# Study of Deformation and Breakup of Submillimeter Droplets' Spray in a Supersonic Nozzle Flow
## 3 Results
### Droplets Size, Shape Oscillations, and Breakup  
Over 600 pairs of shadow images with different interframe delay were acquired in the area from \(x=-15\) mm (upstream) to \(x=+3\) mm (downstream) with the distance of 3 mm between the centers of the visualization regions. The average size and shape of droplets were evaluated from shadow images (see Figure 6). Statics of about \(1.5\times 10^{4}\) droplets were collected for each visualization region. Only droplets in size range of 15-60 \(\upmu\)m that retained spherical or elliptical shape (up to \(d_{max}d=1.8\)) and did not exhibit breakup were taken into account. An average droplet's size, that is an average diameter of a sphere of equal volume:  
\[\langle d\rangle=\langle\sqrt[3]{d_{x}d_{y}^{2}}\rangle \tag{9}\]  
where \(d_{x}\) is a streamwise ellipse dimension and \(d_{y}\)--transversal ellipse dimension, was in the range of 20-24 \(\upmu\)m in the whole measurement area. Most of the droplets (from 95 to 99% in different regions) were in a range of sizes from 15 to 39 \(\upmu\)m. The plots in Figure 6 show that the average droplet size \(<\)\(d\)\(>\) and size variance \(\delta_{d}\) gradually drop along the axis until droplets reach the SW front position, indicating that larger droplets tend to break up. A small increase in \(<\)\(d\)\(>\) and \(\delta_{d}\) immediately after the SW front is likely due to the coalescence of decelerating droplets. Although described variations are almost within the range of single droplet size random measurement error (about \(\pm 1.7\)\(\upmu\)m), for the ensemble-averaged data based on thousands of droplets, the error is essentially negligible. The plot of the elongation defined as  
\[e_{d}=d_{max}/d \tag{10}\]  
indicates the intensity of the aerodynamic load and conforms to the Weber number profile (see Figure 5). Larger droplets exhibit a larger variance of the elongation \(\delta_{e}\) (shown as error bars in the plot) and stronger deformation. Elongation of droplets appears to be insufficient for the transition to breakup if compared with the estimation provided by the Equation (5), which yields a critical elongation \(d_{max\_c}\) of 1.6-1.65 for the transitional _We_ of 10-12. At the same time, some of the droplets in the size range of 45-51 should be close to the transition to the breakup, taking into account the variance of the elongation. Data for the region located at \(x=-6\) stands out in both plots shown in Figure 6, while there is no \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Droplet Size** & **Maximum Weber** & **Ohnesorge** & **t\({}^{\bullet}\)[s]** \\
**[\(\upmu\)m]** & **Number (\(W_{max}\))** & **Number (\(Oh\))** & **t\({}^{\bullet}\)[s]** \\ \hline
15 & 6.6 & \(2.8\times 10^{-2}\) & \(6.1\times 10^{-6}\) \\
30 & 13.3 & \(2\times 10^{-2}\) & \(12.2\times 10^{-6}\) \\
50 & 22.2 & \(15.6\times 10^{-3}\) & \(20.4\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensionless numbers for droplets of different sizes. average velocity or _We_ defect in this region. A possible explanation is the effect of the expansion fan (Prandtl-Meyer expansion) propagating from the nozzle throat and forming a diamond-shape pattern with leading waves intersection around \(x=-6\). This pattern can be noticed in \(V_{y}\) velocity field in Figure 4. Thus, a noticed defect in the plot can be explained by the cease of action of expansion waves, such as wave drag, on droplets. To get more detailed information on droplets evolution along the nozzle, a size distribution for each of the visualization regions was calculated. Size distribution was truncated for \(d<15\) because smaller droplets were poorly resolved, and were not expected to exhibit any breakup or substantial deformation.