# Physical and Mathematical Modeling of the Interaction of Water Droplets and High-Speed Gas Flow
## 1 Introduction  
A large number of studies devoted to the investigation and modeling of two-phase flows are being carried out in the world at the present time. The interaction of high-speed gas flows with the liquid phase has great importance in many areas of industry such as refining and transportation of oil products, power engineering, testing of aero and hydrodynamic technology, aerospace technology and others. The intensification of gas-dynamic breakup of liquid droplets is an important direction in the effective usage of two-phase flows in engineering. A detailed physical and mathematical modeling of the spray and motion of liquid droplets in the gas phase makes it possible to increase the accuracy of predicting the efficiency of the working process in the combustion chambers of power plants, to evaluate the characteristics of two-phase mixing in heat exchangers, technological plants and test benches. Modeling of interaction and mutual movement of different phases is an important aspect of studying two phase flows. Thus, numerical modeling is impossible or its result will significantly differ from experimental research without proper models of gas-dynamic breakup. There are many works focused on the computational and experimental study of individual components of the processes of liquid droplet gas-dynamic breakup when these droplets move in a gas flow. In works [1; 2; 3; 4; 5; 6; 7; 8; 9] studies of gas-dynamic breakup of isolated droplets are presented whereas in practice flows with a high content of droplets that can interact with each other are more common. In this case, the droplets can interact with each other. It can adjust the intensity of gas-dynamic fragmentation. Some regularities of gas-dynamic breakup obtained in the course of mathematical modeling of droplets are described in [10; 11]. However, analyzing the analytical and computational methods presented in [10] it can be assumed that they are limited in application due to applied assumptions, which do not take account of the features of the unsteady flow pattern, time-varying degrees of deformation and acceleration of the droplet. The description of disruptive mechanisms presented in [11] requires additional validation and it is associated with labor-intensive calculations in each individual case, which reduces the demand for this approach in solving of practical problems. At the moment, there are multiple data that the nature of the gas-dynamic droplet breakup is characterized mainly by the Weber, Laplace (or Onezorge), Mach and Reynolds numbers [5; 12]:  
\[Re = \frac{\rho_{g}\cdot D_{32}\cdot\left(V_{g}-V_{l}\right)}{\mu_{g}} \tag{1}\] \[We = \frac{\rho_{g}\cdot D_{32}\cdot\left(V_{g}-V_{l}\right)^{2}}{ \sigma_{l}}\] (2) \[La = \frac{1}{Oh^{2}}\ =\ \frac{Re^{2}}{We} \tag{3}\]  
where \(\rho_{g}\)--gas density; \(D_{32}\)--the average volumetric surface (Sauter's) diameter of the droplet; \(V_{g}\)--gas velocity; \(V_{l}\)--liquid droplet velocity; \(\mu_{g}\)--gas kinematic viscosity; \(\sigma_{l}\)--surface tension coefficient. Hereinafter, the index \"\(g\)\"--corresponds to the parameters of the gas, and \"\(l\)\"--to the parameters of the liquid. The secondary disintegration of the jet, in which gasdynamic breakup of drops occurs, appears due to the fragmentation of liquid droplets due to aerodynamic and hydraulic forces, as well as inertial forces. It is noted, that at low relative velocities of the droplets, their shape is close to spherical and practically no breakup occurs. With an increase in the relative velocity, the intensity of mass loss due to gas-dynamic fragmentation increases. Based on a large amount of scattered experimental data, various authors have made assumptions about the occurrence of gas-dynamic breakup mechanisms and the corresponding prevailing force that affects the character of droplet disintegration. The literature describes several classifications that determine the types of droplet fragmentation. Some generalization of some researches was carried out in [13; 14; 15; 16].