## 3 Results and Discussion
### Aerodynamic Performance
Therefore, in the first half of the third stage rotor flow path, the pressure near the tip of the blade is slightly higher than the vicinity of the blade root. Figure 6: Temperature distribution at the 50% blade height section. Figure 7: Pressure distribution at the 50% blade height section. In order to quantify the pressure change trend in the cascade channel, the pressure distribution curve of the S-CO\({}_{2}\) turbine three-stage cascade meridian plane at the 50% blade height is shown in Figure 9. It can be clearly seen from the figure that the pressure of the working fluid gradually decreases from 14.3 MPa at the turbine inlet to 9.5 MPa, which is the design outlet pressure. It is worth noting that there is a small increase in pressure near the nozzle outlets of each stage. This is due to the gradient between the high velocity fluid on the suction surface and the low velocity fluid on the pressure surface of the nozzle, and these two kinds of fluid are vigorously blended at the trailing edge. The trailing edge will be affected by its front boundary layer, producing a wake vortex. For the S-CO\({}_{2}\) turbine, the shedding vortex generated by local acceleration will produce obvious low-temperature and low-pressure regions, making the transcritical phenomenon in the channel difficult to avoid. Minimizing the transcritical region is the key to improving turbine aerodynamic efficiency. Multi-stage axial turbines have more cascade stages than single-stage centrifugal turbines. As the working fluid flows to the rear cascade, the complex flow field structure makes the flow uniformity more difficult to control, so in addition to the leading edge and trailing edge of the  
Figure 8: Static pressure distribution on the meridian plane. Figure 9: Pressure distribution curve of each cascade meridian plane at the 50% blade height. cascade, the flow separation phenomenon is more likely to occur in the channel, and even the transcritical phenomenon appears. In order to clearly show the transcritical region of the designed S-CO\({}_{2}\) multi-stage axial turbine, the pressure upper limit of the cloud icon ruler of the surface pressure distribution of the third-stage cascade blade is adjusted to the critical pressure of carbon dioxide of 7.38 MPa, as shown in Figure 10.  
Since the carbon dioxide in the whole flow is always at a high temperature, which is much higher than the critical temperature, therefore, it is only necessary to judge whether the carbon dioxide is transcritical by judging whether the pressure is lower than the critical pressure. It can be seen from the figure that the transcritical region is mainly distributed at three locations, the area close to the blade root at the leading edge of the first stage rotor (A), the area near the blade root of the leading edge on the suction side of the third stage stator (B), and the outlet side of the trailing edge of the third stage stator (C). The minimum pressure is about 3.5 MPa, which is located at the bottom of the blade root of the A region, and the range is extremely small and negligible. The pressure in the remaining transcritical areas is higher than 6 MPa. The mixing of the complex vortex system affected by the front cascade and the change of the blade shape curvature of the suction side trigger the flow acceleration. As a result, a large-area transcritical phenomenon occurs on the third-stage stator suction side, as shown in the B region in Figure 10.