## 2 Numerical Analysis
### CFD Models and Governing Equations  
This study used the pressure-based type with absolute velocity formation for the solver. The effect of gravitational acceleration was also included in the simulation. The acceleration value used was 9.81 m/s2. Due to the piston's moving action during the simulation, an unsteady (transient) approach was used. In engine modeling, the time step size is calculated based on the piston movement speed. The time step size value was 4.62963 \(\times\) 10\({}^{-5}\) s. However, due to the complexity of the fuel combustion process, the time step size of the simulation during this period (12 CADs BTDC to 20 CADs ATDC) was set to be reduced to 16 times to ensure solution convergence. Here, 50 maximum iterations per time step were set. Monitoring the residuals showed that the solution reached convergence after 31 to 35 iterations per time step, and the computation was then skipped to the next time step. The PISO scheme with a value of 1 for both skewness and neighbor corrections was applied. For spatial discretization, the Green-Gauss Node-Based gradient and standard pressure were used. Meanwhile, Second-Order Upwind was applied for density, momentum, TKE, turbulent dissipation rate, nuclei, and pollutant calculations. An \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fuel** & **Diesel** & **NG** & **Unit** \\ \hline Density & 730 & 0.657 & kg/m3 @ 25 \({}^{\circ}\)C, 1 at \\ Chemical name/formula & n-Decane/C10H22 & Methane/CH4 & \\ Lower calorific value & 42,000 & 50,000 & kJ/kg \\ Molecular weight & 142.284 & 16.043 & g/mol \\ Boiling point & 174 & \(-\)162 & \({}^{\circ}\)C \\ Auto-ignition temperature & 223 & 537 & \({}^{\circ}\)C \\ Specific heat capacity (Cp) & 2090 & 35.69 & J/mol:K @ 25 \({}^{\circ}\)C \\ Stoichiometric air–fuel ratio & 14.98 & 17.19 & kg/kg \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fuel properties [33; 35; 36]. Figure 3: Schematic of the SA and IP adjustments. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Simulation Case**} & \multicolumn{4}{c}{**Injection Position (IP) [Unit: m]**} \\ \cline{2-5}  & & **0.01** & **0.015** & **0.02** \\ \hline \multirow{2}{*}{**Spray Angle** **(SA)** [Unit: degree]  
absolute criterion of \(10^{-3}\) for convergence criteria was set for continuity, velocity, k, and \(\varepsilon\), while a stricter criterion of \(10^{-6}\) was set for energy and pollutant calculations. These convergence criteria are suitable for combustion and emission simulations of internal combustion engines, as presented in [33]. The standard k-\(\epsilon\) turbulence model was employed to model the fluid flow turbulence inside the engine cylinder. This model is suitable and widely used to model the turbulence inside the cylinder of ICEs [37]. The combustion of fuel in the engine cylinder was modeled by the non-premixed combustion model using a reduced chemical mechanism. The reduced chemical reaction was generated using the probability density function (PDF) method in the ANSYS Fluent solver. This model is suitable for simulating non-premixed combustion, where the oxidizers and fuels are introduced into the combustion apparatus separately [33]. Reaction mechanisms play a very important role in combustion simulations. For a deeper understanding of the reaction mechanism as well as the latest developments in this field, it would be helpful to refer to [38]. To simulate the spraying of the nozzle, the Discrete Phase Model (DPM) was used. This model offers an effective way to correct the spray velocities and initial droplet diameters caused by the cavitation phenomenon. In this model, the Kelvin-Helmholtz Rayleigh-Taylor (KHRT) sub-model was utilized to model the breakup phenomenon of fuel droplets. Meanwhile, the auto-ignition model was employed to simulate the self-ignition of fuel in the engine cylinder.