## 2 Materials and Methods
### Mesh  
The mesh employed is a hybrid mesh formed using the blocks shown in Figure 3. All of the blocks are structured, apart from the two unstructured, marked with a \(u\). These blocks are used to adapt the boundary layer mesh to the near field domain, avoiding bad quality elements, as discussed at the end of this section. These unstructured regions use mainly quadrilateral elements.  
A mesh sensibility study was conducted for the _base_ case in [15]. The chosen mesh from that study had \(171\),\(947\) elements for the largest blockage ratio (\(0.75\)). The mesh employed in [12] for the same geometry and speed had \(77\),\(000\) elements. As the mesh  
Figure 3: Blocking chosen for the mesh. Reference for the _base_ case [15].  
Figure 2: Schematic domain for the _compressor_ case. Not to scale.  
parameters are equal in the _base_ and _compressor_ cases, a new mesh sensibility study has not been conducted for this work.  
Regarding these parameters, the boundary layer on the tube wall has been set as a geometric progression using a first element height of \(5\times 10^{-6}\) m, a progression ratio of 1.25, and 38 layers. For the boundary layer on the capsule, the resolution is increased, using a first layer height of \(10^{-6}\) m, a geometric ratio of 1.2, and 58 layers. These prism layers allow obtaining a \(y^{+}=0.83\) on the vehicle wall and \(y^{+}=1.74\) on the tube wall.  
The capsule boundary-layer geometric progression is maintained until the normal distance to the wall (horizontal or vertical) is 0.37 m, which implies that the largest element on the unstructured region is 1.75 times larger than the last element height on the boundary layer. Then, the space between the two boundary layers has used a linear progression that ensures the same spacing on the border elements between blocks.  
The mesh size on the compressor, turbine, and front and rear faces of the vehicle have a uniform spacing, equal to the last boundary layer height (0.0326 m). On the channel, a bigeometric progression is imposed, using a geometric ratio of 1.05 for both ends.  
To avoid wave reflection, a large element size is imposed near inlet and outlet boundaries. These elements are the results of imposing a geometric progression from the end of the unstructured block to the inlet and outlet boundaries. The growth ratio is 1.012 with 400 layers upstream, and 1.006 with 700 layers downstream.  
Using the mesh parameters already described, the meshes for the different blockage ratios are obtained. They are shown in Figure 4 for the _compressor_ case and in Figure 5 for the _base_. Note that both cases are equivalent in terms of element size and distribution. The captures are taken for the case of \(\beta=0.75\), as it is considered to be the most restrictive due to the proximity of both boundary layers. For the first case, a capture for \(\beta=0.20\) is also provided to better show how the channel between the vehicle and the tube has been discretized.  
Figure 4: Captures for the mesh used in the _compressor_ case and \(\beta=0.75\). Top figure is for the upstream domain, and bottom is for the wake.  
The quality and number of elements for each mesh are collected in Table 1. The quality is always above 0.37 for the _base_ case, and above 0.33 for the _compressor_ case. Moreover, only 1 element has a quality below 0.4 for the first case and only 20 (around 0.01%) for the second case. To better show what this implies for the actual mesh, Figure 6 represents the worse quality elements for the _compressor_ case, whose quality is slightly lower. These elements are mainly near the border between the structured and unstructured blocks, as the mesh has to transition from the boundary layer size to the near field size.  
Figure 5: Captures for the mesh used in the _base_ case. Except the upper one, which is for \(\beta=0.2\), the rest of the images are for \(\beta=0.75\). The top figures show the upstream domain, and the bottom one shows the wake.