PhD, December 2004
Advisors: J. Meiss, K. Julien
thesis.pdf photos
This thesis considers the mixing of passive scalar quantities in three-dimensional flows exhibiting roll switching. The main goal of this thesis is to understand and explore the topology of three dimensional transport/mixing using techniques from dynamical systems theory. Measurements of the mixing quality are then compared with the existence and bifurcations of the coherent invariant structures in the flow.
The models constructed in this thesis are three-dimensional analogies of the Aref blinking vortex flow. Aref found that in two-dimensional incompressible time-dependent flows, alternatively active vortices can be used to efficiently mix a laminar fluid. Barriers to mixing emerge in the form invariant elliptical islands which are theoretically predicted and experimentally observed.
In three-dimensions, the difficulty of the problem increases considerably. However, there exists three-dimensional systems analogous to the blinking-vortex model in the form of vortical roll arrays which switch quasi-periodically. Two of these systems are modeled in this thesis. The first is a set of orthogonal roll arrays which lose stability to rolls rotated by 90 degrees. The phenomenon is observed experimentally in Rayleigh-Benard convection in a binary fluid layer. A simple model of this behavior is constructed. It is found that the scalar paths are restricted to two-dimensional topological spheres. On these surfaces, the dynamics can display all the complexity of area preserving mappings, however the mixing is still two-dimensional.
Rotating Rayleigh-Benard convection in a pure fluid also displays a roll switching phenomenon known as the K\"uppers-Lortz instability. In this case, roll arrays switch in a manner which allows for three-dimensional transport. Two models in the form of a continuous time stochastic representation and a discrete deterministic system are used to simulate the tracers dynamics. Barriers to mixing form as two-dimensional invariant tori and are observed in both the stochastic and discrete case. A high correlation between these models is found thus allowing us to restrict our attention to the discrete case. The topology of the phase space is then compared to the quality of mixing in this system.