# Modeling the Formation of Urea-Water Sprays from an Air-Assisted Nozzle
## 3 Results and Discussion
### Evaluation of Droplet Velocities
The contour of the simulated turbulence indicates that the value chosen for the \(\text{C}_{\epsilon 1}\) parameter of the turbulence model is suitable. As shown in Figure 7c, the choice of \(\text{C}_{\epsilon 1}\) strongly influences the profile of the turbulence kinetic energy. In addition, the radial components of the droplet velocity fluctuations are compared in Figure 11b. The simulation and the experimental results are represented by the velocity fluctuations of the 5 \(\upmu\)m droplets. It becomes apparent that the velocity fluctuations of the droplets are much lower in the simulation, while the shapes of the two curves are qualitatively in good agreement. Even the position of the maxima is correctly predicted. This may be an indication of an underestimation of the turbulent dispersion effects inside the spray plume. One has also to consider that the information of the radial velocity fluctuations of the gas velocity is not provided by the RANS solution. Therefore, they have to be estimated by the turbulent dispersion model using the modeled turbulence kinetic energy, which will lead to uncertainties. More complex turbulence models like Reynolds stress models could be an interesting alternative for a more detailed modeling of the turbulent dispersion phenomena due to the fact that they are able to consider anisotropic turbulence. However, the turbulent transport inside the spray is a significant issue for further investigations. The Stokes number of the droplets is, as generally discussed in Section 2.2.3, a decisive parameter to evaluate a droplet's response to the turbulent motion of the gas phase. Therefore, the Stokes number of the droplets is analyzed by means of the simulation results at the first two measurement positions downstream of the nozzle. Figure 12a illustrates the Stokes number as a function of the droplet size averaged in radial direction from -3 mm to 3 mm at the first measurement position. The radial positions are chosen due to the fact that there are still small droplets detectable, as previously discussed. The Stokes number increases with an increasing droplet diameter, which is mainly driven by a higher relaxation time due to a higher inertia of bigger droplets. The smallest droplets are in a range of a Stokes number of 2.5 and are, therefore, less affected by the turbulent motions of the gas phase, as expected. One reason for this behavior is the small eddy lifetime at this position. Further downstream the nozzle exit, the Stokes number decreases due to an increasing eddy lifetime, shown in Figure 12b for the second measurement position. At this position, the Stokes numbers of the smallest droplets are lower than unity, resulting in a high impact of the turbulent motion on the droplet trajectories. If the findings of [PERSON] et al. [37] are considered, the Gosman and Ioannides model [31], applied for the modeling of the turbulent dispersion, provides good results in a narrow area of Stokes numbers -2. The model should be able to describe the turbulent dispersion shortly after the first measurement position. In the area upstream the first measurement position, the Stokes numbers are higher, leading to uncertainties in the modeling. One has still to consider that, to finally judge the quality of the selected dispersion model, more detailed information about the initial conditions of the droplets at the nozzle exit is necessary.