# Numerical Modelling of the Multiphase Flow in an Agricultural Hollow Cone Nozzle
## 1 Introduction
Experimental or numerical methods can be used to analyze the fluid stream flowing out of the nozzle. Empirical studies utilize tools such as high-speed cameras, laser diffractometers, particle image velocimetry (PIV) equipment [7], and laser doppler anemometry (LDA) apparatus [8]. Droplet sizes and velocities for the three agricultural nozzles were measured using laser diffraction as part of the research presented at [9]. The atomization process in different types of agriculture nozzles using a PIV was analyzed by [10]. This method is also used in the analysis of aerodynamic phenomena [11]. The numerical studies are based on computational fluid mechanics using flow equation modeling and a turbulence submodel. The Reynolds-averaged Navier-Stokes (RANS) equations are usually used because of the speed of the results and satisfactory accuracy [12]. Computationally demanding large eddy simulation (LES) and detached eddy simulation (DES) models are used for modeling complex turbulent structures. A significant advantage of the numerical method is the ability to quickly test multiple nozzle configurations and the low cost of conducting tests compared to the experimental approach. Due to the above-mentioned advantages, the CFD method finds particular application in research related to aviation [13; 14] or the navy [15]. The results of the nozzle simulation tests can be verified through the use of test stand examinations [16]. The CFD method can also be used for modeling spraying stations. An example is presented in the study [17] that analyzed the design of an axial fan used in an agricultural spraying system. The effect of aircraft speed and spray nozzle orientation on droplet distribution at a given height was analyzed by [18]. Based on the calculation results, it was possible to characterize the spraying process and determine its efficiency. A three-dimensional CFD model for evaluating a boom sprayer for use in orchards to reduce droplet drift was presented in [19]. A Lagrangian multiphase flow model was used to track the trajectory of droplets, combined with spray models and accounting for velocity variations at the fan outlet. Spray modeling utilizes two-phase flows, where the liquid is injected into a domain filled with air. A popular method used to model such phenomena is the volume of fluid (VOF) method. The method is widely used, for example, in the modeling of water discharge from a firefighting aircraft [20] or in the analysis of cavitation in orifice flow [21]. The VOF model has also been applied to simulate the interaction between the gas phase (air) and the liquid phase (water) in a multi-fluid swirling mixing atomizer [22], as well as to investigate the influence of geometrical modifications in pneumatic spray nozzles on droplet size distribution [23]. In Ref. [24], the effect of pulsing frequency on spray characteristics of an agricultural nozzle was studied using the VOF approach. The method can also be coupled with user-defined functions (UDFs) for numerical simulations of the trajectories of electrostatically charged droplets [25]. An alternative approach is the discrete phase model (DPM) method. This is an advanced numerical technique that allows for a simulation of multiphase flows in which one of the phases is in the form of discrete particles. These particles can be very small (e.g., in the form of droplets) or larger (e.g., in the form of solid particles). In the DPM method, the dispersed phase is treated as a collection of discrete particles moving within a continuous medium (continuous phase). Each particle is assigned individual properties, such as mass, diameter, and velocity. Particles interact with a continuous medium through aerodynamic forces, gravitational forces, or lift forces. A continuous medium, on the other hand, influences the movement of the particles by modifying the velocity and pressure fields. For modeling the phenomena of breakup and dispersion of droplets in two-phase flow, it is possible to use both methods mentioned above (VOF and DPM) [26].