# Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study

arXiv: https://arxiv.org/abs/2207.14080 · HTML: https://mlsanity.com/paper/2207.14080/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 115 references that are in the map.
- First posted on arXiv 28 Jul 2022.

## Builds on (most cited first, 12 of 12)

- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Task-based Augmented Contour Trees with Fibonacci Heaps](https://mlsanity.com/paper/1902.04805/) (2019)
- [Progressive Wasserstein Barycenters of Persistence Diagrams](https://mlsanity.com/paper/1907.04565/) (2019)
- [Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization](https://mlsanity.com/paper/2106.00157/) (2021)
- [Distributed computation of persistent homology](https://mlsanity.com/paper/1310.0710/) (2013)
- [Persistence Atlas for Critical Point Variability in Ensembles](https://mlsanity.com/paper/1807.11212/) (2018)
- [Uncertainty Visualization of 2D Morse Complex Ensembles Using Statistical Summary Maps](https://mlsanity.com/paper/1912.06341/) (2019)
- [A Structural Average of Labeled Merge Trees for Uncertainty Visualization](https://mlsanity.com/paper/1908.00113/) (2019)
- [Edit Distance between Merge Trees](https://mlsanity.com/paper/2207.08511/) (2022)

## Built on by (most cited first, 11 of 11)

- [Wasserstein Dictionaries of Persistence Diagrams](https://mlsanity.com/paper/2304.14852/) (2023)
- [Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction](https://mlsanity.com/paper/2502.20215/) (2025)
- [Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2307.02509/) (2023)
- [TTK is Getting MPI-Ready](https://mlsanity.com/paper/2310.08339/) (2023)
- [TDA-RC: Task-Driven Alignment for Knowledge-Based Reasoning Chains in Large Language Models](https://mlsanity.com/paper/2604.04942/) (2026)
- [A Practical Solver for Scalar Data Topological Simplification](https://mlsanity.com/paper/2407.12399/) (2024)
- [BondMatcher: H-Bond Stability Analysis in Molecular Systems](https://mlsanity.com/paper/2504.03205/) (2025)
- [Identifying Locally Turbulent Vortices within Instabilities](https://mlsanity.com/paper/2408.12662/) (2024)
- [Topology Aware Neural Interpolation of Scalar Fields](https://mlsanity.com/paper/2508.17995/) (2025)
- [Continuous Edit Distance, Geodesics and Barycenters of Time-varying Persistence Diagrams](https://mlsanity.com/paper/2512.12939/) (2025)
- [DebrisTracer: Reliable Tracking in Hypervelocity Impact Fast Imaging](https://mlsanity.com/paper/2607.15986/) (2026)

## Abstract (arXiv)

This application paper presents a comprehensive experimental evaluation of the suitability of Topological Data Analysis (TDA) for the quantitative comparison of turbulent flows. Specifically, our study documents the usage of the persistence diagram of the maxima of flow enstrophy (an established vorticity indicator), for the topological representation of 180 ensemble members, generated by a coarse sampling of the parameter space of five numerical solvers. We document five main hypotheses reported by domain experts, describing their expectations regarding the variability of the flows generated by the distinct solver configurations. We contribute three evaluation protocols to assess the validation of the above hypotheses by two comparison measures: (i) a standard distance used in scientific imaging (the L2 norm) and (ii) an established topological distance between persistence diagrams (the L2-Wasserstein metric). Extensive experiments on the input ensemble demonstrate the superiority of the topological distance (ii) to report as close to each other flows which are expected to be similar by domain experts, due to the configuration of their vortices. Overall, the insights reported by our study bring an experimental evidence of the suitability of TDA for representing and comparing turbulent flows, thereby providing to the fluid dynamics community confidence for its usage in future work. Also, our flow data and evaluation protocols provide to the TDA community an application-approved benchmark for the evaluation and design of further topological distances.