We study finite-time mixing in time-periodic open flow systems. We describe the transport of densities in terms of a transfer operator, which is represented by the transition matrix of a finite-state Markov chain. The transport processes in the open system are organized by the chaotic saddle and its stable and unstable manifolds. We extract these structures directly from leading eigenvectors of the transition matrix. We use different measures to quantify the degree of mixing and show that they give consistent results in parameter studies of two model systems.
This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
We analyze large-scale patterns in three-dimensional turbulent convection in a horizontally extended square convection cell by means of Lagrangian particle trajectories calculated in direct numerical simulations. Different Lagrangian computational methods, i.e. finite-time Lyapunov exponents, spectral and density-based clustering and transfer operator approaches, are used to detect these large-scale structures, which are denoted as turbulent superstructures of convection.
We study mixing by chaotic advection in open flow systems, where the corresponding small-scale structures are created by means of the stretching and folding property of chaotic flows. The systems we consider contain an inlet and an outlet flow region as well as a mixing region and are characterized by constant in-and outflow of fluid particles. The evolution of a mass distribution in the open system is described via a transfer operator. The spatially discretized approximation of the transfer operator defines the transition matrix of an absorbing Markov chain restricted to finite transient states. We study the underlying mixing processes via this substochastic transition matrix. We conduct parameter studies for example systems with two differently colored fluids. We quantify the mixing of the resulting patterns by several mixing measures. In case of chaotic advection the transport processes in the open system are organized by the chaotic saddle and its stable and unstable manifolds. We extract these structures directly from leading eigenvectors of the transition matrix.
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