Topological flow data analysis for transient flow patterns: a graph-based approach

Abstract

We introduce a method of time series analysis for two-dimensional transient flow patterns based on Topological Flow Data Analysis (TFDA), a new approach to topological data analysis. TFDA identifies local topological flow structures from an instantaneous streamline pattern and describes their global connections as a unique planar tree and its string representation. With TFDA, the evolution of two-dimensional flow patterns is reduced to a discrete dynamical system represented as a transition graph between topologically equivalent streamline patterns. We apply this method to study the lid-driven cavity flow for Reynolds numbers from $Re=14000$ to $16000$, a benchmark problem in the analysis of fluid dynamics. Our approach can extract some physical information from the lid-driven cavity flow: transition of the flow from periodic to quasi-periodic and chaotic; estimation of the period of periodic dynamics; relation between variations in energy and enstrophy and topological changes in flow patterns; statistical properties of intricate flow evolution at higher Reynolds number. In addition, we perform an observational causal inference to analyse changes in local flow patterns in the cavity corner. This work demonstrates the potential of TFDA-based time series analysis to uncover complex dynamical behaviours in fluid flow data from a topological perspective.