## 2 Materials and Methods
### Inlet Boundary Condition  
Concerns about the choice of the boundary condition at the inlet exist due to the simplification of the problem. This issue was analyzed in a previous work conducted by the present authors [15]. In that study, it was concluded that the value of the drag is highly dependent on the upstream boundary condition. Two different boundary conditions were analyzed, total pressure inlet and mass flow inlet, reporting large differences between them.  
In the real case, due to the wave propagation inside a tube, the upstream thermodynamic conditions vary continuously. The consequence is that the drag is not only a function of the actual speed of the capsule \(V_{ref}\), but also of the position of the train inside the tunnel, how fast this speed has been reached, and where the boundary conditions are placed (i.e., tube walls, another vehicle, changes on the tube size...). This effect cannot be reproduced with the current simplification of the problem and makes it not possible to set a constant value for the inlet boundary.  
As a representative average result, the boundary condition chosen at the inlet is imposing the mass flow. This condition is reported to be the most restrictive, and it represents the case in which all the mass flow passes through the channel between the tunnel and the train. This is considered by the authors to be the design point of hyperloop to avoid air compression in front of the capsules, which would considerably increase the drag on a route. As reported in [15], if the total pressure is imposed, the mass flow that should pass through the vehicle may not be guaranteed.  
This is a relevant difference to an open tunnel. While, in a closed tube, if the required air does not pass around the vehicle it ends up being compressed in the front, with open ends, this displaced air could exit the domain without being compressed.  
However, the inlet boundary condition is not the only factor that affects the flow behavior upstream. If too much mass flow is imposed on the inlet, it reaches what is known as the Kantrowitz limit [34]. This limit occurs when the flow reaches sonic conditions (Mach equal to 1) on the smaller cross-sectional area, which is the passage between the tunnel and the capsule. Using an isentropic 1D simplification, the Kantrowitz limit leaves a 2D space in the plane (\(\beta\),\(M_{ref}\)) in which the flow is not blocked and all the mass flow can pass around the capsule without increasing its far field pressure. This solution space is shown in Figure 7.  
Figure 7: Kantrowitz limit as a function of the inlet Mach of the domain \(M_{ref}\) or the blockage ratio \(\beta\). Speeds of 500 km/h and 700 km/h, computed using a static temperature of 288.15 K, are marked with dashed lines.  
The Mach number where the Kantrowitz limit is achieved is known as the critical Mach number. Beyond this value, if the physical mass flow is imposed, the pressure has to increase over \(p_{ref}\) and speed has to decrease under \(V_{Ref}\) to maintain the inlet corrected mass flow. This effect will be discussed in Section 4 alongside the results. There, it will be proved how the addition of a compressor mitigates the induced effects.  
It is important to remark that for the geometries without compressor, at 700 km/h, all the blockage ratios are above the Kantrowitz limit, as the limit is \(\beta=0.18\); and for 500 km/h, the limit is \(\beta=0.36\). For the _compressor_ case, the blockage ratio that reaches this limit is discussed in Section 4, as it cannot be analytically obtained.