3.2.2. Frequency Domain Assessment

As already detected when describing the experimental results, it is in the frequency domain where the benefits of the staggered mesh finite volume method are more apparent. Consider first the results corresponding to the T-junction, shown in Figure 10 in the case of the reflection coefficients. Here, using either MDT or FCT as flux limiter, the staggered mesh finite volume method produces results for the reflection coefficients which overestimate dynamic effects when the excitation is at port 1, but produces a suitable approximation up to 1000 Hz when the excitation is at port 3 and the FCT flux limiter is used. In comparison with this, it is apparent that the conventional pressure loss model (again labeled as 1D in the figure) is unable to fully capture the dynamic features of the results, while still providing a sort of suitable average value, even if all the dynamic issues are lost, as an unavoidable consequence of the quasi-steady assumption underlying the calculation.

**Figure 10.** Comparison between the experimental results and the different models for the reflection coefficients of the T-junction (frequency domain): (**a**) excitation at port 1; (**b**) excitation at port 3. Ports are denoted as in Figure 1.

The corresponding transmission coefficients are shown in Figure 11, where it can be observed that the staggered mesh finite volume method produces results that follow the overall trend of the experimental results, with two exceptions: when the excitation is at port 1 the method underestimates the transmission to port 3, and when the excitation is at port 3 the method is unable to capture the behavior observed between 1000 and 1250 Hz. In the case of the conventional model, it is apparent that in this case it is fully unable to reproduce neither the level nor the dynamic features of the measured data, the only acceptable results being produced when the excitation is at port 1 and that only for very low frequencies.

This essential difference between the two modeling approaches considered is even more apparent in the case of the Y-junction, whose results are shown in Figures 12 and 13 for the reflection and transmission coefficients, respectively. In this case, the results provided by the conventional model are rather similar regardless of the port at which the junction is excited. In all the cases, an acceptable value of the transmission coefficient in the very low frequencies is produced in those propagation paths with smaller change in direction, and also a suitable average value for the reflection coefficient as seen from any of the exciting ports. However, differences in transmission between the two propagation paths are not reproduced in any case and, moreover, the results start to decrease monotonically at about 200 Hz and reach totally unrealistic values for frequencies above 750 Hz in all the cases.

On the contrary, the staggered mesh finite volume method reproduces quite fairly the overall dependency with frequency, but tends to overestimate the influence of the change in direction of the propagation path on the transmission coefficients (and thus to underestimate the value of the corresponding coefficient). With this geometry, this effect is especially evident in the results obtained with the FCT flux limiter for |*<sup>T</sup>*13|, |*<sup>T</sup>*23|, |*<sup>T</sup>*31| and |*<sup>T</sup>*32|, i.e., all the cases in which the side branch (port 3) is involved. On the contrary, the results of the FCT method are affected by a certain overestimation when transmission through the main branch is considered (|*<sup>T</sup>*12| and |*<sup>T</sup>*21|). Accordingly, with the

description given in Appendix B, this difference in behavior between the FCT and the MDT methods can only be due to the effect of the application to the junction itself of the different ways used to handle the information of the neighboring volumes when limiting the flow.

**Figure 11.** Comparison between the experimental results and the different models for the transmission coefficients of the T-junction (frequency domain): (**a**) excitation at port 1, transmission through port 2; (**b**) excitation at port 1, transmission through port 3; (**c**) excitation at port 3, transmission through port 1; (**d**) excitation at port 3, transmission through port 2. Ports are denoted as in Figure 1.

**Figure 12.** Comparison between the experimental results and the different models for the reflection coefficients of the Y-junction (frequency domain): (**a**) excitation at port 1; (**b**) excitation at port 2; (**c**) excitation at port 3. Ports are denoted as in Figure 1.

**Figure 13.** Comparison between the experimental results and the different models for the transmission coefficients of the Y-junction (frequency domain): (**a**) excitation at port 1, transmission through port 2; (**b**) excitation at port 1, transmission through port 3; (**c**) excitation at port 2, transmission through port 1; (**d**) excitation at port 2, transmission through port 3 (**e**) excitation at port 3, transmission through port 1; (**f**) excitation at port 3, transmission through port 2. Ports are denoted as in Figure 1.

In the case of the reflection coefficients, the overestimation of the junction dynamics already observed in the T-junction is also present here when the junction is excited at port 1, but the measured dynamics are quite successfully reproduced when the junction is excited at ports 2 and 3. The characteristic frequencies governing the reflection coefficient are not exactly captured, but the overall amplitude and the influence of the exciting port are reproduced by the numerical results.

### *3.3. Assessment of a Modelling Approach with a Quasi-3D Description of the Junction*

In order to explore the additional potential offered by the staggered mesh method regarding the approximate solution of the three-dimensional flow field inside the junction, such an approach (commonly referred to in the literature as quasi-3D) was finally considered. In Figure 14 the mesh used is shown together, for reference, with that used in the previous subsections. The four volumes at the endpoints of the part shown are then connected to a single volume, thus providing the connection with the one-dimensional computation at the ducts. The mesh chosen is relatively modest, in order to

keep the computation time at reasonable values, but sufficient to show any potential advantages of this description.

**Figure 14.** Meshes used in the staggered-grid method: (**a**) a 0D description of the junction; (**b**) a quasi-3D description.

Additionally, in view of the previous results, only the MDT method will be used as a flow limiter, since overall it has appeared to be more robust and consistent, and only the case of the T-junction will be analyzed in the following, as no new qualitative issues have been identified in the Y-junction that were not present also in the T-junction.
