# 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
In this control volume, the circumferential force \(F_{\rm u}\) could be expressed by the pressure difference between the pressure surface and the suction surface, written as follows:  
\[F_{\rm u}=\left(\int_{0}^{1}p_{\rm PS}(t)dt-\int_{0}^{1}p_{\rm ss}(t)dt\right) c_{\rm FB}cos\lambda_{\rm FB} \tag{3}\]  
where \(p_{\rm PS}\) and \(p_{\rm ss}\) were the pressure of pressure surface and suction surface, respectively, and \(c_{\rm FB}\) was the chord length of FB. Therefore, the momentum equation could be written as follows:  
\[\frac{\rho V_{11}^{2}cos^{2}\beta_{11}}{\sigma_{\rm FB}cos\lambda_{\rm FB}}( tan\beta_{11}-tan\beta_{12})=\int_{0}^{1}p_{\rm PS}(t)dt-\int_{0}^{1}p_{\rm ss}(t)dt \tag{4}\]  
where \(\sigma_{\rm FB}\) was the solidity of FB. As the incoming flow of the FB remained unchanged, the outlet airflow angle of the RB could be expressed as follows:  
Figure 17.0  
In the formula, \(\frac{\sigma_{\rm FB}cos\lambda_{\rm FB}}{\rho V_{11}^{2}cos^{2}\beta_{11}}\) represented the geometric parameter item of the blade, and it was not affected by the RB, so the relation between the outlet airflow angle and the pressure distribution on the blade surface was established, and they had a monotonous negative correlation. The RB had a stagnation effect on the flow exiting the FB, which increased the load on the FB. As shown in Formula (1), \(\beta_{12}\) decreased and the airflow turning angle of the FB increased, indicating that the deviation angle decreased. From Formula (5), it could be seen that the circumferential force of the FB increased. Consequently, the wrapping area between the suction surface and the pressure surface of the tandem FB was larger than that of the single FB, as illustrated in Figure 17.1  
With the increase in incidence, corner separation began to develop progressively at the end-wall region of the FB. Gradually, corner separation loss became the major contributing factor to overall losses. The severity of corner separation was positively correlated with the pressure gradient in both the flow and circumferential directions. The presence of the RB increased the pressure gradient in these two directions, which made the corner separation  
Figure 17.2 Schematic diagram of FB control volume. in tandem blades more significant. Therefore, the tandem FB loss was higher than that of the single FB in the root regions. Figure 19 shows a comparison of the limit streamlines on the blade surface between tandem blades and single FB at DE and NS. It could be seen that the influence area of corner separation for the tandem FB was larger than that of the single FB at NS. The second mechanism was that the RB decreased the pressure at the FB trailing edge. Although the presence of the RB increased the FB load, the gap between the FB and RB resulted in an acceleration of the flow, consequently decreasing the pressure at the FB trailing edge. Under the constraint of the Kutta condition, which stipulates that the velocity around the trailing edge must be tangential to the surface for smooth flow detachment, the pressure on the suction surface of the tandem FB was lower than that on the single FB (as highlighted by the green dashed box in Figure 17). Thus, the reverse pressure gradient on the suction surface of the tandem FB was reduced, which decreased the boundary layer loss on the suction surface. At the mid-span region of the FB, boundary layer loss was a major source of overall loss. The gap between the FB and RB inhibited the growth of the boundary layer thickness, resulting in lower losses for the tandem FB compared to the single FB. This mechanism explained why the FB loss in two-dimensional tandem cascades was smaller than that of the single FB. However, in a three-dimensional tandem stator environment, the loss in the mid-span region also included the influence of the axial velocity distribution ratio (AVDR).