## 2 Methodology  
The objective of this analysis is to develop a transonic linear cascade wind tunnel. The design arises from the need to test numerically and experimentally different intensities of shock wave systems within vaned diffusers, while minimizing computational and experimental resources. With a mass flow imposed on the linear cascade inlet, it was anticipated to achieve at least three operating points on the cascade channels. This is due to the distinct flow characteristics and mass flow passing through every channel. Figure 1 shows the initial configuration of the mixed flow/centrifugal compressor, designed as part of a microgas turbine engine application. To obtain the infinitely long linear cascade, the cylindrical intersection of the blade row needs to be topologically unwrapped into a plane. Figure 2a presents the middle plane  
Figure 1: Centrifugal compressor stage—baseline geometry. of a vaned diffuser from which the geometry of the blades is extracted, while Figure 2b shows a part of the initial linear cascade that resulted from this procedure. Due to the very small vane height (i.e., 4 mm), it was decided to scale up the model \(2.5\times\) in height and \(2\times\) in the other two dimensions. This is acceptable since the unwrapping is equivalent to a 2D blade-to-blade mapping of the original vane, which means that the height of the cascade does not influence the flow features to be investigated. The baseline cascade model defined in Figure 2b was used for a preliminary numerical analysis of the flow structure. To capture different intensities of shock waves, multiple working points were simulated using ANSYS CFX. The flow angle was adjusted to prevent boundary-layer separation, and, eventually, it was defined as a boundary condition. No further changes were made to the geometry to correct the flow pattern in this initial stage. Since the stator flow enters the cascade at a high angle (\(\alpha_{\text{inlet}}=68^{\circ}\)), for practical purposes, the blade row itself was rotated by the said angle \(\alpha_{\text{inlet}}\) in the actual test rig. Therefore, the final experimental configuration considered for the transonic cascade is the one presented in Figure 3.  
Figure 3: Three-dimensional representation of the test rig. Figure 2: Transformation of a vaned diffuser into a linear cascade: (**a**) 50% blade to blade cut the vaned diffuser, and (**b**) the plane-unwrapped blade to show the blade view of the 5 instances of main and splitter vanes. The geometry consists of five groups (main blade and splitter) whose parameters and reference flow conditions are defined in Table 1.  
This geometry was defined to observe the nominal inlet flow angle. However, each blade assembly (main blade and splitter) works on a different mass flow. This approach allows the observation of shockwave structure and behavior over a \(\pm\)9% mass flow interval around the nominal point. This leverages the natural tendency of the rig to induce different local mass flows per individual passage, thereby facilitating the study of the flow differences between the CFD simulations and the experiments. Two sets of simulations were performed. The first simulation assumed full periodicity of the passage and utilized 2\(\times\) instancing of the main blade + splitter (to eliminate the possibility of interface artefacts). The second simulation was performed for the full cascade, with the wall interference and vane inter-dependencies that arise from the experimental setup. The numerical simulations were performed using the commercial software Ansys CFX ([[https://www.ansys.com/products/fluids/ansys-cfx](https://www.ansys.com/products/fluids/ansys-cfx)]([https://www.ansys.com/products/fluids/ansys-cfx](https://www.ansys.com/products/fluids/ansys-cfx)), accessed on 23 August 2023), employing the steady-state RANS approach. The solver model implemented by CFX is a pressure-based implicit coupled solver. To accurately capture shock waves and adequately resolve advection and turbulence model equations, high-order schemes were employed.