## 4 Results and Discussion
### Effects of Different Mediums  
The main purpose of this section is to verify and study the accuracy of the skin-friction coefficient model of different mediums and the flow characteristics in the gap. In practical engineering applications, the geometric dimensions and boundary conditions of the radial impeller machinery are mainly concentrated in Region III and Region IV of the turbulent state; this section verifies 10 cases (as shown in Table 4) of these two regions. Figure 12 shows the comparison between the numerical calculation and the model prediction results of the three mediums with different relative gaps. The results in the figure have been repeatedly verified. As shown in Figure 12a, the relative deviation of the skin-friction coefficient calculated by the Region III model is greater than that of Region IV. The maximum relative deviation of air reaches \(-7.6\%\) when the relative gap is \(0.015\) and the Region III deviation is negative. However, unlike the results of water, the relative deviation does not decrease with the increase of the relative gap. Overall, the relative deviations of the three mediums in Region IV are less than \(2\%\) and there is no regular change. The maximum relative deviation is \(1.9\%\) and the model prediction of Region IV is more accurate. The reason for the large model deviation in Region III may be related to the simultaneous boundary layer merging and separation in the gap. In Region IV with a higher Reynolds number, the boundary layer separation weakens the interaction betw \begin{table}
\begin{tabular}{c c c} \hline \hline
**Region** & **Relative Gap (S/\(R_{0}\))** & \(Re\) \\ \hline \multirow{2}{*}{Region III} & 0.005 & \(1.0\times 10^{6}\) \\  & 0.015 & \(2.0\times 10^{5}\), \(1.0\times 10^{6}\), \(5.0\times 10^{6}\) \\  & 0.025 & \(1.0\times 10^{6}\) \\ \hline \multirow{2}{*}{Region IV} & 0.035 & \(1.0\times 10^{7}\) \\  & 0.050 & \(1.0\times 10^{6}\), \(1.0\times 10^{7}\), \(1.0\times 10^{8}\) \\ \cline{1-1}  & 0.065 & \(1.0\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Boundary conditions of different mediums. Figure 11: Contour of Geometry 1 velocity magnitude distribution at the end wall rim gap. (**a**) \(Re=4.06\times 10^{5}\). (**b**) \(Re=3.77\times 10^{6}\). the stator wall. The deviation rules of the three mediums are basically the same. It is worth mentioning that the relative deviation of critical CO\({}_{2}\) is closer to that of water. The density and other physical properties of critical CO\({}_{2}\) are significantly different from that of air. When the relative gap changes, the velocity distribution and the gap working medium is air, as shown in Figure 13a,b. As mentioned above, when \(S/R_{0}=0.005\), the axial gap is very small and the boundary layers still merge together. When \(S/R_{0}\) increases to 0.025, the boundary layers first separate from the shaft and gradually diffuse along the radial direction. Figure 13c shows the vorticity distribution in the gap cross-section under the condition of air and critical CO\({}_{2}\) (Region III, \(S/R_{0}=0.015\)). By controlling the different rotational speeds, the Reynolds numbers in the two cases are the same. It can be seen that the vorticity of critical CO\({}_{2}\) is greater. Under the conditions of two mediums, high vorticity zones appear near the rim of the end wall. In the rotor different zone, the vorticity distribution of the two mediums is diverse. This may be the reason for the large relative deviation of the air medium. Compared with the relative gap, the influence of the Reynolds number on the windage loss in the gap is more intuitive and effective. The rotational speed of the rotor will make the centrifugal force of the fluid stronger which will have a great influence on the flow of the fluid in the gap. When the Reynolds number is high, the influence of the end wall flange on the flow in the gap will be more significant.