Characterizing 1D Inertial Particle Clustering

TL;DR

This work investigates how projecting 3D inertial-particle clustering into 1D/2D measurements bias Voronoï-based statistics in turbulent flows. By combining 2D experimental data with DNS and applying a projection procedure that mimics quasi-Eulerian probes, the authors establish a measurement-window framework to compare observations across platforms. They develop a theoretical cluster-PDF model conditioned on the number of particles per cluster and introduce a N_PC-dependent test to distinguish turbulence-driven clustering from random fluctuations, showing that 2D/3D Voronoï analyses are more robust than 1D in capturing the right-tail behavior. The findings provide practical guidance for 1D measurements, highlighting lower and upper bounds for the measurement window set by the Kolmogorov scale $\eta$ and the integral length scale $\mathcal{L}$, respectively, and offer a pathway to reconcile disparate observations in particle-turbulence interactions.

Abstract

Clustering is an important phenomenon in turbulent flows laden with inertial particles. Although this process has been studied extensively, there are still open questions about both the fundamental physics and the reconciliation of different observations into a coherent quantitative view of this important mechanism for particle-turbulence interaction. In this work, we study the effect of projecting this phenomenon onto 2D and 1D (as usually done in experiments). In particular, the effect of measurement volume in 1D projections on detected cluster properties, such as size or concentration, is explored to provide a method for comparison of published/future observations, from experimental or numerical data. The results demonstrate that, in order to capture accurate values of the mean cluster properties under a wide range of experimental conditions, the measurement volume needs to be larger than the Kolmogorov length scale, and smaller than about ten percent of the integral length scale of the turbulence. This dependency provides the correct scaling to carry out 1D measurements of preferential concentration, considering the turbulence characteristics. It is also critical to disentangle the cluster-characterizing results from random contributions to the cluster statistics, especially in 1D, as the raw probability density function of Voronoi cells does not provide error-free information on the clusters size or local concentration. We propose a methodology to correct for this measurement bias, with an analytical model of the cluster PDF obtained from comparison with a Random Poisson Process probability distribution in 1D, which appears to discard the existence of power laws in the cluster PDF. We develop a new test to discern between turbulence-driven clustering and randomness, that complements the cluster identification algorithm by segregating the number of particles inside each cluster.

Figures (15)

• Figure 1: a) Sketch of the wind tunnel experimental setup. 1, 2, and 3 refer to the locations of the active grid, the injection rack, and the measurement region, respectively. The measuring region downstream distance was taken from the beginning of the injector rack. The shaded region illustrates the extend of the laser sheet. The transverse square cross-section has dimensions of 750 $\times$ 750 mm$^2$. b) Velocity power spectral density for the active grid (AG) $Re_\lambda \approx 250$, and for the open grid (OG) $Re_\lambda \approx 30$. Both spectra were obtained via hot-wire anemometry. c) Droplet Diameter $D_p$ distribution. The symbol ($\circ$) refers to data from sumbekova2017preferential, and the () line refers to a log fit (parameters shown in the plot legend)
• Figure 2: Undimensional Voronoï tessellation (1DVOA). For a given particle position $Z$ with left, and right neighbour particle $Z_L$, and $Z_R$ respectively, the length of the Voronoï cell is given by $L=\vert Z_R-Z_L \vert /2$.
• Figure 3: a) Sketch of the $2D_{EXP}\rightarrow1D_{\perp}$ particles centers projection for an arbitrary image. MWS is the measuring window size, $\hat{y}$ is the randomly generated vertical coordinate of the axis $\gamma$ over which the points are orthogonally projected. b) Sketch of the $3D_{DNS}\rightarrow1D_{\perp}$ particles centers projection for an arbitrary DNS snapshot. MWS is the measuring window size equal to the cylinder diameter, $\hat{y}$ and $\hat{z}$ are the randomly generated coordinates of the axis $\gamma$ onto which the points are orthogonally projected. c) Average number of projected samples per snapshot from 1D sampling ($2D_{EXP}$ or $3D_{DNS} \rightarrow 1D_\perp$), and its dependency with the measuring window size. At very small MWS with respect to $\mathcal{L}$ the average number of samples captured is small, which is directly linked to lack of clustering recently reported Mora2018.
• Figure 4: a) Probability density function plot of 3DVOA for the DNS data bec2010turbulentbec2010intermittency. Following the criterion of Monchaux et al. Monchaux2010, it is clear that the DNS data contains clustering, as $\sigma_\mathcal{V} \approx 0.62>\sigma_{{3D_{RPP}}}\approx 0.42$ is larger than its equivalent one for a 3D RPP distribution. b) PDF plot of ($3D_{DNS} \rightarrow 1D_\perp$) 1DVOA for several MWS.
• Figure 5: a) 1DVOA standard deviation evolution for the datasets used. The larger the concentration, the higher $\sigma_{\mathcal{V}}$ for fixed measuring window size. The peak location follows the relation MWS$_\star \approx \mathcal{L}/10$, where $\mathcal{L}$ the integral length scale of the flow (see table \ref{['tab:par']}). However, its value depends on $Re_\lambda$, as these and previous studies have shown sumbekova2017preferential . b) 1DVOA standard deviation evolution of projections coming from synthetic random 3D data ($3D_{RPP} \rightarrow 1D_\perp$). $N_P$ stands for the number of points inside the 3D domain for 1000 synthetic snapshots.