Numerical Investigation of Chaotic Advection in Three-Dimensional Experimentally Realizable Rotating Flows
Lackey, Tahirih Charryse
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In many engineering applications involving mixing of highly viscous fluids or mixing at micro-scales, efficient mixing must be accomplished in the absence of turbulence. Similarly in geophysical flows large-scale, deterministic flow structures can account for a considerable portion of global transport and mixing. For these types of problems, concepts from non-linear dynamical systems and the theory of chaotic advection provide the tools for understanding, quantifying, and optimizing transport and mixing processes. In this thesis chaotic advection is studied numerically in three, steady, experimentally realizable, three-dimensional flows: 1) steady vortex breakdown flow in a cylindrical container with bottom rotating lid, 2) flow in a cylindrical container with exactly counter rotating lids, and 3) flow in a new model stirred-tank with counter-rotating disks. For all cases the three-dimensional Navier-Stokes equations are solved numerically and the Lagrangian properties of the computed velocity fields are analyzed using a variety of computational and theoretical tools. For the flow in the interior of vortex breakdown bubbles it is shown that even though from the Eulerian viewpoint the simulated flow fields are steady and nearly axisymmetric the Lagrangian dynamics could be chaotic. Silnikovs mechanism is shown to play a critical role in breaking up the invariance of the bubble and giving rise to chaotic dynamics. The computations for the steady flow in a cylindrical container with two exactly counter-rotating lids confirm for the first time the findings of recent linear stability studies. Above a threshold Reynolds number the equatorial shear layer becomes unstable to azimuthal modes and an intricate web of radial (cats eyes) and axial, azimuthally-inclined vortices emerge in the flow paving the way for extremely complex chaotic dynamics. Using these fundamental insights, a new stirring tank device with exactly counter-rotating disks is proposed. Results show for the first time that counter rotation of the middle disk in a three-disk stirred tank can create a flow with large chaotic regions. The results of this thesis serve to demonstrate that fundamental studies of chaotic mixing are both important from a theoretical standpoint and can potentially lead to valuable technological breakthroughs.