## 5 Description of Modeling Approach
### In-Cylinder Sub-Models and Tuning: Combustion, Turbulence and Pollutant Emissions  
In-cylinder 0D sub-models were integrated within the 1D engine model to refine the prediction of turbulent combustion and of the exhaust emissions. The combustion process is modeled by the fractal approach [19; 20; 21], where the burning rate is written as follows:  
\[\mathrm{dm}_{\mathrm{b}}/\mathrm{dt}=\rho_{\mathrm{u}}\cdot\mathrm{S}_{\mathrm{ L}}\cdot\mathrm{A}_{\mathrm{T}}=\rho_{\mathrm{u}}\cdot\mathrm{S}_{\mathrm{L}}\cdot \mathrm{A}_{\mathrm{L}}\cdot(\mathrm{L}_{\mathrm{max}}/\mathrm{L}_{\mathrm{ min}})^{(\mathrm{D}_{3}-2)}, \tag{6}\]  
where \(\rho_{\mathrm{u}}\) is the unburned gas density, \(\mathrm{A}_{\mathrm{L}}\) and \(\mathrm{A}_{\mathrm{T}}\) the area of the laminar and turbulent flame fronts, respectively, and \(\mathrm{S}_{\mathrm{L}}\) the laminar flame speed (LFS). \(\mathrm{L}_{\mathrm{min}}\) and \(\mathrm{L}_{\mathrm{max}}\) are the minimum and maximum flame wrinkling scales and \(\mathrm{D}_{3}\) is the fractal dimension. \(\mathrm{L}_{\mathrm{min}}\) represents the Kolmogorov scale evaluated under the hypothesis of isotropic turbulence, while \(\mathrm{L}_{\mathrm{max}}\) is proportional to a characteristic dimension of the flame front, i.e., the flame radius in the present study. \(\mathrm{D}_{3}\) is computed by the equation proposed in Reference [22], as a function of the ratio between the in-cylinder turbulence intensity and LFS. The latter is evaluated by using an \"in-house\" developed correlation, based on 1D LFS computations via chemical kinetics solver (CANTERA), accounting for the in-cylinder thermodynamic state (pressure \(\mathrm{p}\) and temperature \(\mathrm{T}\)), equivalence ratio, \(\Phi\), and EGR dilution. In particular, the adopted correlation reported in Equation (7) presents the well-known expression of the power law:  
\[\mathrm{S}_{\mathrm{L}}=\mathrm{S}_{\mathrm{L}0}\cdot(\mathrm{T}/\mathrm{T}_{ \mathrm{ref}})^{\alpha}\cdot(\mathrm{p}/\mathrm{p}_{\mathrm{ref}})^{\beta}\cdot \mathrm{EGR}_{\mathrm{factor}}, \tag{7}\]In Equation (7), \(\mathrm{S}_{\mathrm{L0}}\) is the flame speed at reference conditions \(\mathrm{T}=\mathrm{T}_{\mathrm{ref}}\) and \(\mathrm{p}=\mathrm{P}_{\mathrm{ref}}\), depending on the fuel sensitivity \(\mathrm{S}=\mathrm{RON}\)-\(\mathrm{MON}\) and \(\mathrm{\Phi}\); \(\mathrm{EGR}_{\mathrm{factor}}\) is a reduction term for LFS that accounts for the presence of residual gas in the unburned mixture. Exponent \(\alpha\) depends on \(\mathrm{\Phi}\), \(\mathrm{T}\) and \(\mathrm{p}\), while exponent \(\beta\) includes the dependencies on \(\mathrm{\Phi}\) and \(\mathrm{S}\). It is the case to underline that the above LFS correlation was developed considering a maximum EGR percent mass of 20%; therefore, the employed LFS correlation shows the capability to cover the range of examined EGR rates during the experiments (Table 2). The laminar flame area, \(\mathrm{A}_{\mathrm{L}}\), is computed by an automatic procedure implemented in a CAD software and processing the actual 3D geometry of the combustion chamber. The estimation of \(\mathrm{L}_{\mathrm{min}}\), and \(\mathrm{D}_{3}\) is based on the K-k-T turbulence sub-model, extensively reported in Reference [23].