## 2 Experimental and Computational Numerical Methods
### CFD Digital Twin of Experimental Setup  
In computational fluid dynamics, a computerized replication of an experimental setup is established to enable in-depth studies and examinations. The CFD digital twin, as depicted in Figure 3a, was utilized for the investigation of droplet size and distribution for 4 different cases (N1H1, N1H2, N2H1, and N2H2) and later validated with experimental results. For each simulation case, the corresponding log-normal distribution for the droplet size was used as input. For the N1H1 and N1H2 cases, the distribution from 17 droplet injection points in Figure 4 was used, whereas for the N2H1 and N2H2 cases, the distribution from 30 droplet injection points in Figure 5 was utilized. Figure 4: (**a**) Experimental droplet size frequency distribution with fitted curve. (**b**) Log\(-\)normal distribution fitting with linear fit of \(R^{2}\) for N1H0. Figure 3: (**a**) Digital twin of the experimental setup. (**b**) Polyhedral meshing of CFD setup for N1H1. To conduct the CFD simulations, a volumetric domain was constructed based on the provided digital twin geometry (Figure 3a). This domain was subsequently partitioned into discrete cells to establish a mesh. In all the CFD simulations presented in this study, a polyhedral mesh (Figure 3b) with a base size of 4.5 mm was employed, as depicted in the figure. An implicit unsteady time step model was utilized, taking a time step of 0.005 s across all simulations, with a total physical simulation time of 2 s. The governing equations, namely the continuity Equation (8) and Reynolds-Averaged Navier-Stokes (RANS) Equation (9) [35], which were utilized in all four simulation scenarios within this study, were discretized using the finite volume method. The turbulence model used for the closure of the Reynolds-Averaged Navier-Stokes equation was the k-\(\omega\) Menter SST two-layer model [35]. This model was specifically chosen due to its ability to strike a suitable balance between precision, computational efficiency, and robustness. \[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v_{x})}{\partial x}+\frac{ \partial(\rho v_{y})}{\partial y}+\frac{\partial(\rho v_{z})}{\partial z}=0 \tag{8}\]  
In Equation (8), \(\rho\) represents the fluid density, which is a measure of the mass per unit volume. The variable \(t\) denotes time, and \(v_{x}\), \(v_{y}\), and \(v_{z}\) correspond to the velocity components in the \(x\)-, \(y\)-, and \(z\)-directions, respectively. The term \(\frac{\partial\rho}{\partial t}\) represents the rate of change of the density with respect to time. It accounts for how the density of the fluid changes over time within a given volume. Additionally, \(\frac{\partial(\rho v_{x})}{\partial x}\), \(\frac{\partial(\rho v_{y})}{\partial y}\), and \(\frac{\partial(\rho v_{z})}{\partial z}\) represent the rates of change of the mass flow rate density (\(\rho v\)) with respect to the spatial coordinates \(x\), \(y\), and \(z\), respectively. \[\rho\frac{\partial U_{i}}{\partial t}+\rho U_{j}\frac{\partial U_{i}}{ \partial x_{j}}=-\frac{\partial P}{\partial x_{i}}+\frac{\partial}{\partial x _{j}}\Big{(}2\mu S_{ji}-\rho\overline{u_{j}^{\prime}u_{i}^{\prime}}\Big{)} \tag{9}\]  
The vectors \(u_{i}\) and \(x_{i}\) are the velocity and position, \(\rho\) is the density, and \(s_{ij}\) is the strain-rate tensor, which is expressed as follows:  
\[s_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}\right) \tag{10}\]  
Note that \(s_{ji}=s_{ij}\). Equation (9) is referred to as the RANS equation [35]. The quantity \(-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\) is known as the Reynolds stress tensor, and we denote it by \(\rho\tau_{ij}\), where \(\tau_{ij}\) is the specific Reynolds stress tensor given by:  
\[\tau_{ij}=-\overline{u_{i}^{\prime}u_{j}^{\prime}} \tag{11}\]  
Figure 5: (**a**) Experimental droplet size frequency distribution with fitted curve. (**b**) Log\(-\)normal distribution fitting with linear fit of \(R^{2}\) for N2H0.