## 4 Results and Discussion
### Effects of Radius Ratio  
Figure 16 presents the variation of \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) with respect to the radius ratio \(\eta\) for \(Re\) ranging from \(10^{2}\) to \(10^{8}\) when \(\pi\) is 4.5 and 6.0, respectively. It is found that both \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) slightly rise as \(\eta\) increases, regardless of whether the value of \(\pi\) is 4.5 or 6.0 under different \(Re\) conditions. This increase is attributed to the increased flow instability and vortex generation in gaps as the value of \(\eta\) becomes larger, resulting in larger disk-type and shaft-type windage losses. Further evidence supporting this conclusion will be provided later. In addition, \(C_{\text{f,d}}\) is a great deal larger than \(C_{\text{f,s}}\) for the same \(Re\) and \(\pi\) values, as well as \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) being larger at higher \(\pi\) values, which is consistent with the conclusions drawn from Figure 9. However, the impacts of \(\eta\) on \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) have a negligible effect relative to \(Re\) and \(\pi\), since the growing rate in \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) with \(\eta\) is minimal.  
The \(m\) versus \(\eta\) for \(Re\) =\(10^{2}\) for \(\pi\) ranging from 4.5 to 6.0 is presented in Figure 17. When \(\pi\) = 4.5, the \(m\) increases from 0.0430 kg/s at \(\eta\) = 0.018 to 0.2145 kg/s at \(\eta\) = 0.054, and similarly to the conditions of \(\pi\) = 6.0, the \(m\) grows from 0.0584 kg/s at \(\eta\) = 0.018 to 0.3021 kg/s at \(\eta\) = 0.054 due to larger through-flow area. The rate of the increase in \(m\) is constant for different \(\eta\), indicating that the influencing degree of different \(\eta\) on \(m\) is identical. Moreover, regardless of the values of \(\eta\), the \(m\) grows when \(\pi\) grows from 4.5 to 6.0, which is consistent with the observations in Figure 12.  
Figure 16: Skin friction coefficients \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) versus \(\eta\) under the \(Re\) of \(10^{2}\)-\(10^{8}\) conditions with \(\pi\) of 4.5 and 6.0. (**a**) Skin friction coefficients \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) at \(\pi\) = 4.5. (**b**) Skin friction coefficients \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) at \(\pi\) = 6.0.  
Moreover, Figure 18 depicts the vorticity at inlet regions with \(Re\) of \(10^{2}\), \(10^{5}\), and \(10^{8}\) and \(\eta\) of 0.018, 0.036, and 0.054 under \(\pi\) = 4.5 conditions in order to analyze the flow characteristics for different \(\eta\). It is noted that the red and blue regions indicate the opposite rotational direction of the vortices, but both represent larger vorticity in all contours, and the green regions mean that the vorticity is 0. From all contours, it can be seen that the high-vorticity regions expand with increasing \(Re\) and \(\eta\), which reveals that a larger \(Re\), corresponding to larger centrifugal force, results in enhanced flow instability, and a larger \(\eta\) makes the size of vortices increases, which indicates that the effects of vortex dissipation is the main reason for an increase in \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) with \(\eta\), as presented in Figure 16. Furthermore, it is observed that the vorticity in the vicinity of the walls is considerably higher than that at the center due to the effect of the boundary layer attached to the wall, as evident in all contours. As \(Re\) becomes larger, the boundary layer effect becomes more pronounced.