## 3 Results and Discussion
### Assessment of Modelling Approaches Considering a 0D Description of the Junction
#### 3.2.1 Time Domain Assessment
The values obtained for the mean quadratic errors are summarized in Table 1, where it is confirmed that the best values for the reflection coefficients are those provided by the conventional method, while the FCT method gives the best approach to the transmission coefficients. Figure 8: Comparison between the different models for the T-junction: raw data (top) and differences (bottom plots) in the time domain. **(a)** excitation at port 1; **(b)** excitation at port 3. Ports are denoted as in Figure 1. \(\Delta R(i)\): difference in the pulse reflected at port \(i\); \(\Delta T(i-j)\): difference in the pulse transmitted between ports \(i\) and \(j\). FCT: flux-corrected transport; MDT: momentum diffusion term; 1D: conventional pressure-loss model. Similar comments can be made about the comparison shown in Figure 9 for the case of the Y-junction. Again, the conventional method reproduces better the behavior of the reflected pulses, regardless of the excitation port, and the MDT and the FCT methods exhibit significant differences only when the junction is excited at port 3, following the same trend as for the T-junction. The trend is also very similar for the different transmission paths. Transmission between ports 1 and 2 is acceptably reproduced by all the modeling approaches, regardless of the exciting port, again with a slightly better performance of the conventional model. In those cases in which port 3 is included in the transmission path, the tendency observed is again the same when the junction is excited at port 1 or 2, with a small difference with respect to the T-junction when the excitation comes from port 3: in this case, the lowest values are those provided by the MDT method, most notably in the transmission from port 3 to port 2. Again, the mean quadratic errors were calculated, and the corresponding results shown in Table 2 confirm the previous comments. From these results, it is apparent that the conventional pressure loss model, while is not able to account for all the differences observed between the two transmission paths studied in each test, could still provide a sufficient approximation to the real situation if the focus of the problem is on the reflection properties of the junction and only time domain issues are relevant for the problem under study (for instance, the eventual influence of a reflection at an intake junction on the volumetric efficiency on the engine). The staggered mesh finite volume method appears to be more sensitive to the relative directions of the different branches, mostly when the excitation comes from the side branch (port 3), as should be expected since the momentum equation is actually solved, albeit in an approximate way, at the junction, whereas in the conventional model such effects are included only through their influence on the pressure loss coefficients. \begin{table}
\begin{tabular}{c c c c} \hline
**Path** & **MDT** & **FCT** & **1D** \\ \hline \(R(1)\) & \(1.575\times 10^{-4}\) & \(1.915\times 10^{-4}\) & \(1.273\times 10^{-4}\) \\ \(R(2)\) & \(1.992\times 10^{-4}\) & \(2.172\times 10^{-4}\) & \(1.681\times 10^{-4}\) \\ \(R(3)\) & \(1.648\times 10^{-4}\) & \(2.945\times 10^{-4}\) & \(1.394\times 10^{-4}\) \\ \(T(1\text{--}2)\) & \(1.171\times 10^{-4}\) & \(1.369\times 10^{-4}\) & \(9.186\times 10^{-5}\) \\ \(T(1\text{--}3)\) & \(1.514\times 10^{-4}\) & \(1.322\times 10^{-4}\) & \(1.742\times 10^{-4}\) \\ \(T(2\text{--}1)\) & \(1.992\times 10^{-4}\) & \(2.172\times 10^{-4}\) & \(1.681\times 10^{-4}\) \\ \(T(2\text{--}3)\) & \(1.336\times 10^{-4}\) & \(1.296\times 10^{-4}\) & \(1.141\times 10^{-4}\) \\ \(T(3\text{--}1)\) & \(1.669\times 10^{-4}\) & \(1.949\times 10^{-4}\) & \(1.355\times 10^{-4}\) \\ \(T(3\text{--}2)\) & \(2.117\times 10^{-4}\) & \(1.516\times 10^{-4}\) & \(1.751\times 10^{-4}\) \\ \hline \end{tabular} Abbreviations: FCT: flux-corrected transport; MDT: momentum diffusion term; 1D: conventional pressure-loss model.  
\end{table} 
Table 2: Values of the mean quadratic error: Y-junction.