3.1.2. Frequency Domain Analysis

Here, the results obtained for the transmission and reflection coefficients defined in Equation (1) are analyzed. For brevity, only the modulus of these coefficients will be considered, as this contains significant information about the overall energetic behavior of the junction. The results for the T-junction are shown in Figure 5, where it can be observed that the values obtained in the very low frequency range (below 200 Hz) are fully consistent with the time domain results shown above in Figure 3: when exciting the junction at port 1, it is seen that |*<sup>T</sup>*12| is systematically larger than |*<sup>T</sup>*13| in this frequency range, whereas when the excitation is at port 3 the differences between |*<sup>T</sup>*31| and |*<sup>T</sup>*32| are significantly smaller.

**Figure 5.** Experimental results for the T-junction in the frequency domain. (**a**) Excitation at port 1; (**b**) excitation at port 3. Ports are denoted as in Figure 1. *Ri*: reflection coefficient as seen from port *i*; *Tij*: transmission coefficient between ports *i* and *j*.

At higher frequencies, above 200 Hz, it can be seen that the behavior of |*<sup>T</sup>*12| and |*<sup>T</sup>*13| is essentially flat around mean values of 0.65 and 0.64, respectively, with a maximum deviation from the mean of 0.065 in |*<sup>T</sup>*12| and of 0.05 in |*<sup>T</sup>*13|. On the contrary, in the case of |*<sup>T</sup>*31| and |*<sup>T</sup>*32| their mean values are very similar to those of |*<sup>T</sup>*12| and |*<sup>T</sup>*13| (0.64 and 0.63, respectively) but some relevant acoustic features can be detected in both coefficients between 1000 and 1250 Hz, mostly in the case of |*<sup>T</sup>*31| where the deviation from the mean value reaches a maximum of 0.125, while |*<sup>T</sup>*32| follows the same trend but with a maximum deviation from the mean of 0.08 . This confirms, on one hand, that when the junction is excited at port 3 the two propagation paths are substantially equivalent and, on the other hand, that their behavior is different from that obtained when exciting the junction at port 1.

This second statement is fully supported by the spectra of the reflection coefficients |*<sup>R</sup>*1| and |*<sup>R</sup>*3|: it is apparent that |*<sup>R</sup>*3| is overall larger than |*<sup>R</sup>*1| for frequencies below 200 Hz, as suggested by the results shown in Figure 4, but now without any uncertainty due to the difference in amplitude between the incident pulses used in each test. Additionally, the trend observed is rather different for frequencies above 200 Hz, and most notably above 1000 Hz, where |*<sup>R</sup>*1| shows a certain decreasing tendency whereas |*<sup>R</sup>*2| increases with frequency.

The corresponding results for the Y-junction are shown in Figure 6. Again, results below 200 Hz confirm the time domain tendencies observed in Figure 4. In this frequency range, it is seen that while |*<sup>T</sup>*12| is only slightly higher than |*<sup>T</sup>*13|, when exciting at ports 2 or 3 one finds that the transmission coefficient corresponding to a smaller change in direction (that is, |*<sup>T</sup>*21| when exciting at port 1 and |*<sup>T</sup>*31| when exciting at port 3) is significantly larger than the other one, and that this effect is more noticeable the larger is the change in direction.

**Figure 6.** Experimental results for the Y-junction in the frequency domain. (**a**) Excitation at port 1; (**b**) excitation at port 2; (**c**) excitation at port 3. Ports are denoted as in Figure 1. *Ri*: reflection coefficient as seen from port *i*; *Tij*: transmission coefficient between ports *i* and *j*.

When considering frequencies above 200 Hz, noticeable differences are also observed between the case with excitation at port 1, for which results very similar to those shown in Figure 5a are obtained, with small differences between |*<sup>T</sup>*12| and |*<sup>T</sup>*13| and an almost flat behavior with little dependency on frequency, and the other two cases, in which the transmission coefficients corresponding to the propagation path with the smallest change in direction (|*<sup>T</sup>*21| and |*<sup>T</sup>*31|) are significantly and systematically higher than those implying an important change (|*<sup>T</sup>*23| and |*<sup>T</sup>*32|, respectively) except at the highest frequencies represented.

However, it is in the reflection coefficients where the effect of the change in the excitation port is more apparent. In fact, the results for |*<sup>R</sup>*1| do not differ substantially from those obtained for the T-junction and shown in Figure 5a, neither in the low frequency values nor in the high frequency trend. On the contrary, the high frequency behavior seen in |*<sup>R</sup>*2| and |*<sup>R</sup>*3| is a clear indication of the change produced in the dynamic characteristics of the junction when the excitation port is changed, an effect that could be guessed from the time domain results of Figure 4 but now is fully confirmed. Actually, a well-defined trend of picks and troughs can be observed in both cases, with similar shape but a clear frequency shift, which provides a sort of acoustic signature of the dynamic behavior of the junction. The fact that such a behavior is not apparent in the spectrum of |*<sup>R</sup>*3| for the T-junction shown in Figure 5b indicates that such dynamic issues are suppressed by the symmetric nature of the excitation through a perpendicular branch.

### *3.2. Assessment of Modelling Approaches Considering a 0D Description of the Junction*

Here, modeling approaches in which the junction itself is regarded as a 0-dimensional element, while the flow in the adjacent ducts is assumed to be one-dimensional, will be evaluated. In the context of the staggered mesh finite volume method, this corresponds to the case in which the junction is regarded as a single volume and the adjacent ducts are meshed only in the axial direction. The pressure loss-based model used here falls also within this category since, as commented in Appendix C, the junction branches are connected through an auxiliary 0D element.

Again, separate analyses for the time and the frequency domains are presented.
