Spatial tiling of exact coherent states in large-scale Kolmogorov flow

The dynamical systems picture of turbulence succeeds in capturing both turbulent statistics and describing mechanistic processes within small fluid domains through the computation and analysis of exact coherent structures (ECSs). Extending this framework to larger domains has proved to be challenging, partially due to limits of computational techniques. In this direction, we consider two-dimensional Kolmogorov flow on a ``stacked’’ $2\pi\times4\pi$ domain. We show that the flow in this problem can often be effectively described as two weakly-coupled $2\pi\times 2\pi$ (`small-box’) Kolmogorov flows. In this regime we are able to construct new ECS by vertically stacking small-box solutions. This includes both exact relative periodic orbits (RPOs) which result from vertical stacking of high-dissipation, small-box RPOs with the laminar state and two-tori which result from vertical stacking of distinct RPOs. These solutions are obtained using gradient-based optimization techniques with a multi-point loss function, which seeks near recurrence in masked regions of the flow domain. Finally, we apply these methods to find `turbulent’ trajectories which locally shadow a small-box RPO in half of the domain.

This work is supported by UKRI Frontier Guarantee Grant EP/Y004094/1