# Physical and Mathematical Modeling of the Interaction of Water Droplets and High-Speed Gas Flow
## 2 Research Methodology
### Numerical Simulation of the Gas Phase  
To determine the parameters of the gas phase along the droplet trajectory a numerical simulation of the flow in a model channel was carried out for the three considered nozzles.  
The calculation was carried out in the three-dimensional setting using a program based on solving the Navier-Stokes equations by the finite volume method. The research was carried out within the framework of the RANS approach using the \(\gamma\)-Re\({}_{\mathrm{e}}\) SST turbulence model [32]. Gas properties correspond to air and depend on temperature and pressure. It should be noted that the calculation was performed without the presence of liquid droplets, which can change the flow structure and the distribution of parameters.  
The unstructured mesh with a prismatic layer consisting of five cells was used. Along the oblique shock wave, as well as along the trajectory of the droplets, the mesh was condensed to the characteristic cubic cell size of 1 mm. The fluxes were calculated using the implicit AUSM scheme [33] of the second order of accuracy in space. The boundary conditions at the inlet are the total pressure of 0.1 MPa and the stagnation temperature of 300 K, at the outlet--the static pressure of 0.002 MPa. It was assumed that the walls of the channel and nozzle are adiabatic.  
Figure 3 shows the distribution of Mach numbers for three nozzles without and with wedge-shaped insert. Figure 3a shows measuring regions 1 and 2, for which droplet distributions were recorded in the experiment.  
Figure 3: Mach number fields without (left) and with (right) oblique shock wave. (**a**) nozzle 1, (**b**) nozzle 2, (**c**) nozzle 3.  
Based on the results of calculations several dependences were obtained. These regularities describe the change of the velocity \(\mathrm{V_{g}}\) and density \(\rho_{g}\) of the gas along the trajectories of the droplets (Figure 4). The velocity and density functions are approximated by linear regularities, equations of which are shown in the graphs. The function for the entire breakup area for calculations without wedge corresponds to the blue graph. The piecewise-linear regularities corresponding to the blue curve before the shock wave and the orange one after are used for the calculation with a wedge. The vertical red line shows the shock wave location.  
It should be noted that gradual increase in the gas velocity along the droplet trajectories is a consequence of the presence of a boundary layer formed at the central aerodynamic  
Figure 4: Distribution of velocity (left) and density (right) of the gas phase along the trajectory of the droplets: blue lines-parameters without shock wave; red lines-parameters behind the oblique shock wave. (**a**) nozzle 1, (**b**) nozzle 2, (**c**) nozzle 3.  
nozzle. As it moves the trace from the boundary layer is blurred and the gas is accelerated to the value close to the average flow velocity. This is also the reason of altering flux density.  
An oblique shock wave results in increasing of static pressure and density as the gas velocity decreases. That is why a linear distribution function of parameters can be used in regimes without shock wave while shock wave leads to the necessity of using a piecewise-specified function with a discontinuity at the point of intersection of droplet trajectories with shock wave. The change of velocity at the point where the droplet trajectory is intersected by an oblique shock wave can be about 100 m/s, and the gas density can be about 0.1 kg/m\({}^{3}\). The coordinate of the intersection of the droplet and shock wave trajectories for different Mach numbers is in the range from 0.01 to 0.04 m.