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Some scientists have performed direct numerical simulations (DNS) of homogeneous Rayleigh-Bénard (RB) convection to seek an asymptotic high-Rayleigh number heat transport scaling law Nu ~ Raγ. By applying periodic boundary conditions, the goal has been to focus on the ‘ultimate’ regime where boundary layers are negligible and convective heat transport is limited only by the turbulence in the bulk. The value γ = 1/ 2 obtained in the DNS is consistent with the analytical conjecture established in the past several decades. However, it should be pointed out that the tri-periodic model possesses exact exponentially growing solutions which transport unlimited heat. Such runaway solutions and their possible secondary instabilities are manifest above a critical Ra in the DNS. Thus, the relevance of computations on tri-periodic domains as models for the RB ultimate state is arguable. In this thesis, to understand the secondary instability mechanism, four systems have been constructed by (1) multiple scalings based on the aspect-ratio of the growing modes; (2) a modal truncation of the dynamical equations for the exact exploding solutions; (3) random noisy horizontal velocity fields; (4) a combination of the modal truncation and multiple scalings (two time-scales). Numerical studies of (1), (2) and (3) have revealed that, respectively, the growing modes are unbounded; boundedness or unboundedness is uncertain; unbounded if the noise strength is small. (4) suggests that when the nonlinear terms are dominant in the governing equations, there exist several constants of the motion for the ODEs obtained by modal truncations. This is possibly one explanation of the bursting cycles of the exponentially growing solutions observed in the DNS. The latest DNS results of homogeneous RB convection (obtained from our co-workers) are also reported, which conclude that the secondary instabilities observed in the earlier homogeneous RB DNS might be caused by numerical errors.

Dans la configuration de Rayleigh-Bénard, on considère une fine couche de fluide placée entre deux parois horizontales, chauffée par le bas et refroidie par le haut. Cette couche peut être le siège d'une instabilité si le gradient thermique est suffisamment important : on passe alors de l'état conductif à l'état convectif et on parle de bifurcation primaire pour qualifier cette première transition. Cette mise en mouvement du fluide se fait de manière ordonnée : on constate ainsi l'émergence de différents motifs de convection comme des rouleaux, des carrés ou encore des hexagones. Ces structures vont à leur tour subir des instabilités qualifiées de secondaires qui vont limiter la gamme de nombres d'onde stables. On étudie ici théoriquement ces instabilités d'une part à proximité du seuil de la convection grâce à une approche faiblement non linéaire, d'autre part loin des conditions critiques grâce à une approche fortement non linéaire. Le fluide est rhéofluidifiant, ce qui correspond au comportement rhéologique le plus fréquemment rencontré, et est décrit par le modèle de Carreau. À proximité du seuil, on considère deux situations : la première correspond au cas où les plaques ont une conductivité finie, la seconde à celui d'un fluide thermodépendant. Dans chaque cas, l'influence du caractère rhéofluidifiant sur la nature du motif émergeant à la bifurcation primaire et sur les instabilités secondaires est mise en évidence. Pour étudier les motifs de convection loin des conditions critiques, on a recours à une procédure de continuation permettant de déterminer de proche en proche les caractéristiques de l'écoulement comme les champs de vitesse ou de température ainsi que le nombre de Nusselt. / In the Rayleigh-Bénard configuration, we consider a thin layer of fluid confined between two horizontal slabs which is heated from below and cooled from above. This layer undergoes an instability if the thermal gradient is strong enough: a transition from the conductive state to the convective state and called _ primary bifurcation _occurs. Moreover, it happens in an ordered way: we can notice the emergence of various convection patterns such as rolls, squares or hexagons. In their turn, these patterns undergo _ secondary instabilities _ that limit the range of stable wavenumbers. These instabilities are studied theoretically _firstly near the threshold thanks to a weakly nonlinear approach, and secondly far from critical conditions thanks to a strongly nonlinear approach. We consider a shear thinning fluid, the most common rheological behavior, which is described by the Carreau model. Near the threshold, two situations are considered: the first corresponds to finite conductivity plates, the second corresponds to a thermodependent fluid. In each case, the influence of the shear thinning effect on the nature of the pattern emerging at the primary bifurcation and on secondary instabilities is highlighted. To study the convection patterns far from the critical conditions, a continuation procedure is used to determine, step-by-step, the characteristics of the flow, such as the velocity or temperature fields and the Nusselt number.