## 2 Numerical Study: Physical Understanding of Wave Propagation Mechanism
### Results: Shock Wave Propagation Visualization  
The propagation of the shock wave is shown in the static pressure contour plots given in Figure 6. Cell values were used to avoid discontinuity so as to be smeared graphically by node averaging. A shock wave is a discontinuity created when a disturbance propagates at a faster speed than the speed of sound in a fluid. This abrupt discontinuity changes the characteristics of the medium and gives rise to irreversible pressure changes in the fluid. The same behavior is observed in the given contour plots. First, a small spherical shock can be seen as a centered energy source was created at the contact between the leak and the duct. This shock expanded spherically, dissipated energy (leading to a decrease in amplitude and propagation speed), and traveled till reaching the lower duct wall. At the lower wall, a hard wall shock wave reflection created a new increase in pressure and amplification of the shock strength. The strong shock produced after reflection propagated faster towards both ends of the duct and had a higher amplitude than the incident shock front. This can be explained by the fact that the passage of the first shock increased the velocity of the medium in which the reflected shock would propagate. The reflected shock front would therefore propagate through a medium that was already in motion and catch up with the first shock front with propagation distance. Both waves collapsed together and formed a Mach stem reflection on the lower wall of the duct. The Mach stem intensity was twice as high as the incident shock front [15]. This was observed on the contour plot since the static pressure value of the shock front at the lower wall (where the Mach stem was formed) was higher w.r.t. the upper wall. After a certain propagation distance, the combination of these effect created a complex reflection mechanism (see  
Figure 5: Grid convergence study: (**Left**) wave pressure amplitude; (**right**) wave velocity. \(t_{3}\) in Figure 6. To finish, the presence of a supersonic localized jet with so-called diamond shocks was observed at the exit of the leak. The diamonds were stationary and would not travel through the capillary, and were therefore not important for this study. The propagation of the first shock front can therefore be considered as a phenomenon on its own, since it was not influenced by the filling of the duct due to the jet. It was observed that the flow velocity close to the leak was not increasing in the first instance. Meanwhile, the shock front was created and propagated further without any influence of the incoming flow. Only after the shock wave passage (local irreversible perturbation), the jet started to fill the capillary, and the flow velocity increased. It was concluded that only the behavior of the first shock front, which evolved from a spherical shock front to a normal shock front with traveling distance, was important for the present application. The shock front pressure amplitude at the upper duct wall was extracted at different axial locations and is shown in the left plot of Figure 7 together with a power curve fit on the data (a fit type that resulted in the highest \(R^{2}\) value). The leak is located at \(x=0\) mm. A lowering of the amplitude with traveling distance was observed, and it evolved to an asymptotic value of 300 Pa. The shock wave carried energy that dissipated with traveling distance due to viscosity. It is known that shock waves are much more sensitive to viscous weakening effects in comparison to acoustical pressure waves. According to [15], the shock-boundary layer interaction implies a lowering of the shock wave amplitude and propagation speed. In the right plot of Figure 7, the propagation velocity of the first shock front was computed based on the arrival time and distance traveled by the shock. As expected, the wave propagation speed showed the same decreasing behavior as for the amplitude (see Equation (2)). The velocity evolved to an asymptotic value of 347 m/s.