# Experimental and Numerical Research on End-Wall Flow Mechanism of High-Loading Tandem Stators
## 4 Loss Mechanism of Tandem Stators
### Development and Evolution Mechanism of FB Corner Separation  
While the trend of loss variation for the FB was largely similar to that of conventional blade losses, the development and evolution of corner separation losses in a tandem stator differed significantly due to the presence of the RB. The development and evolution mechanism of the FB corner separation was mainly the influence mechanism of the RB on the development and evolution of the FB corner region. To study the influence of the RB on the FB, numerical simulations were conducted for a single FB (without RB) and compared against the FB within the tandem environment. The same numerical simulation method was used for both the single FB and the tandem stator cases. In the tandem stator case, the parameter extraction position at the FB outlet was identical to that depicted in Figure 12. For the single FB, the extraction position was similarly situated at an equivalent distance from the FB trailing edge. Figure 16 presented a comparison of the spanwise distribution of total pressure loss between the tandem FB and the single FB at NS. In the figure, the solid line (Tand.) represented the tandem FB, while the dotted line (Sing.) represented the single FB. It could be observed that there was no significant difference in total pressure loss in the mid-span region. However, the tandem FB exhibited higher losses in the end-wall regions. The influence mechanism for this difference could be understood by analyzing the blade surface pressure rise at the 10% spanwise location between the tandem FB and the single FB, as shown in Figure 17. The pressure rise was defined as \((p_{\text{local}}-p_{\text{in}})/(1/2\rho U_{\text{m}}^{2})\), where \(p_{\text{local}}\) was the local pressure on the blade surface, and \(p_{\text{in}}\) was the average pressure at the S1 inlet. This figure illustrates that there were two types of mechanisms through which the RB impacts the FB. These two types of mechanisms played different roles in the mid-span and end-wall regions, which made the loss change different from the single FB. Figure 15: Contours of dimensionless axial velocity measured by experiment at the exit of tandem stators at NC, DE, and NS. The first mechanism was that the RB increased the FB load. The influence mechanism was able to be explained by theoretical analysis of the control volume of FB, as shown in Formulas (1)-(5). In order to simplify the analysis, a pitch distance of the FB channel was taken as the control volume, as shown in Figure 18. The subsonic inflow was assumed to be inviscid and incompressible, and the axial velocity at the inlet and outlet was the same. According to [PERSON]'s equation, the relationship between the inlet and outlet pressure and the airflow angle could be established, written as follows:  
\[p_{11}+\frac{1}{2}\rho V_{11}^{2}sin^{2}\beta_{11}=p_{12}+\frac{1}{2}\rho V_{ 11}^{2}cos^{2}\beta_{11}tan^{2}\beta_{12} \tag{1}\]  
where \(\beta_{11}\) and \(\beta_{12}\) were the inlet and outlet flow angle of FB. \(p_{11}\) and \(p_{12}\) were the inlet and outlet pressure of FB. Because the inlet flow was the same between the two front blades (i.e., \(p_{11}\) and \(\rho\) were the same), and the inlet flow angle was almost unaffected by the RB, the \(p_{12}\) was negatively correlated with \(\beta_{12}\). Figure 16: Spanwise distribution of the total pressure loss for tandem FB and single FB. Figure 17: Pressure distribution on the blade surface at the root of the tandem FB and single FB at DE. Meanwhile, from the perspective of momentum theorem, the variation in the circumferential component of momentum across the control volume could be expressed as follows:  
\[\rho sV_{11}cos\beta_{11}(V_{11}sin\beta_{11}-V_{11}cos\beta_{11}tan\beta_{12})=F_ {\rm u} \tag{2}\]  
In the formula, \(s\) represented the FB cascade pitch, \(\lambda_{\rm FB}\) represented the FB installation angle, and \(F_{\rm u}\) was the circumferential force of fluid on the blade.