## 2 Methodology
For the continuity equation and velocity components, the residuals were set to 1 \(\times\) 10\({}^{-6}\). Additionally, the overall imbalance among pressure, energy, and velocity was less than 0.15%. Due to the asymmetry of the full cascade, it was anticipated that distinct operating points would be achieved in each passage. As a result, numerical simulations were conducted for each passage (idealized periodic passage) using the corresponding conditions from the linear cascade (as detailed in Table 2). This validation primarily aimed to confirm whether the same flow structure and pressure losses would be attained under those conditions. Note that there is a higher mass flow per passage (idealized periodic) than the one estimated by the individual sections in the linear cascade (non-idealized periodic), as shown in Figure 7. It is important to acknowledge that this technique suffers from inaccuracies, particularly when dealing with cross sections featuring high velocity and density gradients. In essence, when computing the mass flow passing through a section, the orientation of \begin{table}
\begin{tabular}{c c} \hline \hline
**Position** & **Inlet Mass Flow [kg/s]** \\ \hline Passage 1 & 0.089865 \\ Passage 2 & 0.0898 \\ Passage 3 & 0.08835 \\ Passage 4 & 0.08675 \\ Passage 5 & 0.08475 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Flow conditions for the single-passage cascade. Figure 6: Wall y+ distribution: (**a**) ideal periodic vanes and (**b**) the linear cascade setup, showing the streamline intersection with the cascade geometry of individual cutplanes. the particular virtual boundary in that section influences the result. Even if the cutplane is normal to the vane geometry, there is still the influence of the flow pattern (vortices, density, and velocity distribution), which may lead to inaccuracies. For this particular case, the results were deemed to be sufficiently close to the original estimations, and therefore, the geometric normal cross section was used. This is mainly because the geometry of the cutplanes does not cut through the vortices, as seen in Figure 8. In other cases where differences become more pronounced, one might consider a cross section that is also determined by the flow features observed, particularly if the vortices do not convect or if they occupy a large area of the cutplane. In the first two passages of the cascade, higher operating conditions, along with blade geometry, lead to the generation of shock waves. These, in turn, give rise to the formation of three-dimensional vortices as a consequence of boundary-layer separation. As depicted in Figure 8, the zone impacted by these vortices is situated in proximity to the trailing edge of the blades. Here, vorticity stretching induces flow separation on both sides of the blades. The blade curvature in this vicinity, combined with the high velocity of the three-dimensional fluid flow, creates the optimal conditions for the development of three-dimensional vortices. In the examined scenario, the vortices exhibit diminished magnitude, thus exerting a localized effect primarily in the vicinity of the trailing edge. Consequently,  
Figure 8: Streamline intersection with the cascade geometry of individual cutplanes. Figure 7: Histograms of individual passage flow estimated from the cutplanes of the linear cascade (non-idealized periodic) vs. inlet flow of the idealized periodic passage. as long as the affected region remains restricted to the proximity of the trailing edge, it exerts a minimal impact on the intended objectives of the study.