# Numerical Modelling of the Multiphase Flow in an Agricultural Hollow Cone Nozzle
## 3 Results and Discussion
Figure 16: (**a**) Distribution of velocity and (**b**) pressure in the horizontal plane F-F.  
Figure 15: (**a**) Distribution of velocity and (**b**) pressure in the horizontal plane E-E. The last of the considered G-G planes (Figure 17) runs just past the orifice outlet. As a result of the locally expanding diameter of the orifice, the velocity decreases to 13-14 m/s. Thus, the vacuum value is reduced to approximately 1 kPa. The velocity distribution is analogous to the distribution inside the orifice. This means that the highest velocity occurs at the perimeter of the jet with a circular cross-section. In the core, there is a gradual decrease in velocity to approximately 4 m/s. The velocity profile shown in Figure 17 confirms the typical behavior of a hollow cone spray, with maximum velocity occurring along the perimeter of the jet and a central low-velocity region. This pattern results from the vortex insert geometry, which forces the liquid into a strong rotational motion along the nozzle walls. In practice, due to air entrainment and turbulent mixing, the central underpressure region is gradually reduced as the spray develops, and the axial velocity may increase in the core of the cone. This recovery is not fully captured by the current VOF model, which focuses on the near-field region. Future coupling with a DPM-based approach will allow a more accurate representation of particle dispersion and core flow reattachment in the far field. The observed distributions of velocity, pressure, and turbulent kinetic energy (TKE) clearly indicate that the geometry of the investigated nozzle--particularly the presence of a ceramic vortex insert and a three-channel flow plate--plays a crucial role in shaping the internal flow. The lack of geometric symmetry, the presence of irregularly shaped channels, and the spiral guidance of the liquid through the swirl chamber promote the generation of strong angular momentum and intense vortex structures. As a result, the liquid jet accelerates more rapidly, and a localized pressure drop--consistent with [PERSON]'s principle--occurs in the outlet flow region. Within the swirl chamber, the pressure gradient becomes intensified, generating a negative pressure reaching as low as \(-13\) kPa. This is caused by centrifugal forces acting on the liquid as it follows spiral trajectories. Such conditions favor the local stretching of the liquid jet and facilitate its disintegration just after exiting the orifice. Additionally, the asymmetric arrangement of the inlet channels leads to a non-uniform velocity distribution along the jet perimeter. The observed velocity and TKE maxima near the outer edge confirm the formation of a characteristic hollow cone pattern, where the highest velocities are concentrated at the periphery of the spray while the central region maintains low pressure and limited flow. Figure 9:0 (**a**) Distribution of velocity and (**b**) pressure in the horizontal plane G-G.  
From the perspective of spray engineering, such a configuration promotes the generation of fine droplets in the outer part of the cone, which can increase leaf surface coverage but, at the same time, creates a greater risk of droplet drift due to wind. In future work, it would be advisable to investigate the impact of changes in the geometry of the feed channels--including their number, angle of attack, and curvature--on the stability of the formed cone, the distribution of TKE, and the timing of spray breakup. Figure 18 presents the pressure distribution inside the nozzle and in its immediate vicinity (at a distance of approximately 2 cm from the exit). A stable liquid flow through the nozzle was achieved after approximately 16 ms. The highest pressure gradient occurs in the chamber formed between the ceramic components: the vortex insert and the stem. As a result of the constriction at the nozzle exit, a rapid drop in pressure to less than ambient pressure is created. The obtained negative pressure was 90 kPa.