## 3 Geometry and Numerical Method
### Model Validation
#### 3.3.2 Turbulence Model Validation  
To simulate turbulent flow in the impeller back gap and labyrinth seal, it is crucial to select an appropriate turbulence model for CFD simulations. Three turbulence models, such as SST \(k\)-\(\omega\), realizable \(k\)-\(\varepsilon\), and the Reynolds stress model (RSM), were employed to evaluate the reliability of simulations, and skin friction coefficient \(C_{\text{f,rs}}\) obtained from the experiment of the rotor-stator disk system [19] was used as the benchmark for validation. The rotor-stator disk system, which consists of stationary and rotational walls, is presented in Figure 5, and Table 5 provides its dimensions. In the experiment, the working fluid was water, and the rotational speed was determined by \(Re\), ranging from \(10^{3}\) to \(5.8\times 10^{7}\)  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Shaft radius, \(R_{\text{s,se}}\) & 1.500 & mm \\ Teeth number, \(N_{\text{se}}\) & 2 & - \\ Teeth height, \(H_{\text{se}}\) & 0.775 & mm \\ Teeth width, \(B_{\text{se}}\) & 1.270 & mm \\ Cavity length, \(L_{\text{se}}\) & 1.270 & mm \\ Clearance width, \(C_{\text{se}}\) & 0.105 & mm \\ Pressure at inlet, \(P_{\text{in,se}}\) & 10 & MPa \\ Temperature at inlet, \(T_{\text{in,se}}\) & 320 & K \\ Pressure at outlet, \(P_{\text{out,se}}\) & 3.400–9.200 & MPa \\ Pressure ratio, \(\pi_{\text{se}}\) & 1.086–2.941 & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: Geometry dimensions and experimental conditions of two-teeth labyrinth seal.  
Figure 4: Meridian plane of two-teeth labyrinth seal and comparison of CFD results and experimental data.  
Additionally, the structured mesh for the computational domain of the rotor-stator disk system was generated with the first cell height of \(5.0\times 10^{-4}\) mm near walls, making the maximum \(Y^{+}\) satisfy the requirements of all turbulent models.  
As depicted in Figure 6, the values of \(C_{t,\mathrm{rs}}\) simulated using the three turbulence models were compared to experimental results under different \(Re\) conditions. It was observed that the downward trends of \(C_{t,\mathrm{rs}}\) versus \(Re\) are similar across all turbulence models, and also agree with the experiment. However, at \(Re<10^{5}\), CFD results adopting realizable \(k\)-\(\varepsilon\) and RSM models are closer to the experimental results than the SST \(k\)-\(\omega\) model. Furthermore, the detailed comparison revealed that the realizable \(k\)-\(\varepsilon\) model provides more accurate results than the RS model for \(Re>10^{5}\). The maximum relative deviations of RS and the realizable \(k\)-\(\varepsilon\) and models compared to experimental results are 6.20% and 4.02%, respectively. Meanwhile, the validation in the above section also demonstrated that the realizable \(k\)-\(\varepsilon\) model can predict the leakage flow rate of sCO\({}_{2}\) in the labyrinth seal. Therefore, it is applicable that the realizable \(k\)-\(\varepsilon\) model for CFD simulations be used.  
Figure 5: Model of rotor-stator disk system.  
Figure 6: Comparison between simulated \(C_{t,\mathrm{rs}}\) and experimental results under different \(Re\) conditions.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Inner radius of gap, \(R_{\mathrm{i,rs}}\) & 25.40 & mm \\ Outer radius of gap, \(R_{\mathrm{o,rs}}\) & 249.20 & mm \\ Gap width, \(\delta_{\mathrm{rs}}\) & 6.35 & mm \\ Gap length, \(L_{\mathrm{rs}}\) & 19.05 & mm \\ \hline \hline \end{tabular}
\end{table}
Table 5: Dimensions of rotor-stator disk system.