Optimum air turbulence intensity for polydisperse droplet size growth

The growth of the average size of liquid droplets suspended in a turbulent air flow is of paramount importance in several natural and engineering systems. Here we present an experimental study of the effects of air flow turbulent intensity on the size growth of water droplets in a polydisperse droplet field. For a given initial distribution of droplets in the size range of 0–120 $\mu \mathrm{m}$ diameter, we identify an optimum air turbulent intensity that maximizes the rate at which the average droplet diameter increases with time. The observed trend is understood in terms of droplet collision rate statistics, droplet clustering, and the existence of a crossover diameter, below and above which the number of droplets decreases and increases in time, respectively. We show that the onset of clustering suppresses the intuitive effect of an increase in droplet collision rate with air turbulent intensity, resulting in the existence of an optimum air turbulent intensity that maximizes the average droplet size growth rate due to droplet coalescence. Our results bear consequences for the understanding of warm rain initiation from clouds and the design of engines with improved combustion characteristics.

Figure 1

Schematic of the experimental setup, with all dimensions in millimeters. The $x$ and $y$ axes are along the axial and transverse directions, respectively, with the origin placed at the spray nozzle exit. The region of interest (200 $\mathrm{mm}\le x\le 400$ mm and $-20\mathrm{mm}\le y\le 20\mathrm{mm}$) is indicated by the dashed rectangle.

Figure 2

Near-field conditions with no background flow ($U=0$). (a) Droplet size distribution at $(x,y)=(40,-8)$ and $(40,-16)\mathrm{mm}$. Variation of (b) the mean droplet diameter $d_m$ and (c) the mean droplet axial velocity $u_m$ with the transverse coordinate $y$ at $x=40\mathrm{mm}$.

Figure 3

Far-field conditions with background turbulent flow ($U=0.88\text{m}/\text{s}$, experiment 9 in Table 1). (a) Droplet size distribution at $(x,y)=(200,-8)$, $(200,-16)$, and $(200,-36)$ mm. (b) Variation of the mean droplet diameter $d_m$ with $y$ at different axial locations [see the legend in (d)]. (c) Droplet axial velocity distribution at $(x,y)=(200,-8)$, $(200,-16)$, and $(200,-36)$ mm. (d) Variation of the mean droplet axial velocity $u_m$ with the transverse coordinate $y$ at different $x$ locations in the far field. The legend in (d) is common to both (b) and (d). The vertical dashed lines in (b) and (d) encompass the region of interest $-20\mathrm{mm}\le y\le 20\mathrm{mm}$ from where droplet size growth rates are estimated.

Figure 4

(a) Variation of $I$ and $I_{\text{PDI}}$ with $x$ and (b) mean droplet size variation with $x$ for experiment 9 listed in Table 1. (c) Corresponding variation of $D_m$ with the droplet residence time $t_r$. The dashed lines in (b) and (c) indicate linear fits to the variation of $D_m$ with $x$ and $t_r$, respectively.

Figure 5

(a) Variation of the droplet number density $C$ with $x$ and (b) droplet axial velocity distribution at $x=200$ and 400 mm for experiment 9 listed in Table 1. In (b), for each $x$, the distribution corresponding to all droplets (closed symbols) and the distribution corresponding to droplets of diameter less than 10 $\mu \mathrm{m}$ (open symbols) are shown.

Figure 6

Nonmonotonic variation of droplet size growth rate ($R$ and $R_B$) with turbulent intensity $\mathit{\text{I}}$; $R$ and $R_B$ are calculated using measured droplet velocities and bulk air velocity, respectively, as described in Sec. 2.

Figure 7

(a) Empirically determined, spatially averaged collision rate ${\overline{N}}_c$ varying with turbulent intensity $I$ and droplet diameter $d_i$. The dash-dotted line tracks the location $d^{*}$ where maximum ${\overline{N}}_c$ occurs at a given $I$. (b) Variation of total collision rate $r_{\mathrm{coll}}$ with $I$. (c) Droplet size growth rate $R$ varying with $r_{\mathrm{coll}}$ for $I\le I^{*}$ (closed circles) and $I>I^{*}$ (open circles).

Figure 8

Temporal pair correlation function ($dt=0.1$ ms) calculated at $(x,y)=(300,0)\mathrm{mm}$ in the experiments with (a) $I\le I^{*}$ and (b) $I>I^{*}$. The experiment numbers corresponding to each $I$ value are given in Table 1.

Figure 9

Droplet size distribution at different $x$ for experiment 9. The inset shows a zoomed-in view of the region around the crossover diameter $d_c$.

Figure 10

(a)–(f) Estimating the crossover diameter $d_c$ for different flow conditions. In each plot, the droplet size distribution is shown at five different axial locations: $x=200\mathrm{mm}$ ($\circ$), $x=250\text{mm}$ ($▴$), $x=300\text{mm}$ ($+$), $x=350\text{mm}$ ($·$), and $x=400\text{mm}$ ($•$). The estimated $d_c$ is indicated within every plot, all of which were done with a bin size of $1\mu \mathrm{m}$.

Figure 11

Variation of (a) crossover diameter $d_c$ and (b) its ratio ($d_c/\eta$) with the Kolmogorov length scale $\eta$, with air turbulent intensity $I$. A nonmonotonic behavior is observed with a peak in $d_c$ at $I=11.5\%$.

Figure 12

Variation of turbulent intensities (PDI-based turbulent intensity shown using closed circles and LDV-based turbulent intensity shown using open circles) along $x$ in the measurement region ($200\mathrm{mm}\le x\le 400\mathrm{mm}$) for 12 different flow conditions. The variation of turbulent intensity with $x$ is small, with a relative decay in intensity by approximately $0.8\%$ over 200 mm. Qualitatively similar variation is observed for experiment 1 too. Average values in the range $200\mathrm{mm}\le x\le 400\mathrm{mm}$ are taken as $I$ and $I_{\text{PDI}}$ for all the plots in Sec. 3.

Figure 13

(a)–(e) Representative sequence of images obtained using long-distance microscopy over a $4.5\times 4.5{\text{mm}}^2$ region in an experiment with the same parameters as experiment 9 in Table 1. The time difference between consecutive images is 0.2 ms and a specific droplet coalescence event is highlighted using the dashed box drawn in every image. (f) Individual droplet trajectories obtained from the images shown in (a)–(e), with the topmost circle within each trajectory corresponding to the time instance of (a). Consecutive circles within each trajectory are separated by $0.1\text{ms}$. The two trajectories shown using closed circles correspond to the coalescing droplets highlighted by the dashed boxes in (a)–(e).

Figure 14

Collision rate estimates based on PDI (closed circles) and LDM (open circles) measurements. Error bars for the LDM-based collision rate are based on the standard deviation obtained from five different runs for the same experimental parameters.

Figure 15

Droplet size distribution at different $x$ locations for a bin size of (a) 1 $\mu \mathrm{m}$, (b) $2\mu \mathrm{m}$, and (c) $5\mu \mathrm{m}$ for experiment 5 with $I=10.4\%$. The estimated crossover diameter $d_c$ is indicated within each plot and is not strongly affected by the choice of bin size from $1\mu \mathrm{m}$ to $5\mu \mathrm{m}$. In each plot, the droplet size distribution is shown at five different axial locations: $x=200\mathrm{mm}$ ($\circ$), $x=250\text{mm}$ ($▴$), $x=300\text{mm}$ ($+$), $x=350\text{mm}$ ($·$), and $x=400\text{mm}$ ($•$).