## B07 - Selfsimilar structures in turbulent flows and the construction of LES closures

**Head(s):** Prof. Dr. Gitta Kutyniok (TU Berlin), Prof. Dr.-Ing. Rupert Klein (FU Berlin), Prof. Dr. Nikki Vercauteren (FU Berlin)

**Project member(s):** Abhishek Paraswarar Harikrishnan, Gottfried Hastermann, Johannes von Lindheim

**Participating institution(s):** FU Berlin, TU Berlin

### Project Summary

According to theory, fluid turbulence involves self-similarity, and high-resolution turbulence simulations reveal a hierarchy of rod- and sheet-like structures across a broad range of scales. Moreover, in complex real-life situations turbulence is often strongly intermittent in that it arises randomly within regions of otherwise quiescent flow. An important and challenging example from meteorology is given by stably stratified atmospheric boundary layers (ABL) in which turbulence coexists with internal waves and large-scale quasi two-dimensional flow structures.

This project aims, first, at quantitative characterisations of such self-similar and intermittent turbulence structures based on advanced data analysis and machine learning techniques. Such characterisations, which would generalise established concepts such as “Exact Coherent States” or “(Lagrangian) Coherent Sets”, would allow us to address an immediate but, due to the chaotic nature of turbulence, still open scientific question: How can two different representations of a given turbulent flow, coming, e.g., from a detailed measurement and from a numerical simulation, be compared quantitatively at a more detailed level than that of mean values, Fourier spectra, or one- and two-point statistics? This project’s results should enable a comparison of whether the two flows involve topologically and geometrically similar flow structures with comparable life times, scaling properties, and statistics.

As a long-term goal, structural characterisations of (intermittent) turbulence of this kind shall, secondly, be utilised in the design of new Large Eddy Simulation (LES) closures, i.e., of models which, in numerical simulations of limited spatio-temporal resolution, describe the net effects of small-scale unresolved turbulence on the large-scale resolved part of the flow. The idea is to extrapolate ensembles of flow structures simulated on a coarse LES-grid in scale based on the learned flow characterisations. This will generate a self-consistent model of unresolved fluctuations and can be used to determine effective unresolved scale fluxes of momentum, energy, and the like in the sense of a closure. Thirdly, the simultaneous characterisation of any embedded quiescent large-scale flow patterns would help define a closure that incorporates scale interactions in intermittent contexts like the ABL.

To achieve these goals, Direct Numerical Simulation (DNS) data of turbulent flows as well as observational data from atmospheric turbulence will be analysed using advanced multiscale data analysis techniques. The latter include wavelets, shearlets, turbulent event detection methods, hierarchical low-rank tensor approximations, and approaches combining dictionary learning with the FEM-VARX-techniques developed by Mercator Fellow Illia Horenko. The aim is to identify characteristic patterns which, up to some basic transformations, appear repeatedly across the scale hierarchy in physical, Fourier, or other transform spaces. In a complementary approach, we exploit techniques from deep learning, relying on the multiscale-like structure encoded in the layers of neural networks. Recent own work has for the first time established a mathematical bridge between sparse approximation and neural networks, which will even allow us to compare both approaches on a theoretical level.

At the level of methods developments, the project aims to generalise multiscale shearlet decompositions. Shearlets are closely related to wavelets and curvelets. They differ from wavelets in that they allow multiscale resolution in arbitrary, grid-independent directions – a feature that is important for capturing, e.g., meandering vortices in a turbulent flow. They differ from curvelets in that their fundamental construction is substantially simpler, although equally powerful, and this renders the shearlet technology amenable to rigorous analysis. In contrast to curvelets, shearlets also allow a unified treatment of continuum domain theory and implementation. Their main methodological extension in this project concerns applications in three space dimensions and to vector-valued fields.