Transition to turbulence and mixing in a quasi-two-dimensional Lorentz force-driven Kolmogorov flow
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The research in this thesis was motivated by a desire to understand the mixing properties of quasi-two-dimensional flows whose time-dependence arises naturally as a result of fluid-dynamic instabilities. Additionally, we wished to study how flows such as these transition from the laminar into the turbulent regime. This thesis presents a numerical and theoretical investigation of a particular fluid dynamical system introduced by Kolmogorov. It consists of a thin layer of electrolytic fluid that is driven by the interaction of a steady current with a magnetic field produced by an array of bar magnets. First, we derive a theoretical model for the system by depth-averaging the Navier-Stokes equation, reducing it to a two-dimensional scalar evolution equation for the vertical component of vorticity. A code was then developed in order to both numerically simulate the fluid flow as well as to compute invariant solutions. As the strength of the driving force is increased, we find a number of steady, time-periodic, quasiperiodic, and chaotic flows as the fluid transitions into the turbulent regime. Through long-time advection of a large number of passive tracers, the mixing properties of the various flows that we found were studied. Specifically, the mixing was quantified by computing the relative size of the mixed region as well as the mixing rate. We found the mixing efficiency of the flow to be a non-monotonic function of the driving current and that significant changes in the flow did not always lead to comparable changes in its transport properties. However, some very subtle changes in the flow dramatically altered the degree of mixing. Using the theory of chaos as it applies to Hamiltonian systems, we were able to explain many of our results.