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 $χ(s,t)$ 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 $β_{ECS}=0.14$, $β_{amp}=0.50$, $β_{ugs}=0.36$, placing $64\%$ of total weight on topology-derived features ($β_{ECS} + β_{amp}$). 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.

翻译：搅动流——垂直两相流中的混沌、振荡流型——在超过40年的时间里一直缺乏定量的数学定义。我们首次引入了基于欧拉特征曲面（ECS）的拓扑表征。我们将无监督流型发现问题建模为多核学习（MKL），将两种互补的ECS衍生核函数——时间对齐（χ(s,t)曲面上的L1距离）和振幅统计（尺度均值、标准差、最大值、最小值）——与气相速度相结合。应用于蒙大拿理工学院的37组无标签空气-水实验数据，该自标定框架学习得到权重β_ECS=0.14，β_amp=0.50，β_u_gs=0.36，其中64%的总权重分配给了拓扑衍生特征（β_ECS + β_amp）。在2英寸管中，ECS推断的段塞流/搅动流转变点比Wu等人（2017）的预测值高+3.81 m/s，这量化了现有模型在小直径管中（界面张力和壁-壁相互作用主导流动）低估段塞流持续性的现象。跨设施验证采用德克萨斯农工大学（TAMU）的947张图像，结果表明搅动流的拓扑复杂度是段塞流的1.9倍（p<10^{-5}）。应用于45组TAMU伪实验数据，同一无监督框架在不使用任何标签训练数据的情况下，实现了95.6%的四类分类准确率和100%的搅动流召回率，达到或超过了需要数千个标注样本的有监督基线模型。本研究首次为搅动流提供了数学定义，并证明无监督拓扑描述符能够挑战和修正广泛采用的机理模型。