Space-time methods for the computation and analysis of coherent families

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Turbulent superstructures arise as macroscopic, long-living dynamical objects that dominate the transport and mixing behavior of the flow. From a dynamical systems perspective they can be viewed as so-called coherent sets: transport-dominating structures in chaotic time-variant systems. Recently, tools based on so-called transfer operators have been developed for the quantitative analysis of coherent sets. Transfer operators encode the global dynamical behavior of the system on a functional-analytic level, and coherent sets can be extracted from their dominant eigenmodes.In our project, on the one hand, we will extend the theoretical concept of coherent sets to deal with the statistical nature of turbulence. On the other hand, we will develop efficient Eulerian and Lagrangian computational methods for the identification and persistence analysis of coherent sets in the specific context of turbulent flows. The focus is on transport-oriented, transfer operator based considerations that we will use in combination with partial differential equation theory and manifold learning (diffusion maps). Our different approaches will cover applicability for various kinds of simulation and experimental data (e.g. direct numerical simulation (DNS), large eddy simulation (LES), particle image velocimetry (PIV), or sparse trajectories from particle tracking). Through interdisciplinary collaborations, we will validate our methods and compare them with other identification and analysis techniques.