A persistent homology method with modified filtration to characterize the phase trajectory of a turbulent wake flow

Abstract

The topological features of recurrent phase trajectories of a turbulent wake are studied using a modified persistent homology method. In the general persistent homology computation, the input data are considered as isolated points in a high-dimensional space. Networks with various spatial resolutions are constructed based on these points. When the resolution is low, many edges among neighboring points are created as they satisfy the distance threshold. However, most of these edges do not reflect new topology other than the phase trajectory itself. Therefore, our modified method discards the duplicated edges in the network. Only the phase trajectory and the essential topological connections, which have a local minimum distance in the network, are used to represent the topological structure of a phase trajectory. The homology of the recurrent loop reflects the topology complexity of a trajectory in the phase space, and the first Betti number can be used to classify the trajectories according to the number of self-crossings, which characterizes the trajectory complexity. A significant number of trajectories have only one or a few self-crossings. There are also complex trajectories that contain more than 100 self-crossings. The topological distribution classified using the first Betti number follows a power law.