## 3 Modeling of the Inviscid Blade Force Based on a Linear Cascade
### Model Analysis on the Subsonic Condition  
First, a comparison of the subsonic flow in linear cascade is performed, where mass flow at the outlet is set to ensure the inlet Mach number of M = 0.77. Figures 9 and 10 show the isolines of the pressure and the axial velocity for both methods, and the values marked on them are the absolute values. In these figures, the leading edge (LE) and the trailing edge correspond to the beginning and end of the colorful isolines. One thing worth noting is that the circumferential average results from the 3D results are circumferentially mass average except the pressure, and for the pressure, it is the circumferential area average. Through these figures, it reveals that although parameters at the inlet and outlet are consistent, contours in the internal passage are different. These differences cannot be observed in the total pressure recovery coefficient and the spanwise distribution of parameters. In Figure 9b, the pressure obtained through the 3D method increases slowly, which shows the diffusive process in the vane passage. However, in the CAM, pressure first increases rapidly and then decreases slowly, which is contradicting to real flow. The velocity contours are shown in Figure 10 also reveal the difference between these two methods. The flow passes the LE should accelerate first and then decelerate with diffusion. However, in the result of CAM, the axial velocity decreases as soon as it flows into the blade passage. Figure 11a,b show the distribution of average pressure and axial velocity at the middle span, which also reveals the aforesaid phenomenon. Figure 12a,b shows the distribution of the tangential and the axial components of the inviscid blade force at the middle span obtained in these two methods. Through comparison, it shows that the tangential component of the inviscid blade force has the same trend, which is solved through iteration. Since the tangential component is relatively accurate, the circulation at the inlet and outlet can be guaranteed, which makes the performance parameters from CAM similar to 3D results. However, the axial component has a great difference. Therefore, it is essential to depict the influence of the blade geometry on the inviscid blade force. Figure 10: Isolines of axial velocity for CDA cascade: (**a**) CAM; (**b**) NUMECA. Figure 9: Isolines of pressure for CDA cascade: (**a**) CAM; (**b**) NUMECA. In Formula (15), it is shown that pressures at the suction side and the pressure side are needed to depict the inviscid blade force modeling. However, pressure on the suction and pressure side cannot be acquired in the throughflow method; the term \(\frac{p_{s}+p_{F}}{2}\) needs to be modeled. Figure 13 shows the pressure distribution in the middle span obtained from NUMECA. PP and PS refer to pressures on the pressure side and the suction side severally, and AP means the algebraic mean value of pressure at both sides, which is \(\frac{p_{s}+p_{F}}{2}\). SP means the spatially averaged value from 3D simulation results. From these figures, it can be observed that the trend of AP and SP is nearly the same, especially at the middle and posterior portion of the blade. Moreover, the average spatial pressure from the 3D numerical method has the same meaning as the average pressure proposed in the CAM method. Therefore, it is appropriate to use the average pressure in the CAM method in place of the term \(\frac{p_{s}+p_{F}}{2}\). Figure 11: Distribution of average parameters at the midspan of CDA cascade: (**a**) static pressure; (**b**) axial velocity. Figure 12: Distribution of the inviscid blade force at the midspan of the CDA cascade: (**a**) Tangential component; (**b**) axial component. Figure 13: Pressure distribution at the midspan of the CDA cascade. Therefore, Formula (15) turns to be:  
\[ \begin{split} F_{Bx}&=-F_{Bu}\tan\lambda_{M}+\overline{ \rho}\frac{\partial b}{\partial x}\\ F_{Br}&=-F_{Bu}\tan\gamma_{M}+\overline{\rho}\frac{ \partial b}{\partial x}\end{split}