## 2 Validation of the CFD Approach
### Mesh and Computational Domain  
The computational domain has been built on the basis of the geometrical coordinates of the hub, shroud and blade provided by [PERSON] (2002) [20], for both the impeller and the vaned diffuser. Structured grids have been generated with Turbogrid for the following parts, as reported in Figure 2: (A) single channel of impeller and a portion of the convergent duct, (B) single channel of vaned diffuser and a portion of the vaneless diffuser.  
The computational domain upstream of the rotor is composed by a non-rotating subdomain from the inlet section to the rotor ogive where the rotating domain starts; at this point, an interface plane with no mixing is defined. The impeller channel is discretized with an O-grid that consists of 40 elements along the blade height. The tip clearance is subdivided into 30 mesh elements. The resulting y\({}^{+}\) value is 0.5 at the hub and 0.467 on the blade. The convergent duct is discretized with a coarser H-grid to save grid elements in a zone without complex flow structures. The diffuser channel is also discretized with an O-grid with 45 elements along the height and this yields y\({}^{+}\) = 0.5 at the hub and shroud walls and y\({}^{+}\) = 0.631 on the blade. The above values of y\({}^{+}\) are average values on the corresponding wall. The mesh of the convergent duct + impeller consists of \(2.4\times 10^{6}\) elements, while the mesh of the diffuser has \(0.5\times 10^{6}\) elements.  
\begin{table}
\begin{tabular}{c c c} \hline \multicolumn{3}{c}{**Compressor:**} \\ \hline Rotational speed at design point & N\({}_{0}\) & 35,200 rpm \\ Maximum total pressure ratio & \(\pi_{\text{t,max}}\) & 4.07 \\ Maximum isentropic efficiency (tot/tot) & \(\eta_{\text{stt,max}}\) & 0.834 \\ Mass flow at maximum efficiency & \(\pi_{\text{corr}}\) & 1.956 kg/s \\ \hline Impeller: & & \\ \hline Exit radius & r\({}_{2}\) & 135 mm \\ Number of blades & Z\({}_{\text{I}}\) & 15 \\ Blade backsweep angle at exit & \(\beta_{\text{bs}}\) & 38\({}^{\circ}\) \\ \hline Diffuser: & & \\ \hline Radial gap & r\({}_{4}\)/r\({}_{2}\) & 1.10 \\ Diffuser height & b & 11.1 mm \\ Number of blades & Z\({}_{\text{D}}\) & 23 \\ Vane angle & \(\alpha_{\text{4 SS}}\) & 16.5\({}^{\circ}\) \\ Vane wedge angle & \(\alpha_{\text{V}}\) & 6.615\({}^{\circ}\) \\ \hline \end{tabular}
\end{table}
Table 1: Technical data of compressor for nominal speed and diffuser reference geometry.  
Figure 1: Measurement planes in meridional (left) and blade to blade (right) view ([PERSON], 2002a) [23]. Dimensions are in [mm] and referred to the configuration [\(\alpha_{\text{4 SS}}=16.5^{\circ}\), r\({}_{4}\)/r\({}_{2}=1.04\)].  
All CFD simulations have been carried out with 2 impeller channels and 3 diffuser vanes without the volute.  
This solution gives a ratio of the circumferential pitch of the rotor domain very close to the circumferential pitch of the stator domain (diffuser vanes) with a pitch ratio at the impeller-diffuser interface very close to one (PR = 1.02). With the above condition, the use of periodic flow option in the circumferential direction is also acceptable for unsteady simulations. The whole CFD model domain consists of \(6.3\times 10^{6}\) elements. A mesh sensitivity has been performed to check that the compressor performance undergoes a variation of less than 1% with the different meshes. The mesh sensitivity and mesh parameters have been tested according to previous experience ([PERSON] et al., 2017) [16].