## 2 Methodology
### Physical Model and Numerical Method
\begin{bmatrix}\rho v\\ \rho uv\\ \rho v^{2}+p\\ \rho wv\\ \upsilon(\rho e+p)\end{bmatrix} H= \begin{bmatrix}\rho w\\ \rho uv\\ \rho wv\\ \rho w(\rho e+p)\end{bmatrix} \]  
\[F_{v}= \begin{bmatrix}0\\ \tau_{xx}\\ \tau_{xy}\\ \tau_{xz}\\ k\frac{\partial T}{\partial x}+u\tau_{xx}+v\tau_{xy}+w\tau_{xz}\end{bmatrix} G_{v}= \begin{bmatrix}0\\ \tau_{yx}\\ \tau_{yy}\\ \tau_{yz}\\ k\frac{\partial T}{\partial y}+u\tau_{yx}+v\tau_{yy}+w\tau_{yz}\end{bmatrix}  \tag{2}\]  
\[H_{v}= \begin{bmatrix}0\\ \tau_{zx}\\ \tau_{zy}\\ \tau_{zz}\\ k\frac{\partial T}{\partial z}+u\tau_{zx}+v\tau_{zy}+w\tau_{zz}\end{bmatrix} J= \begin{bmatrix}0\\ \rho f_{x}\\ \rho f_{y}\\ \rho f_{z}\\ \rho(uf_{x}+vf_{y}+wf_{z})\end{bmatrix} \]  
where '\(\rho\)' is the density of the medium, and '\(u\)', '\(v\)', and '\(w\)' are the components of velocity. '\(p\)' represents the pressure, and '\(T\)' is the temperature. Furthermore, '\(k\)' is the laminar thermal conductivity, with '\(f_{x}\)', '\(y_{y}\)', and '\(f_{z}\)' being the components of external forces. The shear stress tensor component is given by the following:  
\[\tau_{xy}=\tau_{yx}=\mu(\frac{\partial u}{\partial y}+\frac{\partial v}{ \partial z})\quad\tau_{yz}=\tau_{zy}=\mu(\frac{\partial w}{\partial y}+\frac{ \partial v}{\partial z})\quad\tau_{xz}=\tau_{zx}=\mu(\frac{\partial w}{ \partial x}+\frac{\partial u}{\partial z}) \tag{3}\]  
where '\(\mu\)' is the dynamic molecular viscosity. The total energy '\(e\)' is defined by the following:  
\[e=\frac{1}{2}(u^{2}+v^{2}+w^{2})-\frac{p}{\rho} \tag{4}\]  
The Spalart-Allmaras turbulence model was applied in the computational setup based on the above equations, with walls treated as adiabatic and non-slip. Meanwhile, ideal gas was assumed for the fluid medium, with dynamic viscosity and specific heat determined via the Sutherland formula. To enhance computational efficiency, a single passage with periodic boundary conditions was chosen for steady-state computation for such uniform inflow compressor characteristic calculations. The radial distributions of the total pressure and the total temperature at the inlet were prescribed according to the publicly available experimental measurement data [29; 30]. Furthermore, a turbulent intensity of 4% and turbulent length scale of 0.09 m were set, with a Full Non-Matching Mixing Plane for rotor-stator interaction interfaces. The spatial discretization scheme was a central difference format, while the temporal discretization parameter was set to local time stepping. To expedite convergence towards a steady state, a multi-grid strategy was incorporated along with implicit residual smoothing. Convergence criteria require the global residuals to be reduced by four orders of magnitude and the mass flow rate discrepancies between the inlet and the outlet to be below 0.1%. At the same time, parameters including mass flow rate, total pressure ratio, and efficiency were expected to be stable or exhibit minor periodic oscillations. As for the derivation of the characteristic curve, an estimated average static pressure served as the initial solution at the outlet prior to triggering the single-precision solver first. Then, the compressor was brought closer to the stall by gradually raising the back pressure. At the stall threshold, back pressure increments were limited to 100 Pa, and the final stable convergence solution was considered the stall boundary. The blade's structured grid was generated by the AutoGrid5 module, as depicted in Figure 2. To minimize the influence of inlet and outlet parameters on the blade passage, the inlet was positioned approximately four times the rotor chord length away from the rotor root leading edge, and the outlet was set around nine times the stator chord length from the stator root trailing edge. Meanwhile, grid points were refined near the walls and the blade leading/trailing edges to improve solution accuracy. The y+ value remained below one in most regions.