## 2 Methodology  
### Physical Model and Numerical Method  
The German Darmstadt transonic single-stage compressor was selected to simulate as shown in Figure 1. This compressor was designed and optimized by the German MTU Aero Engines Company and the German Aerospace Center [29; 30]. Its key parameters are presented in Table 1. The facility operated as an open-loop system, drawing in air from the surroundings, which was channeled through an inlet throttle and a stabilizing chamber before entering the compressor core. After passing through the compressor, the air was expelled back into the environment through an exhaust passage. Powered by an 800 kW Direct-Current motor, the setup incorporated a transmission system that allowed for rotational speeds to reach 21,000 revolutions per minute. To manipulate and establish various operational parameters that align with specific compressor performance curves, the system employed an exhaust control valve equipped with a protective bleed function to adjust the downstream pressure exerted on the compressor stage. The compressor core was built with modularity in mind, allowing for swift and efficient changes among diverse setups, such as altering the rotor, adjusting casing inserts, or updating the instrumentation. The rotor blade was designed with a high degree of curvature proximal to the hub and a small thickness towards the blade tip, whereas the outlet flow of the stator maintained a consistent feature with no evidence of separation across the full spectrum of operational conditions. Furthermore, it also exhibited a uniform distribution of exit angles. This design typifies the front stage of the high-pressure compressor in a turbofan engine. Moreover, the compressor core integrated outlet struts and a modular casing design not only ensure a precise rotor tip gap but also facilitate access to instrumentation. Upstream of the test zone, ambient pressure, inlet temperature, and humidity levels were assessed to establish inlet conditions. The settling chamber served as the site for the derivation of total and static pressure at the inlet. Measured upstream of the compressor core (ME15), the inlet boundary layer profile was utilized to quantify the total pressure losses attributable to the inlet duct. The mass flow rate was ascertained by employing a precision-calibrated bellmouth situated between the settling chamber and the compressor's inlet. At the exit of the stage (ME30), total pressure and total temperature readings were taken using Kielhead probes, which were radially positioned on rakes. In addition, a torque meter setup was used to measure torque and rotational speed, linking the compressor core to the gearbox. The measurement cross-sections of the compressor also include the rotor inlet (ME20) and the rotor outlet (ME21). The performance characteristics of the compressor were calculated based on the average parameters of the stage inlet (ME15) and the stage outlet surfaces (ME30) [31]. The commercial software Numeca equipped with Fine/Turbo solver was taken to simulate the fluid flow in the blade passage. The compressible Reynolds-averaged Navier-Stokes equations were solved based on the conservation equations of mass, momentum, and energy, as shown in Formula (1)-(4). \[\frac{\partial U}{\partial t}+\frac{\partial(F-F_{v})}{\partial x}+\frac{ \partial(G-G_{v})}{\partial y}+\frac{\partial(H-H_{v})}{\partial z}=J \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline In/out flow & Axial–axial \\ Rotational speed & 20,000 rpm \\ Max diameter & 0.38 m \\ Number of rotors & 16 \\ Number of stators & 29 \\ Hub-to-tip ratio & \(\sim\)0.5 \\ Relative Mach number at tip & \(\sim\)1.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key parameters of the compressor. Figure 1: Test rig of Darmstadt transonic compressor [30]. \[U= \begin{bmatrix}\rho\\ \rho u\\ \rho v\\ \rho w\\ \rho e\end{bmatrix} F= \begin{bmatrix}\rho u\\ \rho u^{2}+p\\ \rho uv\\ \rho uv\\ \mu(\rho e+p)\end{bmatrix} G=