## 2 Experimental and Computational Numerical Methods
### Experimental Setup
#### 2.1.1 Acquisition of Falling Droplets  
The shower head with the nozzle plate was placed in the transparent square tube at 0, 200, and 400 mm, respectively, above the high-speed camera's focus frame so that droplets could be captured at different heights from the nozzle plate head. Distilled water was pumped into the nozzle plate head, generating different sizes of droplets based on the size of the nozzle and flow rate. Droplets were ejected vertically downward. The flow rate for each nozzle plate was selected to ensure that the formation of the droplets occurred in the Rayleigh jet breakup regime. In this regime, the predominant breakup mechanism is the surface tension force [29]. The definition of a certain breakup regime is a combination of different variables, such as the liquid velocity inside the nozzle and the density, viscosity, and surface tension of the fluid. In order to ensure the breakup regime, the Reynolds number of the liquid (Re) should fulfill the condition given in Equation (1), where Oh is the Ohnesorge number. This equation was obtained from the relation between Re and Oh in the first transition of the breakup regime (from the Rayleigh jet breakup to the first wind-induced breakup) [29]. The Reynolds number can be related to the volumetric flow rate, as presented in Equation (2), and the Ohnesorge number can be related to the fluid properties, as presented in Equation (3). In these equations, \(n\) is the number of nozzles, \(d_{0}\) is the nozzle diameter, \(\rho_{l}\) is the liquid density, \(\mu_{l}\) is the liquid viscosity, \(\vartheta_{l}\) is the flow rate, and \(\sigma_{l}\) is the liquid surface tension. Generally, the Rayleigh regime occurs with a Weber  
Figure 1: (**a**) Schematic diagram of acquisition of droplets. (**b**) Experimental setup for acquisition of droplets.  
number less than 0.4, as reported in the literature [29], and the average droplet diameter is approximately twice the nozzle outlet diameter (\(d=1.89d_{0}\)) [30].  
\[\text{Re}\leq 45.973(\text{Oh})^{-1.3069} \tag{1}\]  
\[\text{Re}=\frac{4\dot{q}_{l}\rho_{l}}{n\pi d_{0}\mu_{l}} \tag{2}\]  
\[\text{Oh}=\frac{\mu_{l}}{\left(\rho_{l}\sigma_{l}d_{0}\right)^{0.5}} \tag{3}\]  
Afterward, images of droplets falling from the nozzle plate head in the transparent square tube were captured by the high-speed camera for the various experimental cases (N1H0 to N2H2). In this context, N1 represents the nozzle type, including parameters such as the nozzle number and material. The variable H signifies the various distances from the nozzle plate at which the droplet images were taken. For instance, H0 denotes a vertical distance of 5 mm between the nozzle and the high-speed camera, whereas H1 corresponds to a distance of 200 mm, and so on. A comprehensive description of the experimental runs is presented in Table 1. Subsequently, the sizes of the falling droplets were measured using image analysis software, specifically Fiji, after capturing the images with a high-speed camera.  
In this study, we utilized two different types of nozzle plates, and the diameter of the nozzle outlet for each plate was measured using a Keyence Digital Microscope VHY-2000. We determined the average nozzle size (N1: 170 um; N2: 200 um) to calculate the inlet water flow rate for each nozzle plate based on the number of nozzles in each respective plate. More information regarding the nozzle sizes, their corresponding flow rates, and the materials used in their construction is shown in Table 1. N1 featured 17 nozzles, whereas N2 incorporated a total of 30 nozzles within the nozzle plate. Both nozzle plates had a thickness of 0.5 mm.