## 3 Modeling of the Inviscid Blade Force Based on a Linear Cascade
### Model Analysis on the Transonic Condition  
In previous research, when the CAM is applied in the transonic condition, a special shock wave was captured [13; 17; 20]. However, in real 3D transonic flow, the shock wave is normal to the blade geometry other than axisymmetric. Therefore, in the CAM, the shock wave should be eliminated, where a decelerating process exists. Considering this point, the inviscid blade force involved in the transonic cascade also needs to be improved.  
First, a simulation of the transonic cascade is performed through the 3D method. By changing pressure at the outlet, the Mach number at the inlet is restricted to 0.86. The isoline of Mach numbers at the middle span is shown in Figure 17, which reveals that the strong shock wave exists on the suction side.  
Similar to Figure 13, the distribution of pressure at the middle span is also displayed in Figure 18 for this condition. In this figure, it can be seen that the spatially averaged pressure from the 3D result is nearly the same as the algebraic mean value of the pressure side and the suction side. They have slight differences at the LE and the front half part, and its value at the shock wave of the suction side is a little different. However, they are coincident behind the shock wave.  
To examine the influence of these two aforesaid inviscid blade force models on the result, they are both applied in the CAM simulation. The distribution of the pressure and the axial velocity at the midspan are compared in Figure 19. It reveals a huge difference  
Figure 16: Comparison of average parameters at the midspan of the CDA cascade: (**a**) static pressure; (**b**) axial velocity.  
Figure 17: Isoline of Mach number at the midspan of the CDA cascade.  
Figure 18: Pressure feature at the midspan of the CDA cascade.  
between these two models. When the original inviscid blade force is used, the maximum error between CAM and 3D method for pressure along the flow direction is 128.1%, and it is 47.0% for axial velocity. However, when the improved model is used, these two values are reduced to 39.3% and 19.9%, respectively. The most obvious diversity exists at the position where shock wave occurs, and at other positions, these two sets of results obtained from the revised model and NUMECA are nearly the same. Finally, one element needs to be discussed. The maximum error of the static pressure is still 39.3%, which is still large. It is observed that the maximum error exists at the location of the shock. Considering the difficulties in the simulation of the shock with the throughflow method [11; 13; 20; 21; 22] and the improvement of the parameters in the blade passage, the method is regarded as an applicable method. To further improve the accuracy, there are two aspects that may need to consider. The first one is to add a factor in adjusting variable incidence angles, which makes this model more exact for different incidence angles. Moreover, the other one is that the circumferential fluctuation source terms [23; 24] should also be considered in the governing equation since they also contribute to the flow structure.