## 5 Description of Modeling Approach
### In-Cylinder Sub-Models and Tuning: Combustion, Turbulence and Pollutant Emissions
The governing equations for the mean flow kinetic energy, K, the turbulent kinetic energy, k, and the tumble momentum, \(\mathrm{T}_{\mathrm{m}}\), are as follows:  
\[\mathrm{d}(\mathrm{m}\cdot\mathrm{K})/\mathrm{dt}=(\dot{\mathrm{m}}\cdot \mathrm{K})_{\mathrm{inc}}-(\dot{\mathrm{m}}\cdot\mathrm{K})_{\mathrm{out}}- \mathrm{f}_{\mathrm{d}}\cdot\mathrm{m}\cdot\mathrm{K}/\mathrm{t}_{\mathrm{ Tum}}+\mathrm{m}\cdot\mathrm{K}\cdot\dot{\mathrm{\Phi}}/\mathrm{\rho}-\mathrm{P}, \tag{8}\]  
\[\mathrm{d}(\mathrm{m}\mathrm{k})/\mathrm{dt}=(\dot{\mathrm{m}}\cdot\mathrm{k}) _{\mathrm{inc}}-(\dot{\mathrm{m}}\cdot\mathrm{k})_{\mathrm{out}}+(2\cdot\dot{ \mathrm{\Phi}}/3\cdot\mathrm{\rho})\cdot(\mathrm{m}\cdot\mathrm{k}-\mathrm{m} \cdot\mathrm{\
u}_{\mathrm{t}}\cdot\dot{\mathrm{\Phi}}/\mathrm{\rho})+\mathrm{ P}-\mathrm{m}\cdot\epsilon, \tag{9}\]  
\[\mathrm{d}(\mathrm{m}\cdot\mathrm{T}_{\mathrm{m}})/\mathrm{dt}=(\dot{\mathrm{m}} \cdot\mathrm{T}_{\mathrm{m}})_{\mathrm{inc}}-(\dot{\mathrm{m}}\cdot\mathrm{T}_ {\mathrm{m}})_{\mathrm{out}}-\mathrm{f}_{\mathrm{d}}\cdot\mathrm{m}\cdot\mathrm{ T}_{\mathrm{m}}/\mathrm{t}_{\mathrm{tum}}, \tag{10}\]  
In Equations (8)-(10), the first and second terms describe the incoming and the outcoming convective flows through the valves, respectively. The third term for the K and T\({}_{\mathrm{m}}\) equations is the decay contribution due to the shear stresses with the combustion chamber walls by means of the decay function, \(\mathrm{f}_{\mathrm{d}}\), and the characteristic time scale, \(\mathrm{t}_{\mathrm{tum}}\). In Equation (9), third term represents an additive compressibility term, proportional to \((\dot{\mathrm{\Phi}}/\mathrm{\rho})\); the latter is also included in the K equation as the 0 fourth term. \(\mathrm{P}\) describes the energy cascade mechanism, which causes the transfer of kinetic energy from the mean flow to the turbulent flow. Finally, the last term in Equation (9) takes into account the viscous dissipation rate of k into heat. The described turbulence sub-model proved to be able to adequately reproduce the in-cylinder turbulence evolution along the whole engine cycle, also sensing the variations in the engine operating conditions, the valve strategies and the cylinder geometrical parameters [23]. The heat-transfer model both for cylinders and exhaust subsystem is considered, applying a wall temperature solver based on a finite element (FE) approach. For the in-cylinder heat transfer (gas-to-wall), the Hohenberg correlation is implemented into the 1D code while convective, conductive and radiative heat transfer modes are considered for the exhaust pipes. The Hohenberg correlation here utilized is illustrated in Equation (11):  
\[\mathrm{H}=\mathrm{A}\cdot\mathrm{V}\ ^{-0.06}\cdot\mathrm{p}^{0.8}\cdot \mathrm{T}^{-0.4}\cdot(\mathrm{v}_{\mathrm{pm}}+\mathrm{B})\ ^{0.8}, \tag{11}\]  
where V is the volume, \(\mathrm{p}\) is the pressure, \(\mathrm{T}\) is the mean gas temperature and \(\mathrm{v}_{\mathrm{pm}}\) is the mean piston speed. The calibration constants, A and B, are calculated by [PERSON] and here used as 130 and 1.4, respectively. The convective heat transfer coefficients for engine coolants (oil and water) are evaluated by simulations of coolant circuits. They are subsequently imposed as an input in the code, assuming a dependency on the engine speed. Cooling boundary conditions (temperatures for oil and water) are introduced in the model according to the levels recommended by the engine manufacturer. Concerning the model tuning, an integrated 3D/1D approach is adopted for turbulence sub-model [23]; model constants are identified to better reproduce the turbulence outcomes of in-cylinder 3D CFD simulations, under motored conditions and for various speeds and valve strategies [23]. Referring to the combustion model tuning, this sub-model includes three tuning constants differently affecting the distinct phases of combustion process.