Topological Data Analysis of the flow field patterns in vortex wakes generated by pitching and heaving plates*

The application of topological data analysis (TDA) to detect patterns in fluid dynamics as they evolve in the topological space of an unsteady or vortex-dominated flow field, is a relatively new endeavor. The technique used to decipher complex data by detecting and tracking topological features is called persistent homology (PH). We apply methods of TDA to the vortex wakes generated by pitching and heaving flat plates as models of bio-inspired propulsion. Two such datasets are studied: a 2D discrete vortex method simulation and velocity fields from stereoscopic particle image velocimetry. The results are analyzed using cubical PH to establish the relationship between the topological connectivity of the field and dynamic fluid phenomena. For example, analysis of PH features of the vorticity field identifies vortex core centers and vortex boundaries as representatives of the zeroth and first homology groups. The analysis is also applied to different derived properties including the velocity and the finite-time Lyapunov exponent, among others. Other topological characteristics of these flow fields, such as Betti numbers, persistence diagrams/barcodes, and CROCKER plots, are interpreted and connected to the physical structures in the flows. Results show that using TDA and PH to analyze these flow field interactions will enable a better understanding of turbulent flow and may provide a reduced set of features that could be used in network methods and machine learning.