## 2 Research Objects and Numerical Methods
### Structural Design  
The structural design of the nozzles and rotors of the S-CO\({}_{2}\) turbine is done in 3D modeling software ANSYS-BladeGen. The blade profile has different optimal design shapes for different types of turbomachinery, inlet parameters, working fluids, geometrical parameters of cascade, etc. Although the physical parameters of S-CO\({}_{2}\) are quite different from those of gas and steam, researchers have accumulated sufficient experience in blade design of gas turbine and steam turbine for many years. Considering that S-CO\({}_{2}\) is close to gaseous fluid in tension and viscosity, we can still refer to their classical blades as a reference for blade design. The blade design in this study takes the form of straight blades, and the blade profile remains constant along the blade height. In order to reduce the flow loss of working fluid on the pressure side, the pressure side profile was linearized. In addition, the radian of the suction side is also reduced to improve the flow separation of the boundary layer on the suction side. The maximum thickness is about 40% of the leading edge of the midline. The geometry angles of the inlet and outlet of the cascade of the turbine initial design are determined by the reference to the thermal design results. Then according to the flow condition of three-dimensional aerodynamic analysis of the initial design, the geometrical angles of the inlet and outlet of the rear row cascade are corrected stage by stage, which makes the aerodynamic efficiency of the S-CO\({}_{2}\) turbine optimal. Figure 2a-f show schematic diagrams of the final profiles of the nozzles and rotors of each stage.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**NO.** & **Parameter** & **1 st Stator** & **1 st Rotor** & **2 nd Stator** & **2 nd Rotor** & **3 rd Stator** & **3 rd Rotor** \\ \hline
1 & Speed coefficient & 0.96 & 0.96 & 0.96 & 0.96 & 0.96 & 0.96 \\
2 & Reaction coefficient & - & 0.1 & - & 0.15 & - & 0.18 \\
3 & Flow coefficient & 0.95 & 0.95 & 0.95 & 0.95 & 0.95 & 0.95 \\
4 & Installation angle/\({}^{\circ}\) & 40 & 55 & 40 & 50 & 45 & 50 \\
5 & Relative pitch & 0.75 & 0.74 & 0.76 & 0.74 & 0.762 & 0.70 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Pre-fetched parameters.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**NO.** & **Parameter** & **1 st Stator** & **1 st Rotor** & **2 nd Stator** & **2 nd Rotor** & **3 rd Stator** & **3 rd Rotor** \\ \hline
1 & Number of blades & 28 & 40 & 27 & 41 & 30 & 43 \\
2 & Axial width/mm & 20 & 17 & 22 & 18 & 24 & 20 \\
3 & Blade height/mm & 25.88 & 41.05 & 23.62 & 35.15 & 23.19 & 32.26 \\
4 & Pitch/mm & 23.84 & 15.36 & 26.57 & 17.39 & 25.86 & 18.28 \\
5 & Average diameter of & & & & & & \\ stage/mm & & 212.10 & & 225.35 & & 249.02 & \\
6 & Inlet geometric angle/\({}^{\circ}\) & 80 & 50 & 35 & 60 & 60 & 30 \\
7 & Outlet geometric angle/\({}^{\circ}\) & 30 & 30 & 30 & 35 & 20 & 30 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Key structural parameters.  
According to the profiles and flow passage of the key geometric parameters of turbine cascades, a three-dimensional geometric modeling of the fluid domain of the designed three-stage axial turbine is completed, including inlet extension, three-stage cascade flow passage and outlet extension, as shown in Figure 3.