Resolvent-based models for nonlinear solutions of wall-bounded flows and statistical estimation of chaotic systems

Abstract

This thesis explores turbulence from a dynamical systems perspective, focusing on the development of resolvent-based models for both nonlinear solutions and the statistical estimation of chaotic systems. By leveraging the concept of Exact Coherent Structures (ECSs), postulated to serve as invariant skeletons within turbulent flows, the work aims to systematically reduce the complexity of turbulence representation while retaining essential dynamical features. Resolvent analysis, a modal decomposition technique, is employed to construct low-dimensional subspaces that capture the dominant dynamics of wall-bounded turbulent flows, such as rotating Couette flow.

The thesis introduces a novel variational optimisation methodology that operates within these resolvent subspaces to compute invariant solutions of the Navier-Stokes equations in wall-bounded domains. This approach is shown to retain the robustness of the underlying optimisation algorithm while improving computational performance through dimensionality reduction of the resolvent-based subspace. The method is demonstrated for equilibrium and periodic solutions on the rotating Couette flow. Furthermore, the work extends these models to construct large period state-space loops, termed quasi-trajectories, which approximate the statistics of solutions on chaotic attractors without requiring exact solutions to the governing equations. This concept is validated using the Lorenz system.

Key results include the discovery of equilibrium and periodic solutions for rotating Couette flow, the application of resolvent modes for low dimensional modelling, and the statistical validation of quasi-trajectories against direct numerical simulations. These findings represent a step towards “closing the loop” in resolvent analysis, bridging the gap between kinematic modes and practical turbulence modelling. This work opens new avenues for efficient computation and control of turbulent flows and provides a foundation for further exploration of chaotic systems using dynamical systems theory.

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ORCID for Davide Lasagna: orcid.org/0000-0002-6501-6041

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