## 3 Results and Discussion  
### Aerodynamic Performance  
Figure 5 shows the distribution of the streamline on the 50% blade height section and a partial enlargement of the leading and trailing edges of the cascades of each stage. The S-CO\({}_{2}\) working fluid with high temperature and high pressure first enters the tapered nozzle flow path of first stage for expansion and acceleration, and then changes its own flow direction along the flow path. Since the first-stage nozzle suction surface has a large curvature, the fluid on the suction side has a higher flow velocity. After being mixed with the fluid on the pressure surface side near the trailing edge of the first-stage nozzle, the mixed fluid enters the first-stage rotor inlet. When the fluid flows through the leading edge of the first stage rotor, the sharp curvature change causes a local acceleration phenomenon on the suction side. There is a small range of stagnation areas at the leading edges of nozzles and rotors of the second and third stage, where the fluid has a lower flow velocity. It is easy to see from the streamline distribution that the outlet velocity of each cascade is significantly larger than the inlet velocity. Because the working fluid expands in the cascade, the enthalpy decreases, and part of it is converted into fluid kinetic energy. In general, there is no serious flow separation in the entire cascade flow channel, and the flow field is relatively uniform. The temperature and pressure distribution nephograms at the 50% blade height section are shown in Figures 6 and 7, respectively. It can be seen from the overall distribution that from the inlet of the first stage nozzle to the outlet of the third stage rotor, the temperature and pressure show a downward trend. The temperature and pressure distributions at the outlet of each cascade are uniform. In the rotor cascade of each stage, the pressure on the pressure surface is higher than that on the suction surface, thus generating effective differential pressure power, driving the impeller to rotate and output work externally. In the first stage nozzle, since the fluid on the suction side has a higher velocity, the temperature and pressure here are lowered, and after the fluid is mixed with that on the pressure side at the outlet, they rise again. The temperature and pressure on the pressure side have a uniform gradient, which gradually decreases from the inlet to the outlet. In the first stage rotor, the temperature  
Figure 5: The distribution of the streamline on the 50% blade height section. and pressure of the fluid on the pressure side are more uniform, and the fluid on the suction side has a significant temperature and pressure drop only near the leading edge, and then remains substantially unchanged. As the fluid flows toward the rear cascade, the blending of various vortices gradually reduces the uniformity of the gas flow, resulting in local low temperature and low pressure regions on the suction side of the third stage nozzle and rotor. The static pressure distribution of the S-CO\({}_{2}\) turbine three-stage cascade meridian plane is shown in Figure 8. In general, the working fluid expands along the flow direction, and the pressure gradually decreases, reaching a set back pressure of 9.5 MPa at the outlet of the third stage rotor. Meanwhile, the pressure distribution of the cascades along the blade height direction is relatively uniform. Only a low-pressure region is generated near the blade root of the third stage stator, which is caused by the mixing of the shedding vortex on the suction side of the blade with the flow separation of the rim boundary layer. In addition, since the diameter of the third-stage impeller is the largest, the flow field in the flow passage is greatly affected by the centrifugal force, thus the fluid is concentrated to the tip position, generating a pressure gradient along the blade height direction.