## 3 Results and Discussion  
### CFD Off-Design Pre-Analysis  
The design-optimization of a 60,000 rpm S-CO\({}_{2}\) turbine were completed based on our previous research. First of all, the design-optimization approach using Gauss process regression was adopted [43]. Combined with the rapid 1-D thermal design method and the high-precision 3-D aerodynamic analysis method, the preliminary optimization design was obtained. Then, further 3-D aerodynamic optimization was carried out, mainly including inlet and outlet flow angle correction, flow matching of rotor and stator blades, blade profile optimization, etc. These optimization methods have achieved good results in previous research [24; 44; 45]. The detailed thermodynamic design parameters, geometric parameters, performance parameters, and blade profiles of the designed S-CO\({}_{2}\) turbine are shown in Table 2.  
Figure 4 shows the 3-D model and numerical grid of the S-CO\({}_{2}\) turbine. In this study, the single passage (one stator passage and one rotor passage) was used for calculation. In order to improve the mesh quality, the H type mesh was adopted in the inlet and outlet extension sections, while the O type mesh was adopted for blade meshing. The grids were densified in the tip clearance, around the blade and near the wall to obtain accurate flow parameters. The orthogonal angle of the mesh was greater than 15\({}^{\circ}\), which meets the requirements of grid quality. The SST \(k-\omega\) turbulence model was adopted. The value of Y\({}^{+}\) near the wall was about 1, which meets the calculation requirements of the turbulence model. The corresponding boundary conditions were given according to the thermodynamic design parameters. The grid independence was verified to balance the calculation accuracy and efficiency. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter Type** & **Parameter** & **Value** & **Unit** \\ \hline \multirow{6}{*}{Thermodynamic parameter} & Inlet temperature & 600 & \({}^{\circ}\)C \\  & Inlet pressure & 15 & MPa \\  & Outlet pressure & 8 & MPa \\  & Design power & 1000 & kW \\  & Rotating speed & 60,000 & rpm \\  & Number of stator blades & 16 & pc. \\  & Stator inner diameter & 119.7 & mm \\  & Stator outer diameter & 153.2 & mm \\  & Number of rotor blades & 15 & pc. \\  & Impeller inlet blade height & 6 & mm \\  & Impeller outer diameter & 99.7 & mm \\  & Impeller outlet blade height & 15.9 & mm \\  & Tip clearance & 0.2 & mm \\  & Mass flow rate & 11.38 & kg/s \\  & Torque & 162.2 & N-m \\  & Numerical power & 1019 & kW \\  & Isentropic enthalpy drop & 1139 & kJ/kg \\  & Total static efficiency & 89.44 & \% \\ \end{tabular}
\end{table} 
Table 2: Key design parameters. The output power calculated at different grid scales was used as the evaluation basis. When the calculation error between adjacent grid scales is less than 1%, it is considered to meet the demand of calculation accuracy. The final number of selected grid nodes was 420,000. The numerical method was the same as previous research [24; 43; 44; 45]. In previous research, the S-CO\({}_{2}\) compressor with more complex flow was used for numerical verification. By comparing with the numerical and experimental results of other scholars, it can be shown that the numerical method is accurate [45]. For the off-design performance analysis of the turbine, the turbine's inlet temperature, inlet pressure, inlet airflow angle, mass flow rate, and rotating speed were changed change within \(\pm\)15% of the design value. The Latin Hypercube Sampling method was adopted to obtain a total of 1000 off-design conditions. The off-design performance data set of the designed turbine was obtained by CFD analysis of 1000 off-design conditions. Table 2 shows that the output power of the designed S-CO\({}_{2}\) turbine is 1019 kW and the total static efficiency is 89.44%. Figure 5 shows the limiting streamline, pressure distribution, and temperature distribution of the 50% blade height section of the turbine.