## 3 Numerical Methodology and Setup  
The steady-state, three-dimensional Navier-Stokes equations are solved using the commercial software NUMECA. The turbulence model comparison work for the radial inflow turbine in [23] concluded that the Spalart-Allmaras turbulence model was a suitable choice with good convergence and adaptability. Hence, the Spalart-Allmaras turbulence model is selected for the turbulent closure in the current study. The three-dimensional model is generated based on the one-dimensional optimization design geometry, as shown in Figure 2. In the current simulation, one nozzle passage and the rotor blade passage are selected as the computational domain to save the computational cost. The structured meshes are generated using IGG-AutoGrid5, shown in Figure 3. The O4H and HI topology structures are applied for the mesh generation in the nozzle passage and the rotor blade passage. The meshes around the endwall, nozzle, and rotor blade surfaces are refined with an expansion ratio of 1.2 to describe the velocity boundary layers. The computational meshes stratify the numerical simulation standards with an orthogonality angle of less than 20\({}^{\circ}\), an aspect ratio of less than 2500, and an extension ratio of less than 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Initial Value** & **Optimum Value** \\ \hline Velocity ratio \(x_{\text{a}}\) & 0.650 & 0.662 \\ Reaction degree \(\Omega\) & 0.470 & 0.483 \\ Nozzle velocity coefficient \(\varphi\) & 0.95 & 0.97 \\ Rotor inlet absolute flow angle \(\alpha_{1}\) (\({}^{\circ}\)) & 16.0 & 16.0 \\ Rotor exit relative flow angle \(\beta_{2}\) (\({}^{\circ}\)) & 21.4 & 25.0 \\ Wheel diameter ratio \(\overline{D}_{2}\) & 0.430 & 0.417 \\ Rotor blade velocity coefficient \(\psi\) & 0.85 & 0.90 \\ Rotor inlet relative flow angle \(\beta_{1}\) (\({}^{\circ}\)) & 85.55 & 85.00 \\ Rotor exit absolute flow angle \(\alpha_{2}\) (\({}^{\circ}\)) & 78.0 & 75.0 \\ Nozzle exit diameter (mm) & 170.3 & 170.4 \\ Rotor inlet diameter (mm) & 163.8 & 166.8 \\ Rotor inlet blade height (mm) & 3.0 & 3.3 \\ Rotor exit hub diameter (mm) & 43.1 & 48.6 \\ Rotor exit shroud diameter (mm) & 89.8 & 85.6 \\ Specific speed & 13.76 & 13.84 \\ Output power (MW) & 2.056 & 2.118 \\ Total-to-static efficiency \(\eta_{\text{s}}\) (\%) & 83.76 & 85.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Initial and optimum design results. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Specification** \\ \hline Mass flow rate (kg/s) & 16.0 \\ Rotating speed (r/min) & 42,000 \\ Inlet total pressure (MPa) & 24.6 \\ Inlet total temperature (K) & 823 \\ Outlet static pressure (MPa) & 8.5 \\ Output power (MW) & 2.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Design specifications. A grid independence test is performed to verify the accuracy of the numerical results. Figure 4 shows the total-to-static efficiency variation at five computation mesh resolutions. It is observed that the value of total-to-static efficiency is nearly unchanged since the mesh resolution is higher than 1.72 million. Hence, the mesh resolution of 1.72 million is adopted for the following simulations to balance the prediction accuracy and computation time. The mesh resolutions for the nozzle passage and rotor blade passages are 0.89 million and 0.83 million, respectively. [PERSON] et al. [24] pointed out that the thermal-physical properties of the s-CO\({}_{2}\) significantly impact the prediction accuracy for the performance and flow field of the radial inflow turbine. Hence, the real gas thermal-physical properties are generated using a built-in code, TabGen, in the software NUMECA. Because the table generation technique is adopted in the code TabGen, table resolution must be determined in advance. Based on the design specifications for the radial inflow turbine listed in Table 3, the pressure ranges from 5 MPa to 30 MPa, and the temperature ranges from 400 K to 1000 K for the table generation.