Topological Separation of Vortices

TL;DR

This work addresses entangled vortices in region-based vortex extraction for turbulent flows by extending contour-tree-based segmentation with a layering step and a statistical continuity check. It leverages persistence diagrams to select high-value critical-point pairs and uses a layering strategy to propagate seed-based IDs, enabling adaptive two-way splits without relying on global iso-values. A $R_1$-based statistic, computed from nearby vorticity lines, mitigates inaccurate splits and guides whether a split should occur; the approach is demonstrated on turbulent datasets with qualitative improvements over prior methods. While resulting in slower runtime due to exhaustive critical-point analysis, the method yields more robust vortex separation and reduces mis-splits, facilitating more accurate vortex statistics and topology-aware analyses. Key insights include the use of $λ_2$ as a scalar field, the layering mechanism, and the continuity-based split check, which collectively enhance the topological separation of entangled vortices in complex flows.

Abstract

Vortices and their analysis play a critical role in the understanding of complex phenomena in turbulent flows. Traditional vortex extraction methods, notably region-based techniques, often overlook the entanglement phenomenon, resulting in the inclusion of multiple vortices within a single extracted region. Their separation is necessary for quantifying different types of vortices and their statistics. In this study, we propose a novel vortex separation method that extends the conventional contour tree-based segmentation approach with an additional step termed "layering". Upon extracting a vortical region using specified vortex criteria (e.g., $λ_2$), we initially establish topological segmentation based on the contour tree, followed by the layering process to allocate appropriate segmentation IDs to unsegmented cells, thus separating individual vortices within the region. However, these regions may still suffer from inaccurate splits, which we address statistically by leveraging the continuity of vorticity lines across the split boundaries. Our findings demonstrate a significant improvement in both the separation of vortices and the mitigation of inaccurate splits compared to prior methods.

Figures (9)

• Figure 1: (a) shows a vortical region (light-blue) extracted with the region growing adeelhairpin2023 using $\lambda_2$ criterion. Variance in the patterns of vorticity lines (black) indicate the presence of multiple vortices. (b) shows a hairpin vortex (red) correctly getting separated from the rest of the vortices. (c) shows the zoomed in version of the hairpin vortex. (d) shows the inaccurate split of the hairpin vortex.
• Figure 2: (a) shows the vortical region (light-blue) and the underlying iso-surface components (blue, red, green, etc.). (b) shows the assigned colors to the region's cells based on the Euclidean distance from the closest iso-surface component. The highlighted area shows the wrong color (blue) assigned to the cells of the red vortex.
• Figure 3: (a) shows the persistence diagram of two critical point pairs with the highest persistence. Here maximum($\mathbb{M}$), saddle($\mathbb{S}$) and minimum($\mathfrak{m}$) points are represented by red, cyan and blue, respectively. (b) shows the corresponding minimal join tree of the chosen critical point pairs with one $\mathbb{M}$, one $\mathbb{S}$ and two $\mathfrak{m}$.
• Figure 4: (a) shows a single region containing a streamwise and a horseshoe vortex, indicated by the vorticity line (black). (b) shows the join tree embedded within the region indicating the corresponding location of the critical points. (c) shows the segmentation of the region based on the join tree, where green and blue cells correspond to two $\mathfrak{m}$-$\mathbb{S}$ pairs, and the red cells correspond to a $\mathbb{M}$-$\mathbb{S}$ pair.
• Figure 5: (a) shows the initial segments obtained from the minimal join tree. IDs $0$ and $1$ represent the seed segments corresponding to the ($\mathfrak{m}_1$-$\mathbb{S}$) and ($\mathfrak{m}_2$-$\mathbb{S}$) pairs, respectively. $-1$ is the ID of the query segment corresponding to the ($\mathbb{S}$-$\mathbb{M}$) pair. (b) displays the same region after several iterations of the layering process. It is evident that the seed segments have expanded, resulting in fewer cells remaining in the query segment. (c) demonstrates the region completely separated at the conclusion of the layering process.