# Computational Methodology for Knocking Combustion Analysis in Compression-Ignited Advanced Concepts
## 3 Numerical Methodology
### Numerical Model Set-Up  
The numerical model of the engine was developed using the commercial CFD code CONVERGE [37]. The numerical domain, as shown in Figure 1, included the complete single cylinder geometry and the intake/exhaust ports, allowing to perform complete cycle simulations. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Baseline** & \(\pm\) & **[\%]** \\ \hline Combustion products (\%) & 36.70 & 1.0 & 2.7 \\ Temperature (K) & 406.30 & 2.0 & 0.5 \\ Tumble intensity (-) & 4.31 & 1.1 & 25.0 \\ Fuel mass (mg/cyc) & 19.07 & 0.191 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Baseline operation parameters and ranges of variation for the alteration of the IVC conditions. \begin{table}
\begin{tabular}{c c} \hline \hline Engine speed (rpm) & 1500 \\ Torque (Nm) & 49.9 \\ IMEP (MPa) & 1.04 \\ Number Injections (-) & 3 (pilot + main + post) \\ SoE\({}_{\text{main}}\) (cad aTDC) & \(-\)42.0 \\ Injection pressure (MPa) & 85 \\ Intake pressure (MPa) & 0.275 \\ EGR (\%) & 43.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Operating settings used for the model validation. The mesh discretization was done using the cut-cell Cartesian method available in the code. The base mesh size was 3 mm throughout the domain in the reference grid configuration. Three levels of fixed embedding (0.375 mm of cell size) were added to the walls of the combustion chamber, ports and region near the fuel injector, to improve boundary layer prediction and the accuracy of spray atomization, droplet breakup/coalescence, etc. Mesh size in the chamber was reduced by two levels of embedding (0.75 mm of cell size) after the start of combustion, for an improved recreation of the interaction and reflection of the pressure waves while avoiding undesired spatial aliasing effects. The code also employed adaptive mesh refinement (AMR) to increase grid resolution by three levels of additional refinement (up to 0.375 mm minimum cell size) based on the velocity and temperature sub-grid scales of 1 ms\({}^{-1}\) and 2.5 K, respectively. As a result, the total number of cells varied between 1.5 million at BDC and 0.5 million at Top Dead Centre (TDC). This mesh configuration was achieved after a grid sensitivity analysis [27], offering a grid-independent solution when simulating combustion and its produced unsteady pressure field in internal combustion engines. The Mach Courant-Friedrich-Lewy was set to 1.0 during the combustion in order to properly capture the local pressure oscillations caused by combustion. Thereby, the calculation time-step was gathered between 0.05 and 0.5 \(\upmu\)s. This value was also obtained from the previously referred work performed by [PERSON] et al. [27], whose demonstrated that the energy of the unsteady pressure field is highly sensitive to CFL Mach number, but also that the energy does not change when this parameter is close to or below one. The renormalization group (RNG) \(k-\varepsilon\) URANS model [18] coupled with the heat transfer approach proposed by [PERSON] et al. [38] was chosen for simulating the turbulent properties of the flow. For combustion modelling, the SAGE detailed chemistry solver [39] was employed along with a multi-zone (MZ) approach, with bins of 5 K in temperature and 0.05 in equivalence ratio [40]. This combustion model, despite not using an explicit turbulent combustion closure [17], has demonstrated to perform well for simulating spray combustion under URANS schemes in previous works [41]. A reduced chemical kinetic mechanism for primary reference fuels (PRF) based on [PERSON] et al. [42] was used in this work to account for fuel chemistry. A blend of 5% n-heptane and 95% iso-octane was utilized for predicting the physical properties of the gasoline fuel, being a suitable surrogate for predicting the ignition features of the RON95 gasoline used in experiments. Activating iso-octane reactions, the chemical mechanism resulted in a 45 species and 152 reactions. Figure 1: Numerical domain and mesh features of the engine.