## 2 Numerical Analysis
### CFD Models  
The AVL FIRE code with its advanced CFD models has been demonstrated to be suitable for modeling the combustion and emission formations in the cylinder of ICEs, such as diesel, gasoline, and DF engines with a high accuracy [21]. In the present study, the AVL FIRE ESE diesel platform was employed to simulate the working process of the engine from the IVC to EVO. The simulated results were then compared to the measured results to verify the CFD models.  
The k-\(\zeta\)-f is a well-known four-equation turbulence model. It was developed from the k-\(\epsilon\) two-equation model [32]. It has a better stability and a higher accuracy than the original k-\(\epsilon\) model. Therefore, it was used to model the fluid flow turbulence inside the engine cylinder in this study. In the combustion problems, the transport and mixing processes of the chemical species are governed by solving the conservation equations that describe the diffusion, convection phenomenon, component species concentrations, and reaction sources in the system. In this research, the 3-zone species transport model named the 3-Z extended coherent flame model (ECFM-3Z) [26; 33] was employed to model the combustion of the fuels inside the cylinder of the engine. Based on the assumption that in ICEs, the chemical time scales are much smaller than the turbulent time scale, the ECFM combustion concept was applied. The ECFM is applicable to both non-premixed and premixed combustions, based on a laminar flamelet concept. In this concept, the velocity and thickness of the flames are mean values, integrated along the flame front. They only depend on the pressure, temperature, and richness of the fresh gases. The laminar flamelet concept assumes that the chemical reactions take place within relatively thin layers, which separate the unburned gases from the fully burnt gases. With this assumption, the mean turbulent chemical reaction rate is calculated as the product of the laminar burning velocity and the flame surface density [26; 33]. The direct injection of the pilot fuel was modeled by using the diesel nozzle flow model [26; 34]. The diesel nozzle flow offers a simple approach to correct the velocity and the initial diameters of the fuel droplets caused by the cavitation phenomena. This model is presented in detail in Appendix A. The evaporation and breakup of the fuel droplets were respectively modeled by the multi-component and WAVE models [26; 34]. The ignition of the diesel oil inside the cylinder in the diesel and DF modes was simulated by the auto-ignition and diesel ignition gas engine models, respectively [26]. Regarding EGEs, the well-known extended Zeldovich model [26; 35] was utilized to simulate the NO emission formation inside the engine cylinder. This model consists of seven species and three chemical reactions. It has also been demonstrated to be able to accurately predict the thermal NO emission in ICEs over a large range of fuel-air equivalence ratios. For modeling the formation of soot during the combustion of the engine, a kinetic soot mechanism was used [26; 35]. The interaction between the engine combustion walls and the injected fuel droplets was modeled using the Walljet1 model [26; 34]. The CFD models used in this study are summarized in Table 4. For the convergence criteria, in this study, the absolute convergence criteria of \(10^{-3}\) for the continuity, momentum, velocity, k, and \(\zeta\) were set. Meanwhile, a tighter convergence criterion of \(10^{-6}\) was set for the energy equations. These criteria are technically suitable for high-turbulence combustion problems in ICEs [26].