# Physical and Mathematical Modeling of the Interaction of Water Droplets and High-Speed Gas Flow
## 2 Research Methodology
### Description of Data Processing Methods  
The proposed method for processing the experiment is aimed to establish physical and mathematical model for predicting the rate of gas-dynamic breakup of droplets in a supersonic flow. Within the framework of the proposed method, the analysis of the change in the countable distribution of droplet diameters between region 1 and region 2. The change in the mass of the liquid due to evaporation was not taken into account since the temperature of the gas flow in the experiment [29] is lower than the boiling point of water. It should be noted that water crystallization was also not observed in the considered experiments. The input parameters for the used mathematical model were the distributions of the droplet number and velocity modulus in measuring regions 1 and 2, in which the parameters were recorded. The minimum-recorded droplet diameter was 5 \(\upmu\)m in the carried-out experiment, therefore, in the mathematical model the calculation was performed for droplet diameters from 5 to 85 \(\upmu\)m. The velocity and temperature of submicron droplets practically coincided with similar parameters of the gas phase and their total mass did not exceed 10% of the mass of the supplied liquid; therefore, the absence of droplets with submicron diameters in calculations can be considered acceptable. During the analysis, the droplets were divided into groups according to fractions. The range of droplet diameters for each fraction was \(\Delta\) = 3 \(\upmu\)m.  
The used algorithm allows predicting the distributions of drop diameter and velocity in region 2 according to the distribution of parameters in region 1. In the case of an appropriate choice of mathematical regularities of gas-dynamic fragmentation of drops, the calculated and experimental distributions of parameters by drops in region 2 will have minimal discrepancies. According to the results of published works [1; 22; 23] it was found out that the important factors affecting the destruction of droplets are: the initial diameter of the droplets, the relative velocity of the droplets, the presence of shock wave in the structure of the gas flow, as well as the physical and thermodynamic parameters of the liquid and gas. To predict the distribution parameters in region 2, we used an explicit iterative method for calculating the droplet parameters for each elementary segment of the path. The calculation method is similar to the method described in [18]. To implement the numerical algorithm the path length L between the two measuring regions was divided into N elementary segments. The values of the gas parameters (velocity, density, pressure, temperature) for the elementary segment j were considered constant and corresponded to the calculated values in the center of the segment obtained as a result of the numerical simulation of the flow described above. It should be emphasized that at the point of intersection of droplet trajectories with the oblique shock wave the functions had a discontinuity of the first kind. It is taken into account in the work that the coefficients of dynamic viscosity \(\mu_{g}\) and thermal conductivity \(\lambda_{g}\) for gas depend on temperature. The coefficient of surface tension of water \(\sigma_{l}\) is constant \(\sigma_{l}\) = 7.2\(\cdot\)10\({}^{-2}\) N/m. This is a consequence of an insignificant change of the droplet temperature in the flow (the droplet temperature in experiments changes by no more than 20 K). The mathematical model makes it possible to introduce the regularitiesof \(\sigma_{l}\) of the temperature of liquid droplets if necessary. In this case, the droplet temperature can be determined according to [18].