# Numerical Modelling of the Multiphase Flow in an Agricultural Hollow Cone Nozzle
## 2 Research Object and Methodology
Figure 5: Nozzle geometry obtained in Catia V5 software. achieved did not significantly impact the results, and further increasing it would only prolong the computation time. In regions with high-velocity gradients (e.g., in the swirl chamber and just beyond the outlet), further mesh refinement could improve the accuracy of phase interface representation and local turbulence; however, this would come at the cost of increased computational time. Figure 7 presents the computational mesh in two views. In the first view, the mesh of the entire computational domain is visible, while the second view shows the level of element densification, mainly within the internal volume of the nozzle. More complex shapes, particularly the internal volume of the atomizer, were meshed using tetrahedral elements. However, to ensure better control over the mesh density at the atomizer nozzle outlet, a subdomain in the form of a cylinder was created, extending from the atomizer to the end of the computational domain. Within this volume, a mesh based on hexahedral elements was applied. This is visible in Figure 7a as a refined core region of the domain. The horizontal cross-sections of the examined nozzle, for which the calculated parameter values were analyzed, are presented in Figure 8. The cutting planes, labeled sequentially  
Figure 6: The computational domain: (**a**) perspective view and (**b**) view in the \(YZ\) plane. Figure 7: The computational mesh created in Ansys Meshing software: (**a**) general view and (**b**) close-up on the sprayer; (**c**) influence of the mesh size on the observed average velocity across the nozzle outlet. as A-A, B-B, etc., were positioned at characteristic points of cross-sectional changes in the flow channel. The F-F plane is located in the exit orifice, with the smallest diameter of 1.18 mm. Defining the planes in this way was intended to analyze the average flow parameters in the individual nozzle sections. Selected parameters for the developed 3D computational model are shown in Table 1. The calculations were performed in transient mode, indicating the variability of the analyzed parameters over time. \begin{table}
\begin{tabular}{c c} \hline Boundary conditions \\ \hline Fluids & Water (inlet), air (domain) \\ Inlet pressure & 500,000 Pa (relative to ambient pressure) \\ Ambient pressure & 101,325 Pa \\ Mass flow & 0.010 kg/s \\ Water density & 998.2 kg/m\({}^{3}\) \\ Water viscosity & 1.003 \(\times\) 10\({}^{-3}\) kg\({}^{1}\) m\({}^{-1}\) s\({}^{-1}\) \\ Air density & 1.225 kg/m\({}^{3}\) \\ Air viscosity & 1.789 \(\times\) 10\({}^{-5}\) kg\({}^{1}\) m\({}^{-1}\) s\({}^{-1}\) \\ \hline Physics models \\ \hline Space & Three-dimensional \\ Time & Transient \\ Multiphase model & Volume of fluid (VOF) \\ Turbulence model & \(k\)-\(\omega\) SST \\ Gravity, pressure gradient force, two-way \\ Optional models & turbulence coupling, stochastic collision, \\ coalescence, breakup \\ \hline Eulerian phases \\ \hline Water & Liquid, constant density \\ Air & Gas, constant density \\ \hline Solution methods \\ \hline Pressure–velocity coupling & SIMPLE scheme \\ \hline Spatial discretization \\ \hline Gradient & Least squares cell-based \\ Pressure & PRESTOI \\ Momentum & Second Order Upwind \\ Volume fraction & Geo-Reconstruct \\ Turbulent kinetic energy & First Order Upwind \\ Specific dissipation rate & First Order Upwind \\ \hline \end{tabular}
\end{table} 
Table 1: Selected parameters of the numerical model. Figure 8: Vertical cross-section of the tested nozzle with marked horizontal sections in which the values of the calculated parameters were analyzed. The model considers a gravitational acceleration of 9.81 m/s\({}^{2}\) directed along the vertical \(z\)-axis. The calculation solver was set as pressure-based due to the incompressible flow. The First Order Upwind scheme was used for the discretization of turbulent quantities (turbulence kinetic energy and specific dissipation rate) due to its robustness and stable convergence behavior.