## 4 Discussion
For the _base_ case, the flow is so accelerated on the channel that obliques shock waves appear. These waves disappear when the compressor is installed, explaining why the required power is considerably reduced. For \(\beta=0.20\) and \(\beta=0.50\), the flow on the _compressor_ case is subsonic everywhere, as these cases are operating under the Kantrowitz limit (see Figure 14). For \(\beta=0.75\), Figure 12c shows that Mach numbers greater than 1 are locally reached, but shock waves are still avoided. It is important to remark that the flow patterns with the compressor are more uniform among the cases. Qualitatively, they are not affected by the size of the tunnel. Figure 11: Relative variation of the _compressor_ case with respect to the _base_ case for different blockage ratios and speeds. Figure 10: Required power comparison for the different blockage ratios and speeds, at 0.1 atm. On the other hand, the effect on the induced speed is shown in Figure 13, where the actual inlet speed from the simulation is plotted against the blockage ratio for different reference speeds. Note that the _compressor_ case, especially at 500 km/h, can maintain the capsule speed at the inlet. However, for the _base_ case, the upstream pressure increase does not allow the inlet speed to be maintained, which decreases linearly with the blockage ratio. Note that induced speed is nearly the same for the _base_ case for both reference speeds, as this effect is mainly dependent on the geometry. Figure 12: Comparison of the Mach contour for the _compressor_ case (**upper figure**) and the _base_ case (**lower figure**), for 700 km/h and different blockage ratios. Not only the mitigation of the shock waves, but the fact that the induced speed for the _compressor_ case is considerably lower is also an indicator of the improvement present in this case. Then, we analyzed how far each of the cases is from reaching the Kantrowitz limit. This is evaluated by computing the Mach number at the end of the passage between the capsule and the tunnel, which is the throat for these cases. Thus, it is the location where sonic speeds are firstly reached. This is represented in Figure 14, for each reference speed and BR. Not only from CFD, but the isentropic Mach number \(M\) is also obtained, computed using the compressible mass flow equation, comparable to the _base_ case:  
\[(1-\beta)M\bigg{[}1+\frac{\gamma-1}{2}M^{2}\bigg{]}^{-\frac{\gamma+1}{2(\gamma -1)}}=M_{ref}\bigg{[}1+\frac{\gamma-1}{2}M_{ref}^{2}\bigg{]}^{-\frac{\gamma+1}{ 2(\gamma-1)}}. \tag{8}\]  
It can be seen that for 700 km/h and no compressor, the capsule always operates above the Kantrowitz limit, which is what the operator wants to avoid. As previously mentioned, the limit is 0.18 according to the analytic approximation. In this scenario, quite big tunnels (BR less than 0.2) have to be used, which is a BR even lower than for HSR (0.23). For 500 km/h, the Kantrowitz limit occurs at 0.36, which is also not a particularly high value. Figure 14: Mach number at the end of the passage for 500 km/h or Mach 0.408 (**left**) and 700 km/h or Mach 0.571 (**right**). Figure 13: Inlet speed for the different blockage ratios and speeds. The use of a compressor increases considerably these limits, allowing full BR operation at 500 km/h, and limiting the operation to tunnels of \(\beta=0.5\) or lower blockage ratios for 700 km/h. The analytical approximation follows closely the trend of the _base_ case. However, it is required to have more CFD points to accurately reproduce the exact BR where sonic conditions are reached. Note that, although the analytical approximation outputs a Mach equal to 1 on the throat, the CFD value is slightly higher than 1. This was also reported by [PERSON] et al. in [36] after conducting some experiments for a Poiseuille flow in a pipe.