cs.LGApr 7, 2026
SaveTopological Characterization of Churn Flow and Unsupervised Correction to the Wu Flow-Regime Map in Small-Diameter Vertical Pipes
Organizations: Department of Petroleum Engineering, Montana Technological University, USA · Data Science, George Washington University, USA · Department of Mathematical Sciences, Montana Technological University, USA
Abstract
Churn flow-the chaotic, oscillatory regime in vertical two-phase flow-has lacked a quantitative mathematical definition for over 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 ( distance on the surface) and amplitude statistics (scale-wise mean, standard deviation, max, min)-with gas velocity. Applied to unlabeled air-water trials from Montana Tech, the self-calibrating framework learns weights , , , placing of total weight on topology-derived features (). The ECS-inferred slug/churn transition lies m/s above Wu et al.'s (2017) prediction in -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 Texas A&M University images confirms higher topological complexity in churn vs. slug (). Applied to TAMU pseudo-trials, the same unsupervised framework achieves -class accuracy and 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.
Figures & tables
Figure 1: Multiscale topological characterization via morphological dilation. Binary representations of three flow regimes at increasing morphological scales . Each column shows one regime (slug, annular mist, churn) at three dilation levels: (original resolution), (moderate coarse-graining), and (strong coarse-graining). Black regions represent gas phase, white regions represent liquid. The Euler characteristic quantifies topological structure at each scale. Slug flow maintains near-zero across scales (single large gas region). Annular mist exhibits high positive at fine scales (many dispersed bubbles) that decreases as bubbles merge under dilation. Churn flow shows intermediate positive with fragmented structure that persists longer under dilation due to internal cavities. Stacking across scales and time yields the Euler Characteristic Surface , a multiscale descriptor capturing both spatial organization and temporal dynamics.
Figure 2: Construction of the Euler Characteristic Surface. ( Left ) Four consecutive binary frames of churn flow show temporal variation in gas-liquid topology. ( Center ) At time , morphological dilation at four scales ( ) produces the Euler characteristic at each scale. ( Right ) The ECS matrix assembles all values. Heatmap colors indicate topology: blue regions (positive , dispersed gas), green (intermediate), red (negative , connected gas).
Figure 3: Montana Tech Vertical Flow Loop schematic showing the 47 ft vertical test section with three camera positions and air/water injection systems.
Figure 4: End-to-end MKL-ECS classification pipeline. Starting from raw MTVFL video footage, frames are extracted, binarized by adaptive thresholding, and processed through hexagonal morphological dilation at scale levels. The Euler characteristic at each scale and frame is assembled into a matrix ( ecs ). Three distance modalities— temporal-alignment on ecs , amplitude feature distance, and superficial gas velocity distance—are converted to heat kernels and blended via unsupervised Multiple Kernel Learning (Algorithm 1 ), which learns optimal kernel weights and clustering jointly by alternating optimization.
Figure 5: Spatial validation on Texas A&M images. ( a ) Box plot of spatial variance by regime. Churn exhibits the highest variance, confirming topological complexity. ( b ) Histogram of spatial variance for slug (blue) and churn (orange), with churn/slug ratio ( ). ( c ) Mean ecs profile across morphological scales for each regime, with shaded s.d. bands. The distinct scale-dependent signatures confirm facility-independent topological structure.
| Dataset | Analysis | Metric | Result | |
| MTVFL (2 in.) | MKL 3-class | ARI / Acc | / | |
| TAMU (1.89 in.) | Spatial variance | 947 | -value | |
| Churn/Slug ratio | ||||
| MKL 4-class ( ) | 45 | ARI / Acc | / | |
| Churn / Bubbly / Taylor recall | 100% / 100% / 100% |
Table 1: Cross-dataset validation results. MTVFL: trials, -class ( ). TAMU: images / pseudo-trials ( , trace-normalized kernels). The self-calibrating MKL automatically excludes the degenerate kernel on TAMU (all distances zero).
Figure 6: Representative Euler Characteristic Surfaces for each flow regime. Each surface shows for one trial: frame index (x-axis), morphological scale (y-axis), and value (z-axis/color). ( a ) Annular-mist flow produces a nearly flat surface, reflecting the stable, thin liquid film surrounding a continuous gas core. ( b ) Churn flow shows irregular oscillations across all scales, corresponding to the chaotic liquid-gas interaction. ( c ) Slug flow exhibits strongly modulated columns at low scale, corresponding to the periodic passage of large Taylor bubbles.
Figure 7: Binary image thresholding and its relation to the Euler characteristic. Raw greyscale frames (left) and corresponding binary images after thresholding at (right) for three regimes from the MTVFL -in. tubing. Top: Slug flow (14 SCFM) — many small black clusters yield . Middle: Churn flow (44 SCFM) — chaotic gas-liquid interface with comparable black and white clusters yields . Bottom: Annular mist (82 SCFM) — thin liquid film with dispersed droplets yields . The sign and magnitude of carry direct physical meaning: large positive indicates many dispersed gas clusters, while negative indicates a connected liquid film surrounding few large gas pockets.
Figure 8: ECS clustering overlaid on the Wu flow-regime map. Logarithmic axes show superficial gas velocity ( ) vs. liquid velocity ( ). Wu boundary lines: Ls (slug/churn, green dashed), Lm (churn/annular, red dashed). Trial points are colored by ECS cluster assignment (slug: red circles; churn: teal squares; annular mist: amber triangles). Dotted vertical lines show the ECS-inferred boundaries: m/s ( m/s above Wu) and m/s ( m/s).
| Method | ARI | Acc | Sep |
|---|---|---|---|
| ecs kernel only | 0.541 | 1.65 | |
| kernel only | 0.757 | 1.80 | |
| Amp + (no ecs ) | 0.757 | 1.50 | |
| Uniform weights (no opt.) | 0.520 | 0.784 | 11.2 |
| MKL (learned weights) | |||
| RF baseline [ 16 ] (supervised) | — | 0.890 | — |
Table 2: MKL clustering performance on MTVFL. ARI = Adjusted Rand Index; Acc = accuracy vs. Wu ground truth; Sep = intra-to-inter cluster separation ratio. MKL uses entropy regularization selected by stability. Uniform baseline uses equal weights without optimization.
| Iteration | ||||
|---|---|---|---|---|
| 0 (init) | 0.333 | 0.333 | 0.333 | — |
| 1 | 0.159 | 0.537 | 0.303 | |
| 2 (final) |
Table 3: Weight evolution during MKL optimization. Representative run with , showing convergence to final weights in 2 iterations. Bandwidth computed from raw (pre-normalization) distances per Remark 4 .
| Kernels used | ARI | Acc | Sep |
|---|---|---|---|
| ECS only | 0.541 | 1.6 | |
| Amp only | 0.541 | 1.6 | |
| only | 0.757 | 1.8 | |
| ECS + Amp | 0.486 | 1.4 | |
| ECS + | 0.730 | 1.7 | |
| Amp + | 0.757 | 1.5 |
Table 4: Kernel ablation on MTVFL (spectral clustering labels). Weights within each subset are re-optimized by MKL ( ). All rows use spectral clustering labels for fair comparison; the main-result ARI of in Table 2 applies monotone boundary inference to the “All three” configuration.
Figure 9: Bootstrap stability analysis (500 resamples). ( a ) Distribution of ARI across bootstrap resamples. Dashed line: mean (0.461); shaded band: 95% confidence interval [0.17, 0.87]. ( b ) Distribution of accuracy. Dashed line: mean (0.721); shaded band: 95% CI [0.52, 0.95]. ( c ) Per-trial stability (fraction of resamples in which each trial’s label agrees with the full-data clustering). Trial 5 ( m/s, red bar) is the most unstable trial (stability = 0.20), consistent with its position at the slug/churn boundary.
Figure 10: Clustering quality metrics. ( a ) Bootstrap co-assignment probability matrix ( ). Entry is the fraction of 500 resamples in which trials and are assigned to the same cluster. Strong block structure confirms that within-regime co-assignment is robust. ( b ) Confusion matrix against Wu ground-truth labels. 12 of 15 annular-mist trials are correctly identified; the main disagreement is 8 Wu-churn trials assigned to slug, consistent with the m/s boundary shift.
Figure 11: MKL convergence and weight evolution. ( a ) Regularized objective vs. iteration for the best restart. The entropy-regularized alternating procedure converges monotonically (Proposition 6 ) within 3 iterations. ( b ) Kernel weight evolution over iterations. Starting from uniform initialization , the weights converge to in 2 iterations, consistent with Table 3 .
Figure 12: Kernel component ablation (MTVFL, spectral clustering labels). ARI ( left bars ) and accuracy ( right bars ) for all non-empty subsets of the three base kernels. MKL weights are re-optimized within each subset ( ). The kernel provides the strongest single-kernel signal (ARI = 0.44); adding ecs improves ARI to 0.47, and the full three-kernel combination achieves the highest spectral ARI (0.54). The ecs kernel alone is weakest (0.07), confirming that topology requires fusion with other modalities to be discriminative.
Figure 13: Sensitivity to bandwidth and regularization strength. ( a ) ARI as a function of the ECS heat kernel bandwidth , expressed as a multiple of the median heuristic value . The result is stable over orders of magnitude ( to ), consistent with the Lipschitz bound in Theorem 10 (b). ( b ) ARI as a function of entropy regularization strength . The stability-selected value (red dashed) lies within the plateau ( – ) where performance is insensitive to .
Figure 14: Scale band analysis of the ecs . Each row restricts the ecs kernel to a subset of the 30 morphological scale levels. ( a ) Small scales ( ): ARI = 0.189, corresponding to individual bubble resolution — the connected-component count at these scales tracks the number of gas bubbles in the field of view but is noisy and least discriminative. ( b ) Meso scales ( ): ARI = 0.204, corresponding to Taylor bubble structures — the EC signature encodes whether a large connected gas slug is present. ( c ) Large scales ( ): ARI = 0.569 (highest), corresponding to the global liquid film topology — large-scale connectivity captures regime-level structure most effectively.
Figure 15: TAMU image preprocessing pipeline. Each row shows one representative image per regime (bubbly, slug, churn, Taylor). Columns from left to right: raw image with adaptive crop region (red dashed box, applied only when width px); cropped region isolating the pipe interior; grayscale conversion; Otsu-thresholded binary image (threshold shown in red); downsampled to 120 px width for Hoshen–Kopelman computation. The Otsu threshold adapts per image (range – across regimes), replacing the fixed used for MTVFL.
Figure 16: TAMU cross-facility validation: self-calibrating MKL. ( a ) MDS embedding of the blended kernel colored by ground-truth regime. Slug (green squares) and churn (red triangles) form distinct clusters; bubbly and Taylor overlap, consistent with their similar low-complexity topology. ( b ) Confusion matrix (45 pseudo-trials, 4 classes). Churn, bubbly, and Taylor recall = 100%; slug recall = 86% (2 of 14 misclassified as bubbly, consistent with their similar low-complexity topology). ( c ) Blended kernel heatmap sorted by regime, with learned weights —velocity receives zero weight while ECS retains , confirming the framework’s self-calibrating property.
Figure 17: PAC generalization bound as a function of dataset size. Eq. ( 29 ) evaluated for kernels as a function of , with (normalized kernels) and . For (MTVFL, left dashed line) the bound is — vacuous. For (right dashed line, achievable by combining TAMU and LabPetro) the bound tightens to . The empirical ARI values (horizontal bands) are consistent with the bound in all cases.
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