## 3 Experimental Study: Validation of Simulations and Quantification Effect of Friction
### Results: Shock Wave Pressure Jump Observation  
The signal was digitally filtered (zero-phase digital filtering, \(f_{c}=5\) kHz) and the measurement repeated 10 times and averaged in order to improve the signal-to-noise ratio. Figure 11 gives the measurement of the pressure jump at the ten different locations on the duct. As expected, Sensor 2 (located closest to the leak) measured the pressure jump first. Afterwards, both Sensors 1 and 3 measured the pressure jump at the same moment since they were both located at the same distance 40\(D_{d}\) from the leak. The same reasoning can be applied for the other sensors. After the first shock front arrival, consequent pressure steps were observed for the 10 sensors. [PERSON] et al. showed in their work [11] that the pressure steps represent the reflections of the waves on the hard end walls of the duct. The information needed for the localization of the leak resides only in the first shock front arrival; therefore, a zoom of the first pressure step is shown on the right of Figure 11.  
The amplitude measured at Sensor 2 reached 43 Pa. A difference of 7 Pa was observed compared to the theoretical expected value of 36 Pa (\(\approx\)20% relative error on the amplitude), computed based on Equation 3. As was the case for the simulations, an amplitude decrease (\(\approx\)6 Pa) was observed between Sensors 2 and 10. This resulted in a decrease in amplitude due to the friction between the shock wave and the solid surface of 3.3 Pa/m. Note that this amplitude cannot be compared to the simulations, which were performed in 2D. After filtering, the maximum peak-to-peak noise correspond  
Figure 11: Pressure–time measurement at different upper duct wall locations.  
Figure 10: Experimental setup for wave propagation observation.  
this, shock wave reflections (with an amplitude higher than average noise on the signal) were identified after the first shock front arrival. This shows a similar behavior as the one highlighted in the simulations.  
The experimental wave propagation velocity is given in Figure 12. To obtain an accurate arrival time of the wave, a polynomial curve fitting was applied over the curve slopes of Figure 11. The moment of wave arrival was defined as the moment at which the polynomial fit exceeded the maximum noise on the signal. The measurement points on the graph corresponded to Sensors 3-10 located on the right-hand side of the leak. Sensor 1 was located on the left-hand side and is therefore not shown. In turn, Sensor 2 is not depicted in the plot since it was used as a reference point to compute the wave velocities. Due to the inconsistency of the valve dynamics, it gave a more accurate result for the velocity (since the moment at which the wave passed at the location of Sensor 2 was known). The velocity evolved to an asymptotic value (\(\approx\)333 m/s) close to the speed of sound.  
For both simulations and experiments, a decrease in velocity was observed with propagation distance. Quantitatively however, differences were noted. It is important to remark that the length scales considered for the numerical simulations and the experiments were different. In Figure 13, both numerical and experimental results are shown. A simulation domain of \(L=15D_{d}\) around the leak was chosen to avoid excessive computational times. On the other hand, it was difficult to experimentally measure in the region close to the leak. This is why an experimental duct length of \(L=280D_{d}\) was chosen. Next to that, it also allowed observing the effect of viscosity for long propagation distances. A power curve fitting over the complete data is given in the plot (the fit type that resulted in the highest \(R^{2}\) value). The same tendency of decreasing propagation speed was observed for both cases. This change in wave propagation velocity complicates the localization system and should therefore be taken into account for the localization.  
Figure 12: Wave velocity evolution with propagation distance.