## 2 Numerical Modeling  
In this study, numerical simulations were implemented using CONVERGE Version 2.2 [24], which was used to create constant volume models and set the turbulence model, spray model, and sub-model, as shown in Table 1. This model was created to predict the diesel spray performance by simulating the spray shape and the spray penetration distance in both a liquid and a vapor state, as well as the mixing behavior. The shape of the model is defined as a constant volume combustion chamber with a diameter of 105 mm and a length of 105 mm to reduce the grid number and increase computational efficiency, with input from the case study boundary conditions under the experimental conditions [23]. The injector was placed at the top center of the cylinder, as shown in Figure 1. The spray was designed using the Spray A condition (detailed information is shown in the ECN [23]). The C\({}_{12}\)H\({}_{26}\) reaction mechanism was used as a diesel fuel agent like the experimental considerations since the current model estimation of the trends and evolution of vapor penetration are independent of the fuel type [25]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Modeling Tool** & **CONVERGE** \\ \hline Spray models & \\ Drop evaporation model & Frossling model \\ Collision model & No Time Counter model (NTC) collision \\ Collision outcome & O’Rourke collision outcomes \\ Drop drag model & Dynamic drop drags \\ Breakup & KH-RT model \\ Turbulence model & RANS, RNG k-\(\epsilon\) \\ Grid control & \\ Base grid size & 16 mm \\ Finest grid size & 0.25 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modeling and Numerical Parameters. The grid size has a large influence on the penetration distance [26]. When the grid size is too small, a long simulation time will be required. In this study set up, a fixed grid embedded with a minimum grid size for the spray flow field is shown in Equation (1) [24], where the \"Size of cells in scaled grid\" represents the sizes of the cells in the scaled grid in each axial \"Size of cells in base grid\" indicates the sizes of the cells in the base grid setting of each axial, and \"amr_embed_vel_scale\" is the level of embedding. In this study, CONVERGE can easily change the overall grid resolution prior to executing a simulation to assess grid sensitivity, which is useful for reducing the working time for constant stimulation. For simulations in areas or periods that are not of great importance, a rough grid can be used and then adjusted to consider only the important periods by setting the level of embedding for each section to obtain results that are accurate and not unnecessarily time intensive for calculations:  
\[\textit{Size of cells in scaled grid}=\textit{Size of cells in base grid}\star 2^{-amr\_embed\_vel\_scale}. \tag{1}\]  
The computational fluid dynamics simulation of the in-cylinder process has many modifications in the spray modeling. The interaction of spray turbulence modeling has an important influence on the spray penetration and mixture formation prediction, which emerges at the end of the entire combustion process. A recent literature review on fuel spray modeling showed that the use of RANS guidelines is of fundamental importance to ensure sufficient performance of the spray mixing and combustion processes. In this study, we used the RANS method to describe the spray development process. The RANS model was used in conjunction with the RNG k-\(\varepsilon\) turbulence model to determine the effects of smaller movements. A drop evaporation model base on the Frossling model was also used. The liquid penetration length is defined as the maximum distance from the axial position with 99% injected fuel mass at the injection location, while the vapor penetration length is defined as the distance from the farthest location with a 0.01% fuel mass fraction to the nozzle exit. For a fast and accurate collision calculation response, the NTC model with an O'Rourke outcome was used in this study.