# Physical and Mathematical Modeling of the Interaction of Water Droplets and High-Speed Gas Flow
## 2 Research Methodology
### Description of Data Processing Methods
The calculation of the relative velocity of the drop with a diameter \(d\ V_{rel}=V_{l}-V_{g}\) (here \(V_{g}\) is the gas velocity and \(V_{l}\) is the velocity of the liquid in the coordinate system of the experimental setup) was carried out by integrating the differential equation:  
\[m^{d}\cdot\frac{dV_{l}^{d}}{2}\ =\ C_{x}^{d}\cdot\rho_{g}^{d}\cdot F_{m}^{d} \cdot\frac{V_{rel}^{d}\cdot\left|V_{rel}^{d}\right|}{2} \tag{5}\]  
where \(F_{m}\ =\ \pi\frac{(d)^{2}}{4}\)--drop cross-sectional area; \(C_{x}\) droplet drag coefficient; \(l\)--time. Hereinafter, the index \"\(g\)\"--corresponds to the parameters of the gas, and \"\(l\)\" to the parameters of the liquid. The Reynolds, Weber, and Laplace numbers on an elementary path segment \(j\) for a drop were calculated by the formulas:  
\[We_{j}\ =\ \frac{\left(\rho_{g}\right)_{j}\cdot\left(D_{32}\right)_{j}\cdot \left(V_{g}(dL\cdot j)-\left(V_{l}\right)_{j}\right)^{2}}{\sigma_{l}}, \tag{6}\]  
\[Re_{j}\ =\ \frac{\left(\rho_{g}\right)_{j}\cdot\left(D_{32}\right)_{j}\cdot \left|V_{g}(dL\cdot j)-\left(V_{l}\right)_{j}\right|}{\mu_{g}}, \tag{7}\]  
\[Lp_{j}\ =\ \frac{\left(\rho_{g}\right)_{j}\cdot\sigma_{l}\cdot\left(D_{32} \right)_{j}}{\mu_{g}{}^{2}}. \tag{8}\]  
\(D_{32}\)--the average volumetric surface (Sauter's) diameter of the droplets for the fraction under consideration. To calculate the drag coefficient \(C_{x}\), the modified formula from [18] was proposed:  
\[C_{x}\ =\ \frac{24}{Re_{j}}(1+\frac{3}{16}\cdot Re_{j})^{1.2}\cdot(1+0.25\cdot \frac{We_{j}^{0.1}}{Lp_{j}^{0.04}}). \tag{9}\]  
The first two factors in Equation (9) determine the aerodynamic drag coefficient of spherical droplet under various flow conditions, and the third factor is responsible for the effect of the droplet shape deviation from spherical. According to the modes of the experiments carried out it can be assumed that for the cases under consideration fragmentation occurs according to three mechanisms: bag, shear stripping and vibration. The bag type of breakup is characterized by the detachment of fragments from the edges of the droplet and is realized at Weber numbers \(We<1.7\cdot 10^{4}\cdot\text{Lp}\). In this case, the average diameter of the detached fragments, according to [18], is represented by  
\[d_{1}\ =\ 2.4\cdot\frac{d}{Re^{0.5}}\left(\frac{\mu_{l}}{\mu_{g}}\right)^{0.5} \cdot\left(\frac{\rho_{g}}{\rho_{l}}\right)^{0.25}. \tag{10}\]  
In other cases, a shear stripping breakup is assumed due to the development of the Kelvin-Helmholtz instability. From the crests of disturbances (waves) appearing on the surface of the liquid, the separation of fragments begins, the diameters of which d\({}_{1}\) are estimated by a value proportional to the average wavelength \(\lambda\) of these disturbances:  
\[\lambda\ =\ 9\cdot\pi\cdot\left(\frac{\sigma_{l}}{\rho_{l}}\right)^{0.33} \left[\frac{\mu_{l}}{\beta\cdot\rho_{l}\cdot\left(V_{rel}\right)^{2}}\right]^{ 0.66}. \tag{11}\]  
For Weber numbers \(8\leq We\leq 12\), the transition to a relatively low-speed vibration type of gas-dynamic breakup is accepted [16]. At \(We\leq 8\), it was assumed that droplet fragmentation ceased. Thus, the critical Weber number for all calculations in accordance with the above \(\text{We}_{\text{crit}}=8\). The obtained value of the critical Weber number agrees in order with the work [34] (\(\text{We}_{\text{crit}}=5.9\)), and also agrees with the range \(\text{We}_{\text{crit}}=4.20\) also described in [34]. The breakup mechanisms occurring at high Weber numbers (\(We>350\)) [35] (catastrophic breakup) were not realized within the framework of the experiment. It is assumed that the fraction of droplets with a Weber number less than \(\text{We}_{\text{crit}}\) did not change.