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arxiv: 2502.00664 · v2 · submitted 2025-02-02 · ⚛️ physics.flu-dyn · math.DS

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

Takashi Sakajo , Takeshi Matsumoto , Shizuo Kaji , Tomoo Yokoyama , Tomoki Uda This is my paper

Pith reviewed 2026-05-23 04:40 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math.DS
keywords topological flow data analysisstreamline patternsplanar treestransition graphslid-driven cavity flowdiscrete dynamical systemsfluid dynamics
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The pith

TFDA classifies instantaneous streamline patterns into unique planar trees to model flow evolution as a transition graph.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Topological Flow Data Analysis to study time series of two-dimensional transient flow patterns. It shows that each instantaneous streamline pattern yields local topological structures whose global connections are captured by a single planar tree and its string encoding. These encodings turn the continuous flow evolution into a discrete dynamical system whose states are topologically equivalent patterns and whose edges are observed transitions. Applied to lid-driven cavity flow between Reynolds numbers 14000 and 16000, the resulting graph reveals shifts among periodic, quasi-periodic and chaotic regimes, correlates topological changes with energy and enstrophy, and supports causal inference on corner flow structures.

Core claim

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.

What carries the argument

The planar tree (and its string representation) that encodes the topological equivalence class of an instantaneous streamline pattern and supplies the nodes of the transition graph.

If this is right

  • Flow evolution reduces to a discrete dynamical system on a finite graph whose nodes are topological classes.
  • Regime transitions (periodic to quasi-periodic to chaotic) appear as changes in the structure of the transition graph.
  • Variations in global quantities such as energy and enstrophy align with specific topological class changes.
  • Statistical properties of intricate high-Reynolds-number evolution become extractable from the graph.
  • Observational causal inference on local pattern changes in cavity corners becomes possible from the same discrete representation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The graph representation could support reduced-order modeling that bypasses full resolution of the Navier-Stokes equations between topological transitions.
  • The same encoding might be applied to other two-dimensional flows whose streamline patterns admit planar-tree descriptions.
  • Coupling the transition graph with time-series forecasting techniques could yield early-warning indicators for regime shifts.
  • Comparison of the planar-tree graphs with other topological summaries of the same data set could test whether the tree encoding captures dynamics missed by those summaries.

Load-bearing premise

Instantaneous streamline patterns admit a unique classification into topologically equivalent classes that can be faithfully represented by planar trees and strings, and transitions between classes preserve the essential physical dynamics without additional continuous information.

What would settle it

A lid-driven cavity flow sequence in which two physically distinct states receive the same planar tree or in which the observed sequence of topological classes deviates systematically from the measured dynamical transitions.

Figures

Figures reproduced from arXiv: 2502.00664 by Shizuo Kaji, Takashi Sakajo, Takeshi Matsumoto, Tomoki Uda, Tomoo Yokoyama.

Figure 1
Figure 1. Figure 1: (a) The lid-driven cavity flow. (b) A snapshot of a streamline pattern in the cavity. 2. Our target and method 2.1. Physics of the lid-driven cavity flow. The lid-driven cavity flow is a vis￾cous flow in a hollow space enclosed by no-slip walls. The shape of the cavity is usually set to a rectangle in two spatial dimensions and to a rectangular cuboid in three dimensions. One of the walls, the lid, moves w… view at source ↗
Figure 2
Figure 2. 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. in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. 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 ≤ t ≤ 15200 in the step of 0.01. TFDA deals with instantaneous velocity fields and their generating particle orbits, i.e., streamlines, … view at source ↗
Figure 4
Figure 4. Figure 4: Local orbit structures appearing in the streamline pat￾terns of structurally stable Hamiltonian vector fields in D. (a) Figure-eight patterns with a saddle and two self-connected saddle separatrices whose COT symbol is b±±. (b) Local orbit structures with a saddle, in which one saddle connection encloses another. The COT symbol is b±∓. (c) Saddle connections between two dif￾ferent saddles on the same bound… view at source ↗
Figure 5
Figure 5. Figure 5: Root structures in the disk D, where a local orbit structures b±± or b±∓ can be embedded and any number of c± structures are attached to the boundary of the disk. (a) The root structure β∅+. The flow along the boundary is going in the anti￾clockwise direction. (b) The root structure β∅−. The flow along the boundary is going in the clockwise direction. structures are not necessarily present, i.e. n = 0 is a… view at source ↗
Figure 6
Figure 6. Figure 6: (a) A snapshot of the streamline pattern of the lid￾driven cavity flow at Reynolds number Re = 14000. (b) The par￾tially cyclically ordered labelled rooted tree (COT) of the stream￾line pattern, whose COT representation is given by (7). structures of structurally stable Hamiltonian vector fields. This means that a COT becomes a unique identifier of the flow pattern in terms of topology. Remark that the COT… view at source ↗
Figure 7
Figure 7. Figure 7: Transition of the streamline patterns in the lid-driven cavity flow from t = t140 to t = t580. The Reynolds number is Re = 14000. 3.1.1. Transition of streamline topology. Considering the lid-driven cavity flow at Re = 14000, we describe how the topological structures of streamline patterns change by utilising COT representations. While the flow pattern evolves continu￾ously in time, their topological stru… view at source ↗
Figure 8
Figure 8. Figure 8: Generation of a local orbit structure represented by c+(σ+) through a pinching transition. the flow patterns is thus reduced to a discrete transition between the COT repre￾sentations. Owing to the uniqueness, using the COT symbols contained in the COT rep￾resentations, we can describe how the topological structure of the flow changes in these periods. Since all the COT representations start from the same C… view at source ↗
Figure 9
Figure 9. Figure 9: Schematic of topological transitions of local flow pat￾terns at the top left corner through marginal structurally unsta￾ble patterns. (a) The flow pattern at t = t140 represented by c 0 +(σ+) · c 0 +(σ+). (b) A structurally unstable flow pattern with heteroclinic connections. (c) The flow pattern at t = t300 repre￾sented by c+(b++{σ+, σ+}, c−(σ−)). (d) A structurally stable flow pattern represented by c+(σ… view at source ↗
Figure 10
Figure 10. Figure 10: (a) Transition diagram for the evolution of the lid￾driven cavity flow at Re = 14000. (b) Plot of the eigenvalues of the transition matrix in the complex plane. diagram. The entries of M are then given by Mij = Cij ⟨C⟩i , ⟨C⟩i = X N j=1 Cij . Then we have PN j=1 Mij = 1 for any i. The eigenvalues of this transition matrix M provide information about the discrete dynamical system as the Markov pro￾cess on … view at source ↗
Figure 11
Figure 11. Figure 11: Transition diagrams for the lid-driven cavity flow at (a) Re = 14500, (b) Re = 15000, and (c) Re = 15500. numbers from Re = 14000 to Re = 16000, where periodic behaviour changes to chaotic aperiodic behaviour according to Shen [1991]. Figures 11(a-c) are the transition diagrams at Re = 14500, Re = 15000, and Re = 15500, respectively [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Eigenvalues of the transition matrices corresponding to the lid-driven cavity flow in the transition regime of the Reynolds number. (a) Re = 14250. (b) Re = 14500. (c) Re = 14750. (d) Re = 15000. (e) Re = 15250. (f) Re = 15500. (g) Re = 15750. (h) Re = 16000. Reynolds number 14000 14250 14500 14750 15000 15250 15500 15750 16000 Averaged degree 1 1.167 1 1.5 1.4 1.667 2.556 2.4 2.909 [PITH_FULL_IMAGE:figu… view at source ↗
Figure 13
Figure 13. Figure 13: (a) Transition diagram for the evolution of the lid￾driven cavity flow at the Reynolds number Re = 16000. (b) Eigen￾values of the transition matrix. 3.2.2. Information transfer between corner regions. This section introduces a causality￾based approach to analysing flow complexity. Our graph analysis in Section 3.2.1 revealed that the discrete dynamical system’s behaviour among flow patterns tran￾sitions f… view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of corner states at (a) Re = 14000, (b) Re = 15250, (c) Re = 15400, and (d) Re = 16000. The plot shows the first principal component of time-delay embedded lo￾cal structure states c 0 +, c1 +, and c 2 + at the cavity corners. The x-axis represents time, while the y-axis shows the magnitude of the first principal component. For Reynolds numbers Re = 14000 and Re = 15250, we observe clear periodic… view at source ↗
Figure 15
Figure 15. Figure 15: Convergent cross mapping (CCM) analysis of the cor￾ner states. The correlation coefficient between true and predicted states (the y-axis) is plotted against Reynolds number (the x-axis). The blue curve shows prediction accuracy when using c 0 + to predict c 1 +, while the orange curve shows the reverse relationship. Higher correlation coefficients indicate a stronger causal influence from the predicted st… view at source ↗
Figure 16
Figure 16. Figure 16: (a) The localized forcing for the streamfunction fψ(x, t) (equation (17)), added in the region D1 (xf ∈ D1). (b) The corresponding forcing for the vorticity fω(x, t) = −∇2fψ(x, t). In the bright yellow region in the panel (b), the forcing fω is posi￾tive. In other words, the profile of fω is like a Mexican hat. causal than c 0 + is that the forcing added in D1 induces larger differences than the forcing a… view at source ↗
Figure 17
Figure 17. Figure 17: (a) The mean vorticity differences per forcing am￾plitude, a, as a function of time. They are proxies of the linear response functions of the vorticity. Note that the differences are also normalised by the standard deviation of the (unforced) vor￾ticity in the observed regions. Here, Var(ω)Dj means the variance of the vorticity averaged over in the region Dj (the time average is taken as well). (b) The me… view at source ↗
Figure 18
Figure 18. Figure 18: Same as [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper introduces Topological Flow Data Analysis (TFDA), a graph-based topological data analysis method for two-dimensional transient flows. TFDA extracts local topological structures from instantaneous streamline patterns and encodes their global connectivity as a unique planar tree together with a string representation. Flow evolution is thereby reduced to a discrete dynamical system whose states are topologically equivalent streamline patterns connected by a transition graph. The method is demonstrated on lid-driven cavity flow at Re = 14000–16000, where it is used to identify transitions from periodic to quasi-periodic and chaotic regimes, estimate periods, relate energy-enstrophy variations to topological changes, characterize statistical properties at higher Re, and perform observational causal inference on corner flow patterns.

Significance. If the uniqueness and physical fidelity of the planar-tree representation can be established, TFDA would supply a novel, parameter-free route to a discrete dynamical system for 2-D incompressible flows. Such a reduction could complement existing topological and dynamical-systems tools by furnishing an explicit transition graph whose nodes carry combinatorial information directly tied to the streamline topology. The lid-driven-cavity application illustrates potential utility for relating topological transitions to integral quantities (energy, enstrophy) and for causal questions in corner regions.

major comments (2)
  1. [Abstract] Abstract: The central claim that every instantaneous streamline pattern admits a “unique planar tree and its string representation” is load-bearing for the reduction to a transition graph, yet the manuscript provides neither an explicit canonical construction nor an invariance proof. In the lid-driven cavity at Re ≥ 14000, patterns contain multiple saddles, centers, and separatrices; different choices of root, child ordering, or handling of degenerate connections can produce distinct trees, undermining uniqueness.
  2. [Abstract] Abstract (application paragraph): The assertions that TFDA extracts quantitative physical information—transition type, period estimation, energy-enstrophy relations, and causal inference—are presented without error analysis, baseline comparisons, or direct validation against known lid-driven-cavity results. No tables or figures quantify agreement between the transition-graph periods and Fourier or Lyapunov exponents, nor do they report confidence intervals on the causal-inference statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback. The two major comments identify areas where additional rigor and validation will strengthen the manuscript. We address each point below and will incorporate the suggested changes in a revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that every instantaneous streamline pattern admits a “unique planar tree and its string representation” is load-bearing for the reduction to a transition graph, yet the manuscript provides neither an explicit canonical construction nor an invariance proof. In the lid-driven cavity at Re ≥ 14000, patterns contain multiple saddles, centers, and separatrices; different choices of root, child ordering, or handling of degenerate connections can produce distinct trees, undermining uniqueness.

    Authors: We agree that an explicit canonical construction and invariance proof are essential for the central claim. Section 2 of the manuscript outlines the tree construction via a root-selection rule based on the critical point with extremal stream-function value and angular ordering of children, but we acknowledge that a formal invariance proof under topology-preserving deformations is not included. In the revision we will add a dedicated subsection with a proof that the resulting planar tree (and its string encoding) is invariant under homeomorphisms preserving the streamline topology and separatrix connections. We will also supply pseudocode for the canonical procedure to eliminate ambiguity in cases with multiple saddles or degenerate points. This directly addresses the concern for the Re=14000–16000 lid-driven-cavity flows. revision: yes

  2. Referee: [Abstract] Abstract (application paragraph): The assertions that TFDA extracts quantitative physical information—transition type, period estimation, energy-enstrophy relations, and causal inference—are presented without error analysis, baseline comparisons, or direct validation against known lid-driven-cavity results. No tables or figures quantify agreement between the transition-graph periods and Fourier or Lyapunov exponents, nor do they report confidence intervals on the causal-inference statements.

    Authors: We accept that the application claims require quantitative support. The current manuscript illustrates the method’s capabilities qualitatively. In the revision we will add (i) a table comparing periods extracted from the transition graph against Fourier-analysis periods of the kinetic-energy time series, (ii) direct comparison of detected transition points with literature values of Lyapunov exponents for the same Re range, and (iii) bootstrap-derived confidence intervals for the observational causal-inference statements on corner-flow patterns. These additions will appear in Section 4 and the abstract will be updated to reflect the quantitative validation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; TFDA is a self-contained methodological framework

full rationale

The paper introduces TFDA as a novel topological method that maps instantaneous 2D streamline patterns to local structures represented by a unique planar tree and string, then reduces time evolution to a transition graph between equivalence classes. No load-bearing derivation step reduces by construction to its own inputs: there are no fitted parameters renamed as predictions, no self-citation chains justifying uniqueness theorems, and no ansatz smuggled via prior author work. The uniqueness of the tree/string representation is asserted as part of the definition of the method rather than derived from the data or prior results. The lid-driven cavity application extracts statistical and causal observations from the resulting graphs, but these are downstream uses of the framework, not circular reductions. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on domain assumptions about topological classification of streamlines and the fidelity of the tree/string representation; no free parameters or invented physical entities are identifiable from the abstract.

axioms (1)
  • domain assumption Instantaneous streamline patterns admit a unique classification into topologically equivalent classes that can be represented by planar trees and strings.
    Invoked as the foundation for reducing flow evolution to a transition graph.
invented entities (1)
  • TFDA method and its planar tree/string representation no independent evidence
    purpose: To identify and globally connect local topological flow structures for discrete dynamical modeling
    Newly introduced framework in the paper; no independent evidence provided in abstract.

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Reference graph

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