-INSA Lyon BP163, 69131 Ecully cedex, France
PETER A. MONKEWITZ
Affiliation:
Laboratoire de Mécanique des Fluides, Ecole Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland
DENIS MARTINAND
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon-Université Claude Bernard Lyon I-INSA Lyon BP163, 69131 Ecully cedex, France Laboratoire de Mécanique des Fluides, Ecole Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland Present address: Turbulence and Mixing Group, Department of Aeronautics, Imperial College London, London SW7 2BY, UK.
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Abstract

Envelope equations are derived for the convection rolls in the Rayleigh–Bénard–Poiseuille system, taking into account both their slow streamwise and transverse variations. At finite $O(1)$ Reynolds numbers, the stability of finite-amplitude longitudinal roll patterns is accessible to analysis in a moving frame of reference and stability is predicted provided a generalized Eckhaus criterion is satisfied. At lower Reynolds numbers, the analysis allows the analytical determination of the Green function for arbitrary orientations of the instability pattern. It clarifies previous results concerning the purely convective nature of all modes of instability except transverse rolls (for which a convective–absolute transition exists), as soon as the Reynolds number is non-zero.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 502 , 10 March 2004 , pp. 153 - 174
Copyright
© 2004 Cambridge University Press

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