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The computational analysis and the optimization of transport and mixing processes in fluid flows are of ongoing scientific interest. Transfer operator methods are powerful tools for the study of these processes in dynamical systems. The focus in this context has been mostly on closed dynamical systems and the main applications have been geophysical flows.
In this thesis, we consider transport and mixing in closed flow systems and in open flow systems that mimic technical mixing devices. Via transfer operator methods, we study the coherent behavior in closed example systems including a turbulent Rayleigh-Bénard convection flow and consider the finite-time mixing of two fluids. We extend the transfer operator framework to specific open flows. In particular, we study time-periodic open flow systems with constant inflow and outflow of fluid particles and consider several example systems. In this case, the transfer operator is represented by a transition matrix of a time-homogeneous absorbing Markov chain restricted to finite transient states. The chaotic saddle and its stable and unstable manifolds organize the transport processes in open systems. We extract these structures directly from leading eigenvectors of the transition matrix. For a constant source of two fluids in different colors, the mass distribution in the mixer and its outlet region converges to an invariant mixing pattern. In parameter studies, we quantify the degree of mixing of the resulting patterns by several mixing measures. More recently, network-based methods that construct graphs on trajectories of fluid particles have been developed to study coherent behavior in fluid flow. We use a method based on diffusion maps to extract organizing structures in open example systems directly from trajectories of fluid particles and extend this method to describe the mixing of two types of fluids.
