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

Takashi Sakajo, Takeshi Matsumoto, Shizuo Kaji, Tomoo Yokoyama, Tomoki Uda

TL;DR

This paper develops Topological Flow Data Analysis (TFDA), a topology-based framework for time-series analysis of transient 2D flows, by converting instantaneous streamline patterns into a planar rooted tree (COT) and a corresponding string representation. The evolution of flow fields is modeled as a discrete dynamical system on the transition graph of COT patterns, and the lid-driven cavity across $Re$ from $14000$ to $16000$ serves as a benchmark to reveal a transition from periodic to chaotic dynamics. The authors augment TFDA with a Markov-process view on the transition graph and apply Convergent Cross Mapping (CCM) to extract causal relations between corner eddies, demonstrating asymmetric corner influence in chaotic regimes not captured by linear response. The work highlights TFDA’s interpretability and robustness to noise, offering a structured, topology-based alternative to POD/DMD for time-series fluid data, with potential extensions to experimental measurements and, eventually, to three-dimensional flows.

Abstract

We introduce a time-series analysis method for transient two-dimensional 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 at Reynolds numbers ranging from $Re=14000$ to $Re=16000$, a benchmark problem in fluid dynamics data analysis. Our approach reveals the transition from periodic to chaotic flow at a critical Reynolds number when the reduced dynamical system is modelled as a Markov process on the transition graph. Additionally, we perform an observational causal inference to analyse changes in local flow patterns at the cavity corners and discuss differences with a standard interventional sensitivity analysis. This work demonstrates the potential of TFDA-based time-series analysis for uncovering complex dynamical behaviours in fluid flow data.

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

TL;DR

This paper develops Topological Flow Data Analysis (TFDA), a topology-based framework for time-series analysis of transient 2D flows, by converting instantaneous streamline patterns into a planar rooted tree (COT) and a corresponding string representation. The evolution of flow fields is modeled as a discrete dynamical system on the transition graph of COT patterns, and the lid-driven cavity across from to serves as a benchmark to reveal a transition from periodic to chaotic dynamics. The authors augment TFDA with a Markov-process view on the transition graph and apply Convergent Cross Mapping (CCM) to extract causal relations between corner eddies, demonstrating asymmetric corner influence in chaotic regimes not captured by linear response. The work highlights TFDA’s interpretability and robustness to noise, offering a structured, topology-based alternative to POD/DMD for time-series fluid data, with potential extensions to experimental measurements and, eventually, to three-dimensional flows.

Abstract

We introduce a time-series analysis method for transient two-dimensional 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 at Reynolds numbers ranging from to , a benchmark problem in fluid dynamics data analysis. Our approach reveals the transition from periodic to chaotic flow at a critical Reynolds number when the reduced dynamical system is modelled as a Markov process on the transition graph. Additionally, we perform an observational causal inference to analyse changes in local flow patterns at the cavity corners and discuss differences with a standard interventional sensitivity analysis. This work demonstrates the potential of TFDA-based time-series analysis for uncovering complex dynamical behaviours in fluid flow data.

Paper Structure

This paper contains 14 sections, 1 theorem, 19 equations, 18 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our target and method
  3. Physics of the lid-driven cavity flow
  4. Topological Flow Data Analysis
  5. TFDA for the lid-driven cavity flow
  6. Numerical Results
  7. Detailed analysis at $Re=14000$
  8. Transition of streamline topology
  9. Transition diagram of the flow evolution
  10. Comparison across Reynolds numbers
  11. Graph analysis of transition diagram
  12. Information transfer between corner regions
  13. Summary
  14. Linear response function analysis

Key Result

Theorem 1

The topological structure of particle orbits (streamlines) generated by structurally stable Hamiltonian vector field in $\mathcal{D}$ is uniquely represented by a combination of the local orbit structures $b_{\pm\pm}$, $b_{\pm\mp}$, $c_{\pm}$, $\sigma_{\pm}$ and $\beta_\pm$.

Figures (18)

  • Figure 1: (a) The lid-driven cavity flow. (b) A snapshot of a streamline pattern in the cavity.
  • Figure 2: The kinetic energy as a function of time for the three representative Reynolds numbers for (a) $Re=14000$, (b) $Re=15500$, and (c) $Re=16000$.
  • Figure 3: Power spectral densities of the kinetic energy for the three representative Reynolds numbers. (a) $Re=14000$, (b) $Re=15500$, and $Re=16000$. The width of the bins for the frequency is the same for the three cases. The densities are calculated from the energy $E(t)$ in $5200 \le t \le 15200$ in the step of $0.01$.
  • Figure 4: Local orbit structures appearing in the streamline patterns of structurally stable Hamiltonian vector fields in $\mathcal{D}$. (a) Figure-eight patterns with a saddle and two self-connected saddle separatrices whose COT symbol is $b_{\pm\pm}$. (b) Local orbit structures with a saddle, in which one saddle connection encloses another. The COT symbol is $b_{\pm\mp}$. (c) Saddle connections between two different saddles on the same boundary. The COT symbol is $c_\pm$. (d) Isolated boundaries, $\beta_\pm$. (e) Elliptic centres, $\sigma_\pm$. In each panel, $\Box_{b_\pm}$ and $\Box_{c_\pm}$ indicate that the local orbit structures in (\ref{['COTsymbols']}) are embedded in the flow structures as their internal structure.
  • Figure 5: Root structures in the disk $\mathcal{D}$, where a local orbit structures $b_{\pm\pm}$ or $b_{\pm\mp}$ can be embedded and any number of $c_\pm$ structures are attached to the boundary of the disk. (a) The root structure $\beta_{\emptyset+}$. The flow along the boundary is going in the anticlockwise direction. (b) The root structure $\beta_{\emptyset-}$. The flow along the boundary is going in the clockwise direction.
  • ...and 13 more figures

Theorems & Definitions (1)

  • Theorem 1