# Physical and Mathematical Modeling of the Interaction of Water Droplets and High-Speed Gas Flow
## 3 Analysis of Results  
Analysis of the influence of the coefficients \(\widetilde{K}_{1}\) and \(\widetilde{K}_{2}\) on the characteristics of gas-dynamic breakup and comparison of calculations with experimental data allowed us to select the quantitative values of these coefficients, which for each experiment are shown in Tables 2-4. The tables also show the error value \(Er\), which does not exceed 3.6%. Figure 5: The droplet breakup rate by the longitudinal coordinate. Blue line shows parameters without shock wave; red line—parameters with the shock wave. (a) d = 10 micron, (b) d = 20 micron, (c) d = 30 micron. \begin{table}
\begin{tabular}{c c c c c c} \hline \(p_{\mu}\)**MPa** & **Shock Wave** & \(\widetilde{K}_{1}\)**1/c** & \(\widetilde{K}_{2}\) & \(Er_{r}\)**\%** & \(Re\) \\ \hline
0.6 & + & 5000 & 0.27 & 2.8 & 140 \\
1.1 & + & 4400 & 0.2 & 0.8 & 149 \\
1.7 & + & 3400 & 0.15 & 1.6 & 182 \\
0.6 & \(-\) & 3800 & 0.25 & 2.3 & 158 \\
1.1 & \(-\) & 3500 & 0.22 & 2.0 & 162 \\
1.7 & \(-\) & 3200 & 0.2 & 3.6 & 176 \\ \hline \end{tabular}
\end{table} 
Table 2: Coefficients of the mathematical model at M = 2.0. It is seen that the coefficient \(\widetilde{K}_{1}\) tends to decrease with increasing pressure of liquid injection. It is also worth noting that the smaller the flow Mach numbers the values of \(\widetilde{K}_{1}\) are larger. For example, in Figure 6 shows the experimental and simulated droplet distribution for one of the modes (M = 2.5; \(p_{l}=1.1\) MPa) in the absence and presence of an oblique shock wave. For the same regime in Figure 6 shows a comparison of the droplet velocity distributions. It can be seen from the figures that the calculated and experimental distributions of diameters and velocities coincide practically qualitatively and quantitatively. The coincidence is also preserved in the presence of an oblique shock wave, taken into account in the calculation. In Figures 6 and 7 points mark experimental distributions, lines-calculated ones. The difference between calculations and experiments for this case does not exceed 0.8%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(p_{L}\)**MPa** & **Shock Wave** & \(\widetilde{K}_{1}\)**, 1/c** & \(\widetilde{K}_{2}\) & \(Er\)**, \%** & \(Re\) \\ \hline
0.6 & + & 1500 & 0.23 & 1.0 & 49 \\
1.1 & + & 1100 & 0.19 & 0.8 & 53.7 \\
1.7 & + & 850 & 0.13 & 0.9 & 69 \\
0.6 & \(-\) & 600 & 0.29 & 0.7 & 60.2 \\
1.1 & \(-\) & 400 & 0.25 & 0.9 & 64.7 \\
1.7 & \(-\) & 300 & 0.21 & 1.0 & 75 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coefficients of the mathematical model at M = 3.0. Figure 6: The example of fractional distribution without shock wave (**a**) and with shock wave (**b**): 1—experimental distribution in area 1; 2—experimental distribution in area 2; 3—calculated distribution in area 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(p_{L}\)**, **MPa** & **Shock Wave** & \(\widetilde{K}_{1}\)**, 1/c** & \(\widetilde{K}_{2}\) & \(Er\)**, \%** & \(Re\) \\ \hline
0.6 & + & 2400 & 0.22 & 1.0 & 76 \\
1.1 & + & 1250 & 0.2 & 0.8 & 87 \\
1.7 & + & 600 & 0.15 & 1.5 & 110 \\
0.6 & \(-\) & 800 & 0.22 & 1.2 & 96 \\
1.1 & \(-\) & 600 & 0.2 & 0.5 & 103 \\
1.7 & \(-\) & 400 & 0.18 & 1.1 & 116 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coefficients of the mathematical model at M = 2.5. The dependences of the coefficients \(\widetilde{K}_{1}\) and \(\widetilde{K}_{2}\) on the trajectory-averaged Re number for drops were obtained. While maintaining the Mach number, an increase in the Reynolds number is achieved by increasing the injection pressure of the liquid. The Re number was averaged taking into account the mass fraction of each fraction. Figure 8 shows that the obtained distributions of the \(\widetilde{K}_{1}\) coefficient values have a dependence close to exponential. This tendency manifested itself regardless of the presence or absence of shock wave. With an increase in the Mach number of the flow, the coefficient decreased.