Experimental detection of superclusters of water droplets in homogeneous isotropic turbulence

The usual analysis of preferential concentration of particles exploits individual images as shown in fig. 2(a) to identify particle centers and then perform any sort of diagnosis (such as box counting methods, estimates of pair correlation functions, Voronoï diagrams, etc.) [7]. Such an approach has the drawback that the field of view is generally limited to a few centimeters, in order to be able to visualize and distinguish the small particles (which in the experiments considered are typically a few tens micrometers in diameter). As a consequence, it is difficult to derive any large-scale diagnosis with satisfactory statistical convergence over scales approaching or exceeding the integral scale of the carrier turbulence. In the dataset considered here for instance, the visualization window represents a fraction of the integral scale of the flow.

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Here, we propose to take benefit of high-speed imaging acquisitions and of the high wind speed in wind tunnel experiments (with moderate relative turbulence intensity fluctuations) to reconstruct, unsing a frozen field Taylor hypothesis, a long spatial field of particles distribution.

The idea can be simply understood by first considering that images would be acquired with a linear camera whose pixel line is oriented perpendicular to the main stream velocity U. If the repetition rate of the acquisition is high enough, the linear camera allows to reconstruct a global image of the field of particles as it is advected by the mean stream U. In the context of a Taylor hypothesis, this is equivalent to imagine the field to be frozen and the linear camera to be advected at constant velocity U (which is nothing but the principle of a modern scanner). The "frozen" field approximation requires the fluctuations of particles motion to be much lower than the mean stream velocity U. This is generally written in terms of the turbulence intensity $\tau_{\textit{turb}}=\sigma_u/U$, with $\sigma_u$ the standard deviation of turbulent fluctuations, as $\tau_{\textit{turb}}\ll 1$. For turbulence generated with classic passive grids, the turbulence intensity is typically of the order of 3% $(\tau_{\textit{turb}}\sim0.03)$, hence the hypothesis is fully justified. For active grid generated turbulence, the turbulence intensity is higher and close to 20% $(\tau_{\textit{turb}}\sim0.2)$, but existing studies suggest that the Taylor hypothesis still applies [8–10]. In the present case, the turbulence intensity to be considered is $\tau_{turb}^p=\sigma_v/U$, where $\sigma_v$ represents the standard deviation of particles fluctuating velocity. Particles inertia is generally assumed to reduce fluctuations $(\tau_{\textit{turb}}^p < \tau_{\textit{turb}})$, although recent studies suggest that the fluctuation rate of weakly inertial particles may be comparable or even larger than that of the carrier fluid [11–13]. Other studies show however that this situation very likely applies to ensemble statistics of particles, whereas taken individually, trajectories of inertial particles do exhibit less fluctuations than the fluid [14]. Altogether, we therefore expect the Taylor hypothesis to be at least equivalently valid for the inertial particles than for the fluid itself. This is also supported by the visual inspection of the movies, showing a persistent spatial distribution of the particles as they are advected accross the measurement volume, and showing that the out-of-plane motion of the particles is weak, particles staying in the plane for several tens of successive frames.

Here, instead of using an actual linear camera (made of just one line of pixels perpendicular to the main stream), we use a narrow vertical band of a few lines of pixels from the images recorded with a high-speed camera, and previously studied in [4,5]. Note that a band of a few pixels width is required considering that, for the present experimental conditions, the typical streamwise displacement of a particle between 2 successive images is a few pixels (around 7 for the highest wind speed explored). The width of the band used to reconstruct the spatial field is therefore chosen to be twice the mean streamwise displacement between two successive frames ($2 U/F_s$ converted into pixels given the optical magnification), hence at most $N_{px}=14$. This ensures that nearly no particle is lost in the reconstruction process.

The reconstruction is then obtained by stitching the band of N_px pixel lines time step after time step. Using a band of N_px pixel lines, introduces a typical error (compared to the ideal linear camera case, with $N_{px}=1$) on particles pixel position on the reconstructed image of the order of $\delta_{px}\sim N_{px}\tau_{\textit{turb}}^p$ (which is the typical fluctuating displacement of a particle as it crosses the pixel band). Therefore, assuming 20% of velocity fluctuations, the typical position error is at most $\delta_{px}\sim 2.8\ \text{pixels}$ in the worst case $(N_{px}=14)$. Note also that doublets may appear when stitching successive pixel bands, as slower particles may for instance appear once at the beginning of the band in one image and then at the end of the band in the next time step. When it comes to investigate clustering properties, these doublets may introduce artificially small inter-particle distances, hence biasing the clustering diagnoses. The typical separation between particles in such doublets, again of the order of $N_{px}\tau_{\textit{turb}}^p$, remains however smaller than the typical size of droplet images, of the order of 3 pixels, which anyhow limits the intrinsic resolvable separation between classically detected particles in the images. Spurious doublets can therefore be straightforwardly and robustly handled by replacing, in the reconstructed field, pairs of particles closer than a few pixels (we use a threshold of 3 pixels) by one single particle at the center of mass of the doublet.

Figure 2 shows how this procedure is effectively applied. Figures 2(a), (b) show two successive raw images at some time steps t₀ and $t_0+\delta t$ (with $\delta t = 1/F_s$). The green rectangles schematize the pixel band extracted at each time step and stitched together in fig. 2(c). The extracted band is taken at the center of the images, for optimal illumination conditions. Repeating this procedure for all the acquisition times, a long spatial field is obtained (fig. 2(c)). This large-scale field makes it possible to investigate clustering properties at larger scales than those previously accessible by simply looking at the measurement volume, opening new possibilities of analysis. We discuss here one of these possibilities, which addresses the question of superclustering (i.e. the eventual emergence of clusters of clusters). As will be emphasized below, although the vertical dimension of the reconstructed field (which remains the same as that of original images) still limits the maximum size of detectable structures, the extended horizontal dimension significantly improves the statistical detection of such large structures.

In order to validate the reconstruction procedure of the large-scale particles field, we first compare the statistical properties of simple clustering obtained from the new reconstructed field with previous results from the standard analysis of raw images. As in [5,7] we use Voronoï analysis for the clustering diagnosis and characterization. Figure 3 shows results for the probability distribution function (PDF) of normalized Voronoï areas ($\cal{V=A/\left<A\right>}$, where $\cal{A}$ is the Voronoï cell area), for the experiment at $R_{\lambda}=330$ obtained from the new analysis (blue line) and from the standard analysis from single images (black line). The agreement is almost perfect. The new approach reproduces the same deviation from a random Poisson process (RPP) in fig. 3(a), evidence of clustering and the same quasi-lognormal behavior (fig. 3(b)) evidenced in previous works. A weak deviation, probably due to a few residual doublets in the reconstructed field, can only be seen for the smallest Voronoï areas in fig. 3(a). We have also checked that the properties of clusters and voids are robust in both analysis. Clusters are defined as proposed in [6]. The 2 intersections of the PDF of ${\cal{V}}_p$ with the RPP PDF near the center define 2 thresholds (fig. 3(a)): cells with normalized area ${\cal{V}}_p<{\cal{V}}_p^c$ are defined as belonging to clusters, while cells with ${\cal{V}}_p>{\cal{V}}_p^v$ are defined as belonging to voids. Clusters are then obtained as the ensembles of connected regions with ${\cal{V}}_p<{\cal{V}}_p^c$.

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Figure 4 shows, for all the carried experiments, the properties of Voronoï cells (figs. 4(a), (b)) and clusters geometry (figs. 4(c), (d)) obtained from the new analysis. Overall, these results are in very good agreement with those in the previous work using a standard analysis of images in [5]. Figures 4(a), (b) confirm, with the new approach, the diagnosis of clustering for all the datasets, with a robust quasi-lognormal distribution of Voronoï areas. The inset in fig. 4(b) shows the standard deviation $\sigma_{{\cal{V}}_p}$ of normalized Voronoï areas, whose departure from the RPP value $\sigma_{{\cal{V}}_p}^{RPP}=0.53$ quantifies the level of clustering. The average value of $\sigma_{{\cal{V}}_p}\simeq 1.2$ from the new analysis is slightly larger, but still in good agreement with the value of 1.05 reported in [5]. PDFs of clusters area (fig. 4(c)) show a similar collapse as in [5] when normalized by the mean, with a clear peak around $\frac{A_p^c}{\langle A_p^c\rangle}\sim 0.2$, although some scatter is visible for the tails of smallest areas. Similarly, fig. 4(d) shows cluster perimeters as a function of the squared root of its area, with a fractional behaviour of the exponent, identical to what was obtained with the classical analysis of images, evidencing the fractal nature of clusters. Altogether, these agreements validate the reconstruction procedure of the large-scale particle fields.

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We address the question of the clustering properties of clusters themselves. This is made possible from the new large-scale field reconstructed, which includes several tens of thousands of clusters, therefore allowing to study the collective properties of clusters.

The procedure to identify these structures is conceptually very simple: using the large-scale reconstructed fields, clusters are identified (fig. 2(e)) following the same thresholding procedure on normalized Voronoï areas PDF previsouly described. Then, the centres of clusters are defined as the center of mass of the constituting particles, weighted by the local concentration (which for each particle is given by the inverse of its Voronoï area), as represented in fig. 2(e). Once the centers of mass of the clusters are identified, Voronoï tessellations of cluster centers are computed (fig. 2(f)). We then compute the statistics of the areas of the resulting Voronoï cells, to diagnose (in the same way as for the particle themselves) the eventual presence of clustering for the center of clusters, which we call superclustering.

Figures 5(a), (b) represent the PDFs of ${\cal{V}}_{c}$ (the normalized Voronoï areas of cluster centers) and the PDFs of $\log{\cal{V}}_{c}$, centered and reduced, for all the studied parameters. The PDF of ${\cal{V}}_{c}$ shows a clear departure from a random distribution, hence evidencing the presence of clustering of clusters. The inset of fig. 5(b) shows the dependence of the standard deviation of normalized Voronoï area, $\sigma_{{\cal{V}}_c}$, with particles Stokes number (which we recall to be here directly related to the Reynolds number of the flow, as $St=\tau_p/\tau_\eta$ only varies because of the change in $\tau_\eta$ as $Re_\lambda$ is varied). We find that $\sigma_{{\cal{V}}_{c}}\simeq 0.8$ remains almost constant for all sets (inset in fig. 5(b)). It is larger than the value of 0.53 expected for a random distribution (confirming the presence of clustering) and lower than the value obtained for the standard deviation of Voronoï areas of particles. The intensity of clustering, quantified by $\sigma_{{\cal{V}}_{c}}$ therefore diminishes when compared with the value of $\sigma_{{\cal{V}}_{c}}\simeq 1.2$, which shows that superclustering (clustering of clusters) is less pronounced than simple clustering (clustering of particles). Interestingly, fig. 5(b) shows that superclustering maintains a quasi-lognormal distribution extremely similar to that observed for clusters. This quasi-lognormal behavior for Voronoï tesselations of particles has not been explained yet to our knowledge. We show here that this behavior is robust and prevails at the cluster level.

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From fig. 5(a) we determine the threshold ${\cal V}_c^c$ and define superclusters as connected regions of Voronoï cells such that ${\cal V}_c<{\cal V}_c^c$ (similarly, we can define supervoids as connected regions of Voronoï cells such that ${\cal V}_c>{\cal V}_c^v$, although these will not be discussed here). Figure 2(g) shows an example of such superclusters. Figure 5(c) shows that the PDF of the area of superclusters $A_c^c$ has a shape almost identical to the PDF of $A_p^c$ for standard clusters (fig. 4(c)), with the same peak at $\frac{A_c^c}{\langle A_c^c\rangle}\sim 0.2$. Besides, fig. 5(d) shows that superclusters have similar fractal trends than normal clusters. The mean area of superclusters is however (and naturally) larger than that of clusters, as shown in the inset of fig. 5(c): while the mean cluster size exhibits a weak dependence on experimental conditions, with a typical size of the order of 15η, the average size of superclusters increases markedly with particles Stokes number, reaching values as large as 100η (or 0.1L) for the experiments at the highest mean wind speed (corresponding to the highest values of St and $Re_\lambda$ investigated). It is important to note that superclusters can therefore have dimensions comparable to the field of view of original images $(0.3\text{-}0.5L)$, which emphasizes the importance of the proposed large-scale field reconstruction for the study of this phenomenon. To illustrate this, we have repeated the analysis of superclusters using the original images for $Re_\lambda=330$. The resulting values of $\sigma_{{\cal{V}}_c}$ and $\langle A_c^c\rangle$ are shown with red markers in the inset of figs. 5(b), (c), showing that the standard analysis understimates both the level of superclustering ($\sigma_{{\cal{V}}_c}$ is very close to the RPP value) and the size of supeclusters. It is also important to note that although the reconstruction extends the streamwise dimension of the field of particles, the analysis is still limited in the vertical direction. It is therefore possible that even larger structures exist. As a matter of fact, we did identify elongated superclusters with a streamwise dimension exceeding 0.5L. This motivates future studies, with a larger vertical extended field of view, either by using an actual linear camera, or simply by rotating the camera sensor by 90°.