Nonlinear Pattern Formation in Rayleigh-Benard Convection

In this dissertation, we present a study of several aspects of pattern formation in Rayleigh-Benard convection by solving three different generalized two-dimensional Swift-Hohenberg models. This work involves the numerical integration of nonlinear equations and requires extensive supercomputing resources. The first numerical study deals with the question of how stochastic forces affect pattern evolution in non -equilibrium systems. This was investigated for the case of Rayleigh-Benard convection in a Boussinesq system. This convection takes place when the control parameter of the system is ramped above its threshold value. The noise -induced convective pattern near onset is shown to consist of many irregular convective cells which have no relation to the symmetry of the lateral walls. An unresolved issue is the origin of the large stochastic noise observed in the experiment. The second numerical study addresses the question of pattern formation in Rayleigh-Benard convection in a non-Boussinesq system, in which patterns with different symmetries can coexist. We show that the large-scale mean flow plays an important role in the pattern formation in such a system. Such large-scale mean flow has to be incorporated into the Swift-Hohenberg model equation by adding a second scalar field, which is the vertical component of a vorticity potential field. We studied the transition between the conduction, hexagon, roll and spiral states. We also examined the role of defects in the roll-hexagon transition in a non-Boussinesq system. The third numerical study involves the pattern formation in a rotating convective system. This was first considered by Kuppers and Lortz(1969). Very recently some experiments have shown that pattern evolution in the Kuppers -Lortz instability involves a very different mechanism from that originally envisioned. Namely, the instability does not involve a significant decay of roll amplitudes, but rather, defects and grain boundaries play a crucial role in the re-orientation dynamics. We propose a generalized Swift-Hohenberg model that includes spatial variations, which permits us to study the pattern formation for the Kuppers-Lortz instability. We present some preliminary numerical results obtained from solving the model equation, which are in good agreement with the experiment.