## 2 Materials and Methodologies  
### Throughput Analysis Tool  
In this research, the throughflow model used is the CAM to derive the governing equations. For the 3D flow in aviation turbomachinery, the CAM method applied for any parameter is defined as  
\[\overline{q}(x,r,t)=\frac{1}{\varphi_{s}-\varphi_{p}}\int_{\varphi_{p}}^{ \varphi_{s}}q(x,r,\varphi,t)d\varphi \tag{1}\]  
where \(q\) is an aerodynamic parameter, \(\overline{q}\) is the circumferential value, \(\varphi_{s}\) and \(\varphi_{p}\) are angular values for suction side and pressure side individually. In a vane passage shown in Figure 3, the integral equation starts from the suction side to the pressure side. A blockage factor due to the circumferential blade thickness takes on the form:  
\[b=\frac{\varphi_{s}-\varphi_{p}}{2\pi/N} \tag{2}\]  
where \(N\) is the number of blades. After the definition of the CAM, aerodynamic parameters can be decomposed into circumferential average and spatial fluctuation:  
\[q(x,r,\varphi,t)=\overline{q}(x,r,t)+q^{\prime}(x,r,\varphi,t) \tag{3}\]  
Figure 3: Schematic diagram for the passage of rotor and stator. Figure 2: Schematic diagram for the inviscid blade force. Based on the CAM and the main operation rule, the N-S equation in the relative frame can be derived into:  
\[\frac{\partial\overline{U}}{\partial t}+\frac{1}{br}\frac{\partial}{\partial x} \left[br\left(\overline{F}-\overline{F_{\text{p}}}\right)\right]+\frac{1}{br} \frac{\partial}{\partial r}\left[br\left(\overline{G}-\overline{G_{\text{p}}} \right)\right]=\overline{S}+F_{\text{B}}+F_{F} \tag{4}\]  
where  
\[\overline{U}=\left[ \begin{array}{c}\frac{\overline{\rho}}{\rho \overline{w}_{x}}\\ \frac{\overline{\rho}\overline{w}_{u}}{\overline{\rho}\overline{w}_{u}}\\ \frac{\overline{\rho}\overline{w}_{u}}{\overline{\rho}\overline{w}_{u}}\\ \frac{\overline{\rho}\overline{w}_{u}}{\overline{\rho}\overline{w}_{u}}+\frac{ \overline{\rho}\overline{w}_{u}\overline{w}_{u}^{\prime\prime}}{\overline{ \rho}\overline{w}_{u}\overline{w}_{u}}+\frac{\overline{\rho}\overline{w}_{u} \overline{w}_{u}^{\prime\prime}}{\overline{\rho}\overline{w}_{u}\overline{w}_ {u}}\\ \frac{\overline{\rho}\overline{w}_{u}}{\overline{\rho}\overline{w}_{u}}+\frac{ \overline{\rho}\overline{w}_{u}^{\prime}\overline{w}^{\prime\prime}}{\overline {\rho}\overline{w}_{u}^{\prime}}\end{array} \right],\ \overline{G}=\left[ \begin{array}{c}\frac{\overline{\rho} \overline{w}_{r}}{\overline{\rho}\overline{w}_{x}\overline{w}_{r}+\overline{ \rho}\overline{w}^{\prime\prime}xw^{\prime\prime}}\\ \frac{\overline{\rho}\overline{w}_{r}\overline{w}_{r}}{\overline{\rho} \overline{w}_{r}\overline{w}_{r}}+\overline{\rho}\overline{w}^{\prime\prime} \overline{w}^{\prime\prime}}\\ \frac{\overline{\rho}\overline{w}_{u}\overline{w}_{r}}{\overline{\rho} \overline{w}_{r}\overline{w}_{r}+\overline{\rho}\overline{w}^{\prime\prime} \overline{w}^{\prime\prime}}\end{array} \right],\]  
\[\overline{F_{\text{p}}}=\left[ \begin{array}{c}\frac{0}{\overline{\tau}_{xx} }\\ \frac{\overline{\tau}_{xx}}{\overline{\tau}_{xx}\overline{w}_{x}+\overline{ \tau}_{xx}\overline{w}_{r}}+\frac{\overline{\tau}_{xx}\overline{w}_{u}}{ \overline{\tau}_{xx}\overline{w}_{x}+\overline{\tau}_{rr}\overline{w}_{r}+ \overline{\tau}_{uu}\overline{w}_{u}}-\overline{q_{r}}\end{array} \right], \overline{G_{\text{p}}}=\left[ \begin{array}{c}0\\ \frac{\overline{\rho}\left(\overline{w}_{u}+\omega r\right)+\overline{\rho} \overline{w}_{u}^{\prime\prime}\overline{w}^{\prime\prime}\overline{w}^{\prime \prime}\overline{w}^{\prime}\overline{w}^{\prime}\overline{w}^{\prime}\overline {w}}\\ \frac{\overline{\rho}\overline{w}\overline{w}_{r}}{\overline{\rho}\overline{w}_{r} \overline{w}^{2}r}\end{array} \right],\]  
where \(U\) is the vector of conservative variables, \(F\) and \(G\) are the convective flux vectors, \(F_{\text{p}}\) and \(G_{\text{p}}\) are the viscous flux vectors, \(S\) is the source term, \(F_{B}\) is the inviscid blade force and \(F_{F}\) is the viscous blade force.