3.2.1. Time Domain Assessment

First, in Figure 7, direct comparison between the experiments and the method with the momentum diffusion term (MDT) as flux limiter is given for the case of the T-junction. The figures at the top provide a direct representation of the results obtained, whereas in the figures at the bottom the differences between the experimental and numerical results are represented (these are labeled Δ *R* and Δ *T* for reflected and transmitted pulses, respectively). In general, the model reproduces the experimental results within reasonable limits, but with a superimposed oscillation due to the development of the pulse from station 0 to station 1 (refer to Figure A1) and which is a consequence of the way in which the inlet boundary condition has been set. The scale of the vertical axis in the differences plots has thus been chosen so as to allow proper comparison for the times not affected by those oscillations.

**Figure 7.** Experimental vs. modeled results for the T-junction: raw data (top) and differences (bottom) in the time domain, momentum diffusion term (MDT) method. (**a**) excitation at port 1; (**b**) excitation at port 3. Ports are denoted as in Figure 1. <sup>Δ</sup>*R*(*i*): difference in the pulse reflected at port *i*; Δ*T*(*i* − *j*): difference in the pulse transmitted between ports *i* and *j*.

From the differences plots, it is apparent that the numerical results tend to underestimate the actual measured values in the trailing part of the pulses, for *t* > 0.85 s, the differences being larger in general for the case of the reflected pulse. The situation is rather more complex for the previous instants, with different trends observed for the transmitted and reflected pulses, and with a noticeable influence of the port at which the junction is excited.

The results obtained for the rest of the modeling approaches considered are compared in Figure 8, where for clarity the experimental results are not shown in the top figures, but the bottom figures have been expanded to allow proper analysis of the behavior observed in each of the propagation paths. It is apparent that the conventional pressure loss method (labeled 1D in the figure) is much less dispersive than any of the staggered-grid methods, and thus better suited for this particular calculation setting. This is particularly true in the case of the reflected pulses, where the conventional method approaches the measured values considerably earlier. It is also apparent that while no significant differences can be found between the MDT and the FCT methods in the reflection seen from port 1, this is not the case when the junction is excited at port 3: the FCT method exhibits larger differences, except in the last part of the reflected pulse.

**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. <sup>Δ</sup>*R*(*i*): difference in the pulse reflected at port *i*; Δ*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.

Regarding the different transmission paths, it can be observed that, while relatively small differences between all the methods are seen in the case of transmission from port 1 to port 2 (the performance of the conventional method being slightly better), significant differences appear at intermediate times in all the cases in which port 3 is involved. There is a clear trend in the results obtained in these cases, in the sense that the conventional method produces the lowest values, the FCT method the highest values, and those produced by the MDT method lie in between. However, the maximum differences are observed at time instants in which the amplitude of the transmitted pulses is relatively high.

As an additional criterion for the comparison of the performance of the different modeling approaches, the mean quadratic error corresponding to the differences shown in Figure 8 was computed. A time window with 0.82 < *t* < 0.95 was chosen to avoid the large oscillations and to focus on those times for which the differences between the methods are more apparent.

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.


**Table 1.** Values of the mean quadratic error: T-junction.

Abbreviations: 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.


**Table 2.** Values of the mean quadratic error: Y-junction.

Abbreviations: FCT: flux-corrected transport; MDT: momentum diffusion term; 1D: conventional pressure-loss model.

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.

**Figure 9.** Comparison between the different models considered for the Y-junction (time domain): (**a**) excitation at port 1; (**b**) excitation at port 2; (**c**) excitation at port 3. Ports are denoted as in Figure 1. <sup>Δ</sup>*R*(*i*): difference in the pulse reflected at port *i*; Δ*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.
