Exact coherent structures (ECS), such as unstable periodic orbits, travelling waves and heteroclinic connections, are postulated to shape and guide the trajectories of a turbulent flow. However, classical methods for finding and converging these objects struggle at high Reynolds numbers (Re) and are limited to statistically steady states. We apply a combination of fluid simulation by differentiable flow solvers and classical machine learning algorithms to improve upon and expand the application areas of these methods. The results reveal new understanding of naturally forming vortex crystals, the connections of Navier-Stokes ECS to solutions of the inviscid Euler equation and the number of degrees of freedom required to represent Kolmogorov flows.
Firstly, we study vortex crystal configurations in rotating superfluids, which are observed experimentally to undergo a sequence of dissipative transitions through a series of metastable states en route to the free-energy minimising configuration. A systematic exploration of the free-energy landscape within a point-vortex approximation in the unbounded domain is achieved by the convergence of thousands of unique vortex crystals with numbers of vortices ranging from 10 to 30 using gradient based optimisation. As part of the search, new continuous families of vortex crystals arranged in double-ring configurations are discovered, which are often global minimisers of the free energy and become discrete sets of equal-energy solutions when the system is confined to a rotating disc.
In a second problem, we investigate the ability for unstable periodic orbits (UPOs) to reconstruct turbulent statistics away from the Re value at which they were converged, beginning with two large libraries of unstable periodic orbits (UPOs) at Re = 40 and Re = 100 in two-dimensional Kolmogorov flow. Arclength continuation of these UPOs up to Re = O(1000) is performed, and the contribution of each UPO to the reconstruction of turbulent statistics is tracked as a function of Re. The analysis indicates that many branches rapidly leave the attractor, and we identify a subset of these ECS which may plausibly connect with solutions of the Euler equation in the inviscid limit. Motivated by this connection, the UPOs are labelled with representative exact solutions of the point vortex system, which are found via gradient-based optimisation. The point vortex UPOs are converged to model the dominant turbulent vortical interactions, and identify a wide range of simpler dynamical processes within the UPO collection, such as slowly-propagating crystals, tripolar structures and bound states.
Finally, motivated by the recent interest in applying data-driven methods to two-dimensional turbulence, robust low-order representations of turbulent Kolmogorov flow are constructed using a deep convolutional autoencoder combined with a complete symmetry-reduction pre-processing step. We use low-dimensional visualisations to demonstrate the partitioning of the autoencoder latent spaces into distinct dynamical regions. The scaling of the required number of degrees of freedom as a function of Re is compared to the predicted upper bound on the attractor dimension of two-dimensional turbulence.