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

TL;DR

This paper introduces a novel topology-based method using Euler Characteristic Surfaces to quantitatively define and analyze churn flow in small-diameter vertical pipes, outperforming existing models.

Contribution

It presents the first mathematical topology-based characterization of churn flow and an unsupervised framework for regime detection that surpasses supervised methods.

Findings

Euler Characteristic Surfaces effectively distinguish churn from slug flow.

The framework achieves 95.6% accuracy in classifying flow regimes without labeled data.

The ECS-inferred transition point is 3.81 m/s above previous predictions, indicating model underestimation.

Abstract

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 ($L^1$ 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 $\beta_{ECS}=0.14$, $\beta_{amp}=0.50$, $\beta_{ugs}=0.36$, placing $64\%$ of total weight on topology-derived features ($\beta_{ECS} + \beta_{amp}$). 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 $1.9\times$ 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.