Structuration de la convection mixte en milieu poreux confiné latéralement et chauffé par le bas : effets d'inertie

Abstract

Nous étudions la naissance de la convection dans un milieu poreux chauffé par le bas en présence d'un écoulement horizontal, et plus particulièrement l'influence de l'inertie poreuse et du rapport de forme transversal a du milieu. Nous montrons que l'état de conduction est déstabilisé au profit de rouleaux longitudinaux fixes (R.L) si a est entier et au profit de rouleaux propagatifs purement transversaux (R.T) si a est inférieur à une valeur limite a c < 1. Pour a > a c et non entier, la convection naît sous la forme de structures tridimensionnelles (3D) oscillatoires pour a > 1 ou sous la forme de R.T pour a c < a < 1 pourvu que le nombre de Reynolds ne dépasse pas une valeur critique Re * K . Ces structures sont remplacées par des R.L lorsque Re K > Re * K .

Weakly nonlinear interaction of mixed convection patterns in porous media heated from below

“…It has been demonstrated in [5,9] that for a Rayleigh number larger than a critical value Ra c which depends on Reynolds number Re K the basic state (5) is unstable. The linear theory shows that in this case convective configurations with exponentially growing amplitudes will develop in the form of: Fig.…”

“…[22] for technical details. At the leading order the O(ε) equations are identical to the equations solved for the linear stability analysis [5,9]. The solution at this order is given in Appendix A.…”

“…Consider a small spatially localized perturbation of the conductive state (5). Under the linear part of envelope equations (21), (22), the amplitude evolution of T modes and L rolls is dominated by the exponential factor [27] (G A ,…”

“…The global frequency, the wavelength and the phase velocity of the travelling rolls were also computed in [2]. From the linear point of view, temporal stability analysis of the problem of mixed convection in porous media of the kind performed by some investigators (Prats [3], Rees [4], Delache et al [5]) consider perturbations which amplify in time, starting from an initial spatially periodic perturbation, i.e. they assume that the perturbation wave number k is real while its frequency ω is complex.…”

“…The temporal linear stability analysis performed in [5,9] concluded that there exists a critical Reynolds number Re * K based on the permeability of the medium in such a way that for Re K < Re * K the T modes propagating with phase velocity equal to Pe become unstable first at Ra 3D c . While for Re K > Re * K the most unstable disturbances are L rolls (Ra c < Ra 3D c ).…”