## 2 Numerical Analysis
### CFD Models  
The commercial simulation software AVL FIRE R2022 (2022 R2, AVL List GmbH, 8020 Graz, Austria) with its advanced computational fluid dynamic (CFD) models had been proved to be well-suited for modeling the combustion of the fuel and emission formations inside the cylinder of IECs with very high accuracy [38]. In the present study, the ESE (engine simulation environment) diesel platform provide by AVL FIRE R2022 was utilized to simulate the engine working process starting from the IVC (intake valve closing) to the EVO (exhaust valve opening).  
The k-\(\zeta\)-f turbulence model was employed to simulate the fluid flow turbulence inside the engine cylinder. This is a 4-equation model developed from the well-known k-\(\varepsilon\) 2-equation model. It, therefore, has a higher precision and better simulation stability than the original k-\(\varepsilon\) 2-equation model [39]. In the CFD analysis, the mixing process and transport of reactants and products (chemical species) in a combustion problem is modeled by solving conservation equations, such as energy, mass, momentum, species conservations, etc. In this study, the 3-Z (three-zone) extended coherent flame species transport model (ECFM) [40; 41] was used for the simulation of the combustion of fuel in the engine cylinder. This model is improved from the coherent flame model (CFM), so it contains not only all the features, but also the enhancements of the CFM. It was successfully used in many previous works with very high result accuracy [8; 9; 42; 43]. Therefore, this study used this model for combustion modeling. This model is detailed and can be referenced in Appendix B of our previous study [8]. The fuel direct injection process was modeled by the Diesel Nozzle Flow sub-model [41; 44]. This model offers a simple method for correcting the initial velocities and diameters of the injected droplets owing to cavitation. The model is presented in detail in Appendix A. The evaporation and breakup processes of injected fuel droplets have been modeled by the Dukowicz and WAVE sub-models [41; 44], respectively.  
The extended Zeldovich mechanism [41; 45] was employed to model NO formation in the engine cylinder. This mechanism is composed of seven chemical species and three chemical reactions. It had been proved to be able to predict the thermal NO emission formations in ICEs accurately over a wide fuel/air equivalence ratio range in many works [10; 11; 46; 47; 48; 49; 50]. The formation of soot in the engine cylinder was modeled by the kinetic soot mechanism [41; 45]. The extended Zeldovich and kinetic soot mechanisms are detailed and can be respectively referenced in Appendix C and D of our previous research [8]. The droplet-wall interaction between the injected fuel droplets and combustion chamber walls has been modeled by the Walljet1 sub-model [41; 44]. This model had been developed from the Naber and Reitz spray/wall impingement model [51]. Other CFD models in this study can be referred to in the references [41; 44; 45]. The CFD models employed for the simulations in this study are summarized in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Description** \\ \hline Turbulence & k-\(\zeta\)-f \\ \hline \multirow{3}{*}{Combustion} & Standard species transport model \\ \cline{2-3}  & 3-zone extended coherent flame model (ECFM-3Z) \\ \hline \multirow{3}{*}{Emissions} & NO & Extended Zeldovich mechanism \\ \cline{2-3}  & Soot & Kinetic soot mechanism \\ \hline \multirow{2}{*}{Ignition} & Auto-ignition \\ \cline{2-3}  & Droplet breakup & WAVE \\ \hline \multirow{3}{*}{Atomization} & Droplet evaporation & Dukowicz \\ \cline{2-3}  & Droplet-wall interaction & Walljet1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: CFD models summary.