Data-driven classification of sheared stratified turbulence from experimental shadowgraphs

We propose a dimensionality reduction and unsupervised clustering method for the automatic classification and reduced-order modeling of density-stratified turbulence in laboratory experiments. We apply this method to 113 long shadowgraph movies collected in a “stratified inclined duct” experiment, where turbulence is generated by instabilities arising from a sheared buoyancy-driven counterflow at Reynolds numbers $\mathrm{Re}\approx 300–5000$, tilt angles $\theta =1^{\circ }–6^{\circ }$, and Prandtl number $\mathrm{Pr}\approx 700$. The method automatically detects edges representative of discrete density interfaces, extracts a low-dimensional vector of statistics representative of their morphology, projects these statistics onto a two-dimensional phase space of principal coordinates, and applies a clustering algorithm. Five clusters are detected and interpreted physically based on their typical interface morphology and an examination of representative frames, revealing distinct types of turbulence and mixing: laminarizing, braided, overturning, granular, and unstructured, as well as some intermediate types. The ratio of time spent in each cluster varies gradually across the $(\mathrm{Re},\theta )$ space. At intermediate values of $\mathrm{Re}\theta$, intermittent turbulence cycles between clusters in phase space and reveals at least two distinct routes to stratified turbulence. These insights demonstrate the potential of this method to reveal the underlying physics of complex turbulent systems from large experimental datasets.

Figure 1

Experimental data. (a) Sketch of the setup and shadowgraph measurements used to visualize the density field. (b) The 113 experiments in the space of input parameters $(\mathrm{Re},\theta )$ labeled with their human classified regime. The dashed lines roughly highlight regime transitions. Three arrows indicate the locations of the sample shadowgraph frames $\tilde{I}(x,z)$ presented in (c). The original images are $\approx 3400\times 450$ pixels.

Figure 2

Dimensionality reduction pipeline used to assign a cluster label to each shadowgraph frame. The spaces and successive mappings of the new automated classification approach (second column from left, in black) are compared to the traditional human approach (left, in gray). The bracketed numbers indicate the dimension of each space, highlighting a dramatic reduction in dimensionality.

Figure 3

Sample binary edge frames ($\mathcal{E}$ space), delineating the locations of sharp density interfaces (black), mapped from the shadowgraphs ($\mathcal{S}$ space) of Fig. 1 in the three human-classified regimes: Holmboe (H, top), intermittent (I, middle), and turbulent (T, bottom). Edges are here rendered much thicker than detected for better visualization.

Figure 4

(a) Joint probability density functions (p.d.fs) of the 10 statistics characterizing the morphology of density interfaces [see Eq. (3)]. The example inset (in blue) shows correlations between ${\mu }_r$ (the mean aspect ratio of density interfaces) and ${\sigma }_r$ (the variability in their aspect ratio). (b) Cumulative variance captured by the principal components (or singular values); the red arrow highlights that $k=2$ captures 79% of the total. (c) Coordinates of the first two principal unit vectors $\mathbf{v}$ in terms of the original basis of morphology statistics.

Figure 5

The five distinct clusters (colored, labeled L-G) detected by the OPTICS algorithm, corresponding to local minima in the reachability distance (defined in Appendix pp1-s3). Unclustered data are colored black. The distance monotonically increases above the $y$-axis limit of 0.5 for the last $\approx 1000$ unclustered points.

Figure 6

Clusters (as identified and colored in Fig. 5) plotted in the space of the two dominant principal components of the density interface morphology statistics. The relation between the two principal component coordinates and the 10 morphology statistics is illustrated in Fig. 4.

Figure 7

(a) Density morphology properties [vector $\mathbf{m}$, see Eq. (3)] of cluster centroids (black squares, Fig. 6), expressed as normalized deviations from the total distributions. (b) Histograms of the distributions and the contribution of each cluster in physical, nondimensional space $\mathcal{M}$.

Figure 8

Shadowgraph and corresponding edge images for the centroid of each cluster (colored) and additional examples of frames between clusters. The input parameters $(\mathrm{Re},\theta )$ and time $t$ are indicated in the top-right corner of each frame. Frames are scaled to have equal height, resulting in a range of lengths.

Figure 9

Comparison between the location of clusters in principal component space (approximated by their convex hull perimeter) and the three human-classified regimes (semitransparent colored circles): Holmboe wave (left), Intermittent (center) and Turbulent (right). Insets denote the fraction of frames in each cluster.

Figure 10

Ratio of frames (time spent) in each cluster for each of the 113 experiments organized in parameter space $(\mathrm{Re},\theta )$. Some overlap occurs between bars at neighboring Re values. The top row and right column show overall p.d.f.s. The full time series of 21 representative experiments are shown in Fig. 11.

Figure 11

Temporal dynamics in experiments at increasing Re (from bottom to top) and three tilt angles: $\theta =2^{\circ }$ (left column), $\theta =4^{\circ }$ (middle), and $\theta =6^{\circ }$ (right). Individual frames are marked by transluscent black symbols and cluster boundaries (convex hulls) are denoted in color. Time series indicating which cluster the frames belong to, for $t\ge 100$, are shown by a vertical bar on the right. Unclustered frames are colored based on the nearest cluster. Note the involuntary variations in recording time between experiments.

Figure 12

Detailed transitional states present in the two distinct intermittent cycles identified in Fig. 11 (left and right column, third row). We contrast the G-B-L dynamics (left column) to the O-B-L dynamics (right) during a single cycle. Time series are provided in the top bar of each column (colored by nearest cluster) while frames 1–6 illustrate shadowgraphs at six representative times.