## 2 Materials and Methods
### Set Up  
The simulation was conducted using a 2D axisymmetric model to reduce the computational cost, maintaining the accuracy of the results. The rotation axis is the _axis_ line shown in Figures 1 and 2. Previous studies also used this hypothesis for both trains at atmospheric pressure [7] and at low pressure [12; 17; 23]. Some authors also compared the accuracy of their results using a 3D and a 2D axisymmetric model. [PERSON] et al. [6] compared the 2D model with experimental results at real scale and atmospheric pressure, concluding that the wave problems were properly reproduced. [PERSON] and [PERSON] [30] performed a similar experiment in 2017, focusing on the transonic entrance of a train in a tunnel with the 2D model. For lower pressures, [PERSON] et al. [14] compared simulations for a sphere, concluding that, at transonic speeds, the maximum error on the drag was about 4%. Regarding the thermodynamic parameters, both the specific heat ratio \(\gamma\) and ideal gas constant for the air \(R\) are considered invariant with the pressure. Regarding the viscosity, the Sutherland model has been used as in [9; 12; 14]. The scheme employed is a second-order spatial discretisation, using \(k-\omega\) SST model as turbulence model as used in [14]. [PERSON] et al. in [12] compared the results for different turbulence models, including \(k-\omega\) SST, \(k-\epsilon\), Spalart-Allmaras, and a laminar case for the same _base_ scenario presented in this work. They reported less than 4% of difference among all the models simulated. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Case** & **BR** & **Min. Quality** & **Mean Quality** & **Elements** \\ \hline _Base_ & 0.20 & 0.3727 & 0.9975 & 297,971 \\ _Base_ & 0.35 & 0.3727 & 0.9968 & 234,251 \\ _Base_ & 0.50 & 0.3727 & 0.9963 & 201,683 \\ _Base_ & 0.60 & 0.3727 & 0.9960 & 187,523 \\ _Base_ & 0.70 & 0.3727 & 0.9958 & 176,195 \\ _Base_ & 0.75 & 0.3727 & 0.9957 & 171,947 \\ \hline _Comp_ & 0.20 & 0.3372 & 0.9982 & 298,121 \\ _Comp_ & 0.35 & 0.3372 & 0.9977 & 234,446 \\ _Comp_ & 0.50 & 0.3372 & 0.9973 & 201,901 \\ _Comp_ & 0.60 & 0.3372 & 0.9971 & 187,751 \\ _Comp_ & 0.70 & 0.3372 & 0.9969 & 176,431 \\ _Comp_ & 0.75 & 0.3372 & 0.9968 & 172,186 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quality and mesh size for the different cases. Figure 6: Lowest-quality (<0.5) elements on the _compressor_ case for \(\beta=0.75\). Furthermore, [PERSON] et al. in [31] conducted a comparison of the different turbulence models concerning experimental results of train aerodynamics. They concluded that the \(k-\omega\) SST model provided the best surface pressure among the available RANS models. The solver is compressible pressure-based. A Riemann solver has been also tested, whose results are shown in Section 3.1. Regarding the steadiness of the case, a steady solver has been selected to reduce the computational cost, as only time-averaged values are to be obtained. An unsteady simulation has also been performed to check the validity of this hypothesis, whose results are shown in Section 3.1. The use of a steady solver is valid since wave propagation, which was not studied, is a time-dependent effect. Only steady-state values for the drag and thrust need to be computed. The different boundaries conditions on the domain, marked in Figures 1 and 2, are as follows:  
* Inlet: the mass flow \(\dot{m}_{ref}\) and the total temperature \(T_{t,ref}\) are imposed. Both are based on the capsule speed \(V_{ref}\), reference pressure \(p_{ref}\), static temperature \(T_{ref}\), and the tunnel area \(\pi h^{2}/\beta\): \[\dot{m}_{ref} =\frac{p_{ref}}{RT_{ref}}\frac{\pi h^{2}}{\beta}V_{ref},\] (3) \[T_{t,ref} =T_{ref}+\frac{V_{ref}^{2}}{2c_{p}}.\] (4)
* Tube: moving and adiabatic wall. The speed on the wall is equal to the speed of the capsule, \(V_{ref}\), in a ground reference frame. * Outlet: pressure outlet, set to the same pressure as the reference pressure of the tube \(p_{ref}\). * Vehicle: static and adiabatic wall.