A flow complexity estimation method based on modified persistent homology method

This research focuses on the recurrent flow patterns of a wake flow downstream of two cylinders in tandem. A modified persistent homology method is employed in this research. In the traditional persistent homology computation, input data is treated as isolated points in a high-dimensional space. In contrast, this study introduces a filtration process that considers only the topological connections that are local minima, eliminating duplicated edges present in the usual Vietoris-Rips filtration.

Analyzing the homology of the recurrent loop provides insights into the topological complexity of a trajectory in the phase space. Specifically, the first Betti number is utilized to categorize trajectories based on the number of self-junctions, which reflects the trajectory's complexity. Numerous trajectories exhibit only one or a few self-junctions, while others showcase more intricate patterns with dozens of self-junctions. By classifying the phase trajectories in this manner, an estimate of the flow's degree-of-complexity can be obtained using the excess entropy method.