# Numerical Modelling of the Multiphase Flow in an Agricultural Hollow Cone Nozzle
## 2 Research Object and Methodology
However, it is acknowledged that this choice introduces increased numerical diffusion, which may affect the resolution of turbulent structures and the accuracy of local turbulence parameters. The use of higher-order schemes, such as Second Order Upwind, will be considered in future studies to improve prediction quality in regions of high gradient flow. The multiphase model was the VOF method, with volume fraction parameters specified as explicit. It involves tracking the fraction of the volume of each phase in each computational cell. The fraction takes values from 0 to 1, where 0 means that the cell is completely filled with the first phase and 1 with the second phase. Intermediate values indicate that there is an interface between phases in the cell. For each phase, the transport equation for the volume fraction is solved. This equation characterizes the change in time and space of the proportion of a given phase in each cell. In the VOF model, it is crucial to track the interfaces between the defined phases. This is done by solving the continuity equation that considers the volume fraction of the phases. For the \(j\)-th phase, the equation takes the Equation (1). On the other hand, the volume fraction equation for the primary phase is calculated based on Equation (2) [46]. \[\frac{1}{\rho_{j}}\bigg{[}\frac{\partial}{\partial t}\big{(}\alpha_{j}\rho_{j} \big{)}+\
abla\cdot\big{(}\alpha_{j}\rho_{j}\overline{\
u}_{j}\big{)}\bigg{]}= S_{\alpha_{j}}+\sum_{i=1}^{j}\big{(}in_{ij}-\dot{m}_{ji}\big{)} \tag{1}\]  
where \(\rho_{j}\)--density of the \(j\)-th phase; \(\alpha_{j}\)--fluid's volume fraction of the \(j\)-th phase; \(\overline{\
u}_{j}\)--\(j\)-th phase velocity; \(S_{\alpha_{j}}\)--source term, the default value is 0; \(\dot{m}_{ij}\)--mass transfer from phase \(i\) to phase \(j\); \(\dot{m}_{ji}\)--mass transfer from phase \(j\) to phase \(i\). \[\sum_{i=1}^{j}\alpha_{i}=1 \tag{2}\]  
As mentioned above, an explicit time formulation, which is time-dependent, was adopted in the calculations. The volume fraction is determined in a discrete manner using Equation (3). \[\frac{\alpha_{j}^{n+1}\rho_{j}^{n+1}-\alpha_{j}^{n}\rho_{j}^{n}}{\Delta t}V+ \sum_{f}\Big{(}\rho_{j}U_{f}^{n}\alpha_{j,f}^{n}\Big{)}=\bigg{[}S_{\alpha_{j }}+\sum_{i=1}^{j}\big{(}in_{ij}-\dot{m}_{ji}\big{)}\bigg{]}V \tag{3}\]  
where \(n+1\)--index for new time step; \(n\)--index for previous time step; \(\alpha_{j,f}\)--face value of the \(j\)-th volume fraction; \(U_{f}\)--volume flux through the face, based on normal velocity; \(V\)--cell volume. A sharp interface was chosen, which corresponds to models with a clearly defined interface between the phases. Two phases were defined: water and air. The surface tension coefficient for the interaction between the phases was 0.072 N/m, corresponding to the conditions for pure water. The use of pure water as the working fluid in this study was intended to create reference conditions for analyzing the fundamental mechanisms of two-phase flow in the conical nozzle. This approach allowed for minimizing the influence of additional chemical variables, such as adjuvants, which are commonly used in agriculture and, as mentioned in the introduction, significantly affect the surface tension of solutions. A continuum surface force model was chosen, which interprets the surface tension as a continuous, three-dimensional effect at the interface rather than as a boundary value condition at the interface. The inside of the domain was filled with air. A \(k\)-\(\omega\) SST turbulence model was adopted with the production limiter option enabled. At the inlet, a turbulence intensity of 1% and a turbulent viscosity ratio of 5 were set. Pressure-velocity coupling was defined using the SIMPLE algorithm. It uses the relationship between velocity and pressure corrections to enforce mass conservation and obtain a pressure field. For pressure discretization, the PRESTOI scheme was adopted, while the Geo-Reconstruct scheme, which is based on geometric information, was used for interface tracking.