Hierarchical clustering method of volumetric vortical regions with application to the late stage of laminar-turbulent transition

A hierarchical clustering method for volumetric vortical regions is proposed. Point clouds in vortical regions are extracted from an instantaneous flow field, using a series of parallel planes normal to a direction, and are grouped between consecutive planes. Cross-sectional distributions of points are then stacked in the direction. Each group of extracted points is identifiable as an individual entity that is referred to as a $p$ cluster, with the letter $p$ replaced by the terms proto, sub, super, and hyper, according to the level of clustering. The spatial distribution and temporal evolution of $p$ clusters are visualized and automatically tracked. The present method is applied to the late stage of the K-type natural transition of a boundary-layer flow. Our method clearly captures, with admitting a detailed analysis, pairs of vortex structures located in the viscous sublayers near the roots of hairpin vortices. Moreover, it allows the algorithmic identification of dominant mathematical terms in the enstrophy and helicity equations, and a categorization of the extracted points based on local dynamics characterized by sets of the dominant terms. Construction of ensemble-averaged quantities using the present method, an extension of the method to omnidirections, and comparison between the present method and conventional vortex analysis methods are also discussed.

Figure 1

Vortex structures that appear during laminar-turbulent transition and associated skin-friction coefficients along streamwise Reynolds number. Panels (a) and (b) show vortex structures and skin friction coefficient ($C_f$), respectively. (a) Vortex structures. Vortices are visualized by the isosurface of SIVGT with $Q=8\times {10}^{-4}$. The color signifies streamwise Mach number. ($\mathrm{b})C_f$. Superscript (a) shows a correlation by Cousteix [35].

Figure 2

Hierarchical clusters and its subsets considered in this paper. These clusters are represented in a unified manner as $p$ clusters, where $p$ is replaced with proto, sub, super, or hyper (a simple cluster has no prefix). (a) Fundamental clusters of various levels. Clusters are built up from left to right, and the cluster level increases from lower to upper, accordingly. (b) Subclustering within fundamental clusters.

Figure 3

Overview of the present hierarchical vortex clustering.

Figure 4

Clusters generated on the plane $x=558.9{\delta }_{\text{in}}$. Regions of $|{\omega }_x|>0.1$ and resultant clusters are shown. The surface contour is ${\omega }_x,$ and the points are protoclusters. Different clusters are shown in different colors. The limitation of the number of recognizable colors means that some different clusters are shown in the same color.

Figure 5

Histogram of the number of mesh points at which ${\omega }_x$ belongs to each divided range at $t=t_1$. Red lines: ${\omega }_x=\pm 0.1$.

Figure 6

Effects of varying ${\omega }_{x,0}$ on the resultant distribution of skeletons at $t=t_1$. Open and filled symbols denote the skeletons of the vortical regions with positive and negative ${\omega }_x$, respectively. The color indicates the height from the wall. (a) ${\omega }_{x,0}=0.03,$ (b) ${\omega }_{x,0}=0.1,$ (c) ${\omega }_{x,0}=0.19.$

Figure 7

Backward subclustering. (a) Before backward division. (b) After backward division.

Figure 8

Backward propagation of multiple correspondence. (a) Backward subclustering on the $j\mathrm{th}x$ plane according to the forward multiple correspondences with the ($j+1)\mathrm{th}$ $x$ plane. (b) Emergence of new forward multiple correspondences between the ($j$-1)th and $j\mathrm{th}x$ planes, and backward subclustering on the ($j$-1)th $x$ plane.

Figure 9

Schematic explanation of the construction of a skeleton time surface. (a) Finding the closest point to ${\mathbf{x}}_p$, (b) Selection of the closest SSGS from candidate SSGSs. Solid and dotted curves represent SSGSs at $t=t_1$ and $t_2$, respectively, (c) Generation of a surface element satisfying the admissible condition from the candidate triangles among $\{{\mathbf{x}}_{\alpha },{\mathbf{x}}_{\alpha +1},{\mathbf{x}}_{\beta }^{\prime },{\mathbf{x}}_{\beta +1}^{\prime }\}$. The gray element is the generated surface element $T_1$.

Figure 10

Clustering on the plane going through point ${\mathbf{P}}_{\mathbf{0}}$. (a) Projection of point P to plane $\mathcal{P}$ going through the selected point ${\mathbf{P}}_{\mathbf{0}}$ and having the normal vector $\mathbf{n}$. Here, $\mathbf{n}$ is the vector averaged around point ${\mathbf{P}}_{\mathbf{0}}$. (b) Clustering of points $P_1^{\prime }$, $P_2^{\prime }$, and $P_3^{\prime }$ on plane $\mathcal{P}$, the CG, and the averaged vector at the CG.

Figure 11

Vortex ring and its point cloud with vectors. (a) Isosurface. (b) Vortex ring. Points are plotted every 200 points for visualization purposes.

Figure 12

Superclusters and skeletons constructed by the x-, y-, and z-directional clusterings. (a) Superclusters and skeletons constructed by the x-directional clustering. Red: ${\omega }_x>0.1$, Blue: ${\omega }_x<-0.1.$ (b) Superclusters and skeletons constructed by the y-directional clustering. Red: ${\omega }_y>0.1$; blue: ${\omega }_y<-0.1.$ (c) Superclusters and skeletons constructed by the z-directional clustering. Red: ${\omega }_z>0.1$; blue: ${\omega }_z<-0.1.$ (d) Accumulated plots of the skeletons constructed by the x-, y-, and z-directional clusterings.

Figure 13

Skeleton extracted by the clustering method based on a vortex axis.

Figure 14

Protoclusters in vortical regions, and superclusters and skeletons constructed by the clustering based on a vortex axis. Different colors in (b) and (c) show different superclusters and skeletons. The limitation of the number of recognizable colors means that some different superclusters (skeletons) are shown in the same color. (a) Protoclusters extracted by the $\mathrm{\Delta }$ criterion with the imaginary parts of eigenvalues greater than 0.1. (b) Superclusters. (c) Skeletons.

Figure 15

Supercluster and their skeletons at $t=t_1$. Superclusters are, in particular, shown by both isosurfaces and points. Only a few superclusters are shown for clarity. The skeletons of superclusters with ${\omega }_x>{\omega }_{x,0}$ are shown in red and pink, and those with ${\omega }_x<-{\omega }_{x,0}$ are shown in blue and cyan.

Figure 16

Hyperclusters constructed from superclusters at $t/({\delta }_{\text{in}}/c_0)=0,1.6$ and $3.2$. Regions of ${\omega }_x>{\omega }_{x,0}$ are shown in red, pink, and yellow for $t/({\delta }_{\text{in}}/c_0)=0,1.6$, and $3.2$, respectively. Regions of ${\omega }_x<-{\omega }_{x,0}$ are shown in blue, cyan, and yellow green for $t/({\delta }_{\text{in}}/c_0)=0,1.6$, and $3.2$, respectively. The limitation of the number of recognizable colors means that different superclusters are shown in the same color.

Figure 17

Distribution of ${\omega }_x$ on the streamwise series of $x$ planes and the skeletons of superclusters with ${\omega }_x>{\omega }_{x,0}$ at $t=t_1$. Only the regions with ${\omega }_x>{\omega }_{x,0}$ are shown for the series of $x$ planes.

Figure 18

Superclusters extracted by the omnidirectional clustering at $t=t_1$ (green: supercluster A; light blue: supercluster B). Different colors show different superclusters. The limitation of the number of recognizable colors means that some different superclusters are shown in the same color.

Figure 19

Resultant superclusters (small spheres) and skeletons (large spheres) of superclusters A and B obtained by both omnidirectional and unidirectional clustering. Different colors show different superclusters. The limitation of the number of recognizable colors means that some different superclusters are shown in the same color. (i) Superclusters (left: omnidirectional clustering; right: unidirectional clustering). (ii) Skeletons (left: omnidirectional clustering; right: unidirectional clustering). (a) Supercluster A. (b) Supercluster B.

Figure 20

Comparison of the present method with the Kida-Miura method, LCS, and the community model for an instantaneous flow field of the initial stage. (a) Resultant superclusters generated by the omnidirectional clustering based on a vortex axis for protoclusters extracted by the $\mathrm{\Delta }$ criterion. (b) Resultant superclusters generated by the $x$-directional clustering for protoclusters extracted by the $\mathrm{\Delta }$ criterion. (c) Vortical points extracted by the Kida-Miura method ($D<-100$). (d) Scalar contours of FTLE at each cross section. (e) Vortical points divided by the community model.

Figure 21

All extracted skeletons for the initial, middle, and late stages ranging from ${\text{Re}}_x=300.79-575.1$ at $t=t_1$. (i) Before hairpin vortex generation. Skeletons are obtained by alternate clustering in the $x$ and $z$ directions on the basis of ${\omega }_x$ and ${\omega }_z$, respectively. Small open circles and small spheres are skeletons of positive and negative ${\omega }_x$, respectively. Big spheres are skeletons of negative ${\omega }_z$. The threshold values for ${\omega }_x$ and ${\omega }_z$ are 0.001 and $-0.3$, respectively. (ii) After hairpin vortex generation. Skeletons are obtained by alternate clustering in the $x$ and $z$ directions on the basis of ${\omega }_x$ and ${\omega }_z$, respectively. Open circles and small spheres are skeletons of positive and negative ${\omega }_x$, respectively. Pink and purple spheres are skeletons of positive and negative ${\omega }_z$, respectively. The threshold value for ${\omega }_x$ is 0.01. The threshold value for ${\omega }_z$ depends on whether ${\omega }_z$ is positive or negative because of bias in the statistical mean value; if positive, the threshold is 0.01, if negative the threshold is $-0.5$. (a) Initial stage. (b) Middle stage. Open circles and small spheres denote the skeletons of positive and negative ${\omega }_x$, respectively. The threshold value for clustering is 0.1. (c) Late stage. Open circles and small spheres represent the skeletons of positive and negative ${\omega }_x$, respectively. The threshold value for clustering is ${\omega }_{x,0}$.

Figure 22

All near-wall skeletons with the magnitude of ${\omega }_x$ greater than ${\omega }_{x,0}$ and their associated off-wall skeletons at $t=t_1$. (a) and (b) show the cases of near-wall vortical regions with positive and negative vorticities, respectively. Open and filled symbols denote the skeletons of the vortical regions with positive and negative ${\omega }_x$, respectively. The color indicates the height from the wall. (a) Near-wall skeletons with ${\omega }_x>{\omega }_{x,0}$. (b) Near-wall skeletons with ${\omega }_x<-{\omega }_{x,0}$.

Figure 23

Heights and ${\omega }_x$ values of the skeletons of near-wall vortical regions along $x$ at $t=t_1$. Only the skeletons of near-wall vortical regions with ${\omega }_x>{\omega }_{x,0}$ and those of the associated off-wall vortical regions are shown. Different skeletons are denoted by different symbols. The same symbols in different colors indicate skeleton pairs. The same symbols in (a) and (b) denote the same skeletons. Red and blue symbols indicate skeletons with ${\omega }_x>{\omega }_{x,0}$ and ${\omega }_x<-{\omega }_{x,0}$, respectively. (a) Skeleton heights along $x$. (b) ${\omega }_x$ values of skeletons along $x$.

Figure 24

Local slopes and products between the local slopes of the skeletons of near-wall vortical regions and the associated off-wall vortical regions at $t=t_1$. Only the skeletons of near-wall vortical regions with ${\omega }_x>{\omega }_{x,0}$ and those of the associated off-wall vortical regions are shown. By $b_1$ and $b_2$, we denote the local slope of a near-wall skeleton and the associated off-wall skeleton in the $x\text{−}y$ plane, respectively. (a) Local slopes of the skeletons of near-wall vortical regions and the associated off-wall vortical regions. Red and blue symbols indicate skeletons with ${\omega }_x>{\omega }_{x,0}$ and ${\omega }_x<-{\omega }_{x,0}$, respectively. (b) Products of the local slopes of the skeletons of near-wall vortical regions and the associated off-wall vortical regions.

Figure 25

Skeleton time surfaces constructed for $t=0-3.2{\delta }_{\text{in}}/c_0.$

Figure 26

Histogram of categories at equilibrium points on $x$ planes closest to the intersections between the $x$ planes and near-wall skeletons at $t=t_1$.

Figure 27

Typical case of a spiral sink, showing two-dimensional velocity vectors within the analysis domain, intersection points (pink circles), and the skeleton of the near-wall vortical region (red) and its associated off-wall vortical region (blue) at $t=t_1$.

Figure 28

Streamwise array of cross-sectional streamlines and velocity vectors obtained from the $y$ and $z$ components, i.e., $(v,w)$, within $x$ planes at $t=t_1$ for the typical case of a spiral sink. Panels (a)–(f) show each streamwise location in the range $x=543.49{\delta }_{\text{in}}-547.05{\delta }_{\text{in}}$. Red and blue points are the trajectories of the skeleton elements on each cross section. Black points are the intersections of the skeletons with each cross section. (a) $x=543.49{\delta }_{\text{in}}$. (b) $x=544.20{\delta }_{\text{in}}$. (c) $x=544.92{\delta }_{\text{in}}$. (d) $x=545.63{\delta }_{\text{in}}$. (e) $x=546.34{\delta }_{\text{in}}$. (f) $x=547.05{\delta }_{\text{in}}$.

Figure 29

Histograms of the number of protoclusters at which each combination of mathematical terms becomes the MC at $t=t_1$. (a) and (b) show the cases for the enstrophy and helicity equations, respectively. (a) Enstrophy equation. (b) Helicity equation.

Figure 30

Paired skeletons for near-wall vortical regions with ${\omega }_x>{\omega }_{x,0}$ and the locations in the near-wall superclusters where each combination of the terms is the MC for the enstrophy equation at $t=t_1$. Panels (a)–(f) show different combinations. (a) Baseline. (b) Term 1. (c) Term 2. (d) Term 5. (e) Terms 2 and 5. (f) Terms 1, 2, and 5.

Figure 31

Paired skeletons for near-wall vortical regions with ${\omega }_x>{\omega }_{x,0}$ and the locations in the near-wall superclusters where each combination of the terms is the MC for the helicity equation at $t=t_1$. (a)–(f) show different combinations. (a) Baseline. (b) Term 1. (c) Term 6. (d) Term 7. (e) Terms 1 and 7. (f) Terms 6 and 8.