Droplet arrival pattern, clustering and settling velocity in a polydisperse droplet field

Abstract

Understanding the onset of clustering is an essential consideration in many engineering and natural systems. High operating costs and sophisticated methods associated with the existing spatial measurement techniques in clustering analysis call upon the use of low-dimensional measurements. In this experimental study, we study the 1D temporal droplet diameter data in a polydisperse droplet field with a background turbulence, to directly demonstrate a viable relationship between inertial clustering and the droplet arrival pattern. Upon the onset of clustering, a substantial increase is observed in the occurrence of ordinal patterns containing monotonically increasing sub-patterns, where smaller droplets are followed by bigger droplets, and a clear bifurcation in the occurrence of these patterns with downstream location is observed. These trends are understood in terms of droplet settling velocity. The mean flow dominated background turbulence enhances the settling velocity of clustered droplets, which induces a size–velocity correlation of the droplets and, as a result, increases the occurrence of ordinal patterns containing monotonically increasing sub-patterns. This study shows the potential of using low-dimensional measurements for the qualitative understanding of complex flows in different natural and engineering systems.

Appendices

Appendix A: Monte Carlo simulations

In order to verify the accuracy of the methods employed for ordinal pattern analysis, a simple Monte Carlo simulation has been employed. The simulation attempts to replicate the atomization process in sprays, where large ligaments become unstable and break into spherical droplets through a process known as shedding. These droplets are being carried downstream by the flow and contribute to the overall spray pattern. To achieve this, droplets of various sizes ranging from 1 to 120 $\upmu$m are produced within a small two-dimensional rectangle box, where the width of the box is much larger than its height.

For a shedding frequency of 10 Hz, on average, the ligaments formed in the spray break up into spherical droplets at a rate of 10 times per second. The number of droplets generated inside the box for a given time is randomly generated based on the number density. The typical average number density for the hollow cone spray is approximately 2000 cm$^{-3}$. However, to account for the spatial and temporal variations in number density that occur in real sprays, the number density is randomly generated within a range of 1500–2500 cm$^{-3}$. This range ensures that the simulation captures the variability in droplet generation that occurs in actual sprays.

The Monte Carlo simulation includes the initial velocities of droplets based on the predominant size–velocity correlation existing near the spray nozzle. Specifically, the axial velocities $u_i$ are simulated as $u_i=Sd_i+2.5$, where S falls in the range of 0.05$-$0.1. It is assumed that all droplets reach their terminal velocity within a distance of $x= 200$ mm from the nozzle exit, which is in line with the behavior of real sprays, where droplets achieve their terminal velocity within a few millimeters downstream of the nozzle exit.

The lateral velocities of droplets are determined by their diameters, with smaller droplets showing larger lateral deviations compared to the larger droplets. This difference is due to the fact that larger droplets have higher settling velocities, while smaller droplets have lower St values. In this study, droplets with diameters larger than 50 $\upmu$m and smaller than 50 $\mu$m have lateral velocities equal to 10% and 30% of their initial velocity, respectively.

To determine the droplet arrival time series, the droplets are arranged in ascending order based on their arrival time to reach a small square box of side length 0.2 mm located 400 mm downstream of the nozzle exit. The resulting plot of the distribution of grouped ordinal patterns from a single simulation is shown in Fig. 6a. It is worth noting that the value of $n_{\textrm{I}}$ for the non-clustering case is around 26%, which is similar to the non-clustering experimental cases.

The present simulation does not aim to simulate turbulence-induced clustering in a polydisperse droplet field. However, for the same experimental settings, by using 3D Voronoï analysis on tomographic imaging, we observed the formation of clusters with a size of 10$\eta$ at a temporal frequency of approximately 5 Hz, where $\eta$ represents the Kolmogorov length scale (see Table 1). Further analysis indicated that the clustering is inertial in nature, rather than sub-Kolmogorov. Detailed cluster analysis using Voronoï analysis can be found in Shyam Kumar et al. (2023). Moreover, we found that the clustered droplets exhibit enhanced settling compared to individual droplets, as illustrated in Fig. 5. These results support earlier findings on the effect of inertial clustering on droplet settling rates (Aliseda et al. 2002).

In order to include the effect of clustering, small clusters with a size of 10$\eta$ and containing droplets with diameters larger than 70 $\upmu$m are introduced into the simulation with a temporal frequency of approximately 10 Hz, in addition to the droplets generated through shedding. The number of droplets in each cluster is randomly varied between 100 and 200. Each droplet in a cluster is assigned an initial velocity based on its size, and a common settling velocity within the range of 1–2 m/s is also added (Aliseda et al. 2002). To account for the turbulence-induced enhancement of clustering, the deceleration of clustered droplets is delayed, with the droplets only reaching their terminal velocity at $x=300$ mm instead of $x=200$ mm. In the simulation with clustering, the grouped ordinal pattern distribution is plotted in Fig. 6b. The corresponding $n_{\textrm{I}}$ is around 45%. Although there is a significant offset between the experimental and simulation-based $n_{\textrm{I}}$, the trend in $n_{\textrm{I}}$ with clustering is consistent. This offset may be due to the simplification of the real problem, specifically the neglect of the 3D effects and other complex mechanisms present in particle-laden flows.

Multiple simulations are conducted for both the cases (with and without clustering), and the PDFs for $n_{\textrm{I}}$ are plotted in Fig. 6c. The results indicate that the peak value of $n_{\textrm{I}}$ is approximately 24% for the non-clustering simulations and 38% for the clustering simulations. The lack of overlap between the two PDFs highlights the effectiveness of $n_{\textrm{I}}$ as an indicator for identifying clustering in a polydisperse droplet field with a background mean flow dominated turbulence.

Fig. 6

Sample grouped ordinal pattern distribution obtained from the Monte Carlo simulation without clustering (a) and with clustering (b). c PDFs of $n_{\textrm{I}}$ from multiple simulations for cases without (green) and with clustering (red)

In Fig. 7, the effect of clustering formation rate on the occurrence of group I ordinal patterns is demonstrated. With an increase in cluster formation rate, the number of group I patterns is also found to increase.

Fig. 7

PDFs of $n_{\textrm{I}}$ from multiple simulations for cases without (green) and with clustering formation rates of 10 Hz (red), 5 Hz (magenta) and 1 Hz (black). With a decrease in the clustering formation rate, the PDF shifts toward the left, indicating the importance of the clustering formation timescale on the arrival pattern statistics

Appendix B: effect of non-overlapping method

In arrival statistics calculations, the increased relative frequency of group I ordinal patterns could be caused by a higher sampling rate than the signal bandwidth (Little and Kane 2017), which we have taken care by using the same sampling rate for all the acquisitions by varying the optimum threshold voltages in the PDI system. However, along with sampling rate, the approach used for selecting consecutive elements that construct a section decides $n_{\textrm{I}}$. There are two approaches available, namely the overlap and non-overlap methods. In the overlap method, every possible consecutive element of length m is used to construct a section (see Fig. 2). This method is common, and there are overlaps of elements between consecutive sections. In the non-overlap method, the elements of consecutive sections are not allowed to overlap. That is equivalent to using only the mth section of an overlap condition. To estimate the effect of the method used, the grouped ordinal pattern distribution is estimated using the non-overlapping method for the non-clustering and clustering cases at $x = 400$ mm and plotted in Fig. 8. For comparison, the grouped ordinal pattern distribution estimated from the overlapping case is replotted here. For the non-overlap estimation, every 5th section is used from the time series, so that no section will overlap. The small variation in the number of ordinal patterns for the three groups between the overlapping and non-overlapping cases suggests the reliability of the results, irrespective of the method used to construct each section.

Fig. 8

Grouped ordinal pattern distribution estimated using overlapping (in blue) and non-overlapping (in orange) method at $x= 400$ mm for a non-clustering (Exp. No. 3) and b clustering (Exp. No. 12) cases