## 2 Materials and Methods
### CFD-FVM Model (Step 2.1)
The base dimension for the elements was set to 2.500 mm (i.e., at a refinement level of 2 with reference to the internal scale), with local refinements at the interfaces arriving down to 1.250 and 0.625 mm (corresponding, respectively, to a refinement level of 3 and 4 for the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Quantity**} & \multirow{2}{*}{**Unit**} & \multicolumn{4}{c}{**Engine Operative Point**} \\ \cline{3-6}  & & **P0** & **P1** & **P2** & **P3** \\ \hline Torque (\(M\)) & Nm & 8.09 (\(\pm\)0.22) & 8.15 (\(\pm\)0.31) & 6.79 (\(\pm\)0.25) & 5.33 (\(\pm\)0.25) \\ Rotational Speed (\(n\)) & rpm & 2400 (\(\pm\)4) & 1970 (\(\pm\)13) & 2440 (\(\pm\)14) & 2450 (\(\pm\)7) \\ \hline \(T_{1}\) (grommet) & K & 450 (\(\pm\)2) & 448 (\(\pm\)2) & 438 (\(\pm\)2) & 430 (\(\pm\)2) \\ \(T_{2}\) (clamp) & K & 540 (\(\pm\)2) & 529 (\(\pm\)2) & 503 (\(\pm\)2) & 480 (\(\pm\)2) \\ \(T_{3}\) (well) & K & 680 (\(\pm\)2) & 673 (\(\pm\)2) & 633 (\(\pm\)2) & 591 (\(\pm\)2) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental data gathered in correspondence at four operative points for the engine (NB: the reported data are the average values, and the expanded uncertainty values referred to steady-state conditions only). mesh). The refinement of the mesh for the fluid volume near the solid walls is required by this software to create a sufficient number of very small elements able to cope with the inclined solid walls and, hence, to model correctly the boundary layer. The used software applies automatically the [PERSON] hypothesis about the mixing length in turbulent boundary layers and the k-\(\varepsilon\) turbulence model [58]. Before running the simulations, mesh independence was proven by raising significantly the number of elements used for the general discretization of fluid and solids domains up to +44% and +37%, respectively, then refining the mesh in correspondence with TC2 and TC3 in two further subsequent steps, respectively characterized by +92% and +198% of elements, obtaining in all these cases temperature values for the TCs differing on average by 1.0% from the reference configuration (1.5% for TC2). Figure 5: (**a**) 3D-CAD lateral view of the exhaust duct geometry; the asterisks indicate the positions of the TCs tips. (**b**) Used mesh as visible on a section passing through the exhaust pipe axis; the color green is associated to a higher refinement of the base mesh (in light blue) for the fluid. The exhaust pipe inlet is not visible due to the presence of the pipe-to-head connector. Figure 6: (**a**) Close-up view of the mesh for the first part of the exhaust duct, within the pipe-to-head connector and the engine head; the grey circle is the inlet surface of the exhaust line, placed in the same position of the exhaust valve. (**b**) Close-up view of the mesh in correspondence with the thermowell. The exhaust gas was approximated as air [59], and related properties were then recalled by the software library (\(c_{p}/c_{p}\) = 1.399; molecular mass = 0.2896 kg\(\cdot\)mol\({}^{-1}\); \(c_{p}\), dynamic density, thermal conductivity variable with the temperature). AISI 316L was chosen as the material for all the parts of the exhaust duct (pipe and connector) and also as material for the engine head as the real material was unknown (see tuning procedure in the following Sections). As very frequently occurs when setting up numerical simulations, the used software program needs a lot more input parameters than the parameter actually available/known in reality, especially in cases like this one in which a component (the engine head) has been deliberately simplified. For example, although there is the possibility of using internal libraries for the physical properties of the materials involved, often the missing information is just the exact type of material (in the present case for the simulacrum of engine head, different from the actual engine head and, therefore, using also a material non-necessary equal to the real material).