## 2 Numerical Modeling
To increase the efficiency of our droplet collision calculations, we used a dynamically simulated gas technique for spraying. For this technique to work effectively, it must be able to handle a common case, where the number of droplets in each particle varies. The configuration can work effectively with general cases under different conditions. For the spray breakup model in this study, the Kelvin-Helmholtz instability (KH) and Rayleigh-Taylor instability (RT) models were used. The KH model size constant is defined as proposed by [PERSON] [27]. The wave breakup formulation was used to model the liquid breakup process. The wavelength (\(\Lambda_{\text{KH}}\)) and the breakup size constant (B\({}_{0}\)) determined the child droplet sizes. The drop radius equation is calculated as follows:  
Figure 1: Model characteristics of cylinders. \[r_{\rm KH}=B_{0}\Lambda_{\rm KH}. \tag{2}\]  
By requiring that \(B_{0}\) equal 0.61 for the droplet breakup regime that has the characteristics of \(25<\) We \(<50\), under a higher injection velocity, the \(B_{0}\) will be in accordance with the applications offered by [PERSON] et al. [28]. The KH model velocity constant was determined to be 0.188, which is the basic value commonly used by researchers. During the breakup, the parent droplet parcel radius (r) is continuously reduced until it reaches a stable droplet radius (r\({}_{\rm KH}\)) according to the following equation:  
\[\frac{dr}{dt}=\frac{r-r_{\rm KH}}{\tau_{\rm KH}},\ r_{\rm KH}\leq r \tag{3}\]  
where \(\tau_{\rm KH}\) is the breakup time, given by:  
\[\tau_{\rm KH}=\frac{3.276B_{1}r}{\Lambda_{\rm KH}\Omega_{\rm KH}} \tag{4}\]  
where \(\Omega_{\rm KH}\) is the KH wave with the maximum growth rate. The KH model breakup time constant (B\({}_{1}\)) determines the primary breakup time. In this study, \(B_{1}\) equals 21, which is more accurate than the recommended value (recommended value equals 7) in other references. The RT breakup length equation is given by:  
\[L_{\rm b}=C_{\rm bl}\sqrt{\frac{\rho_{\rm l}}{\rho_{\rm g}}}d_{0} \tag{5}\]  
where C\({}_{\rm bl}\) is the RT model breakup length constant, \(\rho_{\rm l}\) is the fuel density, \(\rho_{\rm g}\) is the ambient gas density, and d\({}_{0}\) is the orifice diameter. The RT model breakup length constant (C\({}_{\rm bl}\)) generally equals 1.0. It is necessary to switch the liquid breakup parameter from the KH model to the RT model. Using this approach, the breakup time can be determined by the RT model breakup time constant (C\({}_{\tau}\)). The value of C\({}_{\tau}\) should be less to reduce the breakup delay, as shown in the RT breakup time equation below:  
\[\tau_{\rm RT}=C_{\tau}\frac{1}{\Omega_{\rm RT}} \tag{6}\]  
where \(\Omega_{\rm RT}\) is the RT wave with the maximum growth rate. The RT model size constant (C\({}_{\rm RT}\)) used to determine the scaled wavelengths and the radius of the RT breakup with a higher value increases the predicted RT breakup radius size:  
\[r=\pi\frac{C_{\rm RT}}{K_{\rm RT}} \tag{7}\]  
where K\({}_{\rm RT}\) is the wavenumber. In this study, C\({}_{\rm RT}\) equals 0.1, in which the parent radius decreases continuously until it reaches the constant value in Equation (3), as in the KH model. The setting of the spray breakup model provides an additional configuration for the breakup model constants. Notably, different breakup model constants are used depending on the test conditions and the numerical calculation tools. The spray dispersion angle is another important input variable. The spray distribution angle defines the fluctuation of air in the injected fuel due to the impulse exchange between gases and liquids. The spray distribution angle will change under different injection intervals (start of injection, the transient regime, the stable regime, and the EOI) due to changes in the nozzle sac flow, momentum, and the interactions between air and the fuel. The actual spray distribution angle detected by each method is different and cannot be compared.