# Topological Characterization of Churn Flow and Unsupervised Correction to the Wu Flow-Regime Map in Small-Diameter Vertical Pipes

arXiv: https://arxiv.org/abs/2604.06167 · HTML: https://mlsanity.com/paper/2604.06167/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 45 references that are in the map.
- First posted on arXiv 7 Apr 2026.

## Builds on (most cited first, 4 of 4)

- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [Euler Characteristic Surfaces](https://mlsanity.com/paper/2102.08260/) (2021)
- [Automated Flow Pattern Classification in Multi-phase Systems Using AI and Capacitance Sensing Techniques](https://mlsanity.com/paper/2502.16432/) (2025)
- [Interpretable Classification of Time Series Using Euler Characteristic Surfaces](https://mlsanity.com/paper/2603.15079/) (2026)

## Abstract (arXiv)

Churn flow-the chaotic, oscillatory regime in vertical two-phase flow-has lacked a quantitative mathematical definition for over 40 years. We introduce the first topology-based characterization using Euler Characteristic Surfaces (ECS). We formulate unsupervised regime discovery as Multiple Kernel Learning (MKL), blending two complementary ECS-derived kernels-temporal alignment (L1 distance on the chi(s,t) surface) and amplitude statistics (scale-wise mean, standard deviation, max, min)-with gas velocity. Applied to 37 unlabeled air-water trials from Montana Tech, the self-calibrating framework learns weights βECS=0.14, βamp=0.50, βugs=0.36, placing 64% of total weight on topology-derived features (βECS + βamp). The ECS-inferred slug/churn transition lies +3.81 m/s above Wu et al.'s (2017) prediction in 2-in. tubing, quantifying reports that existing models under-predict slug persistence in small-diameter pipes where interfacial tension and wall-to-wall interactions dominate flow. Cross-facility validation on 947 Texas A&M University images confirms 1.9× higher topological complexity in churn vs. slug (p < 10-5). Applied to 45 TAMU pseudo-trials, the same unsupervised framework achieves 95.6% 4-class accuracy and 100% churn recall-without any labeled training data-matching or exceeding supervised baselines that require thousands of annotated examples. This work provides the first mathematical definition of churn flow and demonstrates that unsupervised topological descriptors can challenge and correct widely adopted mechanistic models.