**2. Materials and Methods**

Two junctions, shown schematically in Figure 1, were manufactured. A T-junction and a Y-junction were considered, in order to allow the analysis of the effect of the angle of the side branch. An internal diameter of 51 mm was used in all the branches of the junctions.

**Figure 1.** Junctions considered in the study.

While several formalisms may be used for the representation of the transient response of a system, the most intuitive one for the present case of a junction is that based on the consideration of wave components, so that the junction is actually regarded as a multi-port. In this framework, for a junction such as that represented in Figure 2, one has three excitations and three responses, and writing the relations between them directly in matrix form in the frequency domain, one has:

$$
\begin{bmatrix} B\_1 \\ B\_2 \\ B\_3 \end{bmatrix} = \begin{bmatrix} R\_1 & T\_{21} & T\_{31} \\ T\_{12} & R\_2 & T\_{32} \\ T\_{13} & T\_{23} & R\_3 \end{bmatrix} \begin{bmatrix} A\_1 \\ A\_2 \\ A\_3 \end{bmatrix}. \tag{1}
$$

where, as indicated in Figure 2, *Ai* represents the wave component moving towards the junction in port *i* and *Bj* represents the wave component moving away from the junction in port *j*. Regarding

the matrix elements, *Ri* denotes the reflection coefficient as seen from port *i* whereas *Tij* denotes the transmission coefficient between ports *i* and *j*. All these magnitudes are functions of the frequency *f* .

**Figure 2.** Wave components acting on a multi-port.

In this way, one has a reflection coefficient for each of the pipes arriving at the junction, and transmission coefficients for all the possible transmission paths, indicated by the corresponding subscripts. The experimental setup and the corresponding measurement procedure are described in detail in [32,33], and here only a brief overview is given in Appendix A.

Two different modeling approaches were evaluated: a staggered mesh finite volume method and, as a reference, a more conventional pressure loss-based model. The staggered mesh finite volume method used is described in detail in [34], where a flux-corrected transport (FCT) technique was used in order to suppress the spurious oscillations that these numerical methods exhibit in the vicinity of discontinuities in the flow variables. It was found that satisfactory results were obtained through the combination of dissipation via damping together with the phoenical form of the anti-diffusion term. As an alternative, the momentum diffusion term described in [35] was also used. A brief summary is given in Appendix B.

The pressure loss-based model uses a conventional one-dimensional finite volume model with a collocated mesh, derived from the code available in [36]. The junction is modeled as a small volume with three connections to which different pressure loss coefficients are assigned, and in which the mass and energy conservation equations are solved. For the connection to the duct where the incident pressure pulse propagates it has been assumed that the total pressure loss is zero, whereas for the other two connections their corresponding pressure loss coefficients are computed following the expressions given in [15,16]. The procedure is summarized in Appendix C.
