## 3 Results and Discussion
### Effects of Radius Ratio on Skin Friction Coefficient and Flow Characteristics  
Figure 11 describes the skin friction coefficient \(C_{\mathrm{f}}\) versus \(Re\) for the radius ratio \(\eta\) ranging from 0.056 to 0.225, corresponding to the geometric model from Case I to Case IV, at \(Ra=0.8\)\(\upmu\)m. The values of \(C_{\mathrm{f}}\) are listed in Table 4. Similar to Figure 7, \(C_{\mathrm{f}}\) gradually decreased when \(Re\) increased from \(10^{2}\) to \(10^{7}\) for all cases. The rate of decrease versus \(Re\) was constant when \(Re<10^{3}\), but it slowly decreased when \(Re\geq 10^{3}\), which led to \(C_{\mathrm{f}}\) remaining constant for \(Re>10^{5}\). In addition, it was also found that \(C_{\mathrm{f}}\) and the variation rate increased with \(\eta\), which indicated that \(C_{\mathrm{f}}\) was seriously influenced by larger \(\eta\). The reason is that the TC flow was prominent and more Tarloy vortexes occurred at larger \(\eta\), resulting in increased \(C_{\mathrm{f}}\). According to Equation (13), the relative deviations at \(\eta=0.056\)-0.225 were calculated and are shown in Figure 12. It was found that the relative deviations increased with \(Re\) and \(\eta\) when \(Re\leq 10^{5}\), but they decreased with \(Re\) and increasing \(\eta\) for \(Re>10^{5}\). This indicated that \(C_{\mathrm{f}}\) was more easily influenced by rough walls for larger \(\eta\) under lower \(Re\) conditions, but the opposite occurred for significantly higher \(Re\). This is because the flow was stable at low \(Re\), and the interaction between rough walls and working fluid influenced windage loss; hence, it was more prominent in a larger gap width, corresponding to a larger \(\eta\). However, the flow was extremely unstable at high \(Re\) and became the primary source of windage loss; hence, in a smaller gap width, meaning a smaller \(\eta\), the flow was more  
Figure 10: Averaged velocity in the radial direction with \(Re=10^{4}\) and \(Ra=0.8\)\(\sim\)6.3 \(\upmu\)m.  
significantly impacted by rough walls, resulting in larger relative deviations. Additionally, when \(Re=10^{2}\), the fact that the relative deviations ranged from 0.6% to 2.9% also indicated that the \(C_{\text{f}}\) of rough walls was similar to that of smooth walls for extremely low \(Re\), as shown in Figure 8.  
Figure 13 shows the streamlines and vorticity contours near one end wall at the meridional plane with \(Re\) of \(10^{2}\), \(10^{4}\), and \(10^{7}\) at \(\eta=0.056\)-0.225. It can be seen that no Taylor vortexes occurred when \(Re=10^{2}\) for each case, which indicated that the flow was not impacted by rough walls no matter how wide the gap. This was also the most crucial reason that the \(C_{\text{f}}\) of rough walls was almost the same as that of smooth walls, as shown in Figure 13. When \(Re\) was equal to \(10^{4}\) and \(10^{7}\), the size of Taylor vortexes increased with \(\eta\), automatically leading to fewer vortexes. This was because a larger \(\eta\) means a larger gap width, resulting in Taylor vortexes forming more easily and developing in the gap. Further, it was noted that the size of vortexes near the end wall increased as \(Re\) increased, indicating that the effect of the end walls was more permanent for higher \(Re\). Moreover, it was also found that when \(Re=10^{2}\), the vorticity distribution was near the end walls and it increased with \(\eta\). The reason was that the centrifugal force exerted on the fluid in the gap was exceedingly small at lower \(Re\), so the effect of the stationary end walls could not be ignored; thus, the effect of the end walls was more prominent under larger \(\eta\) and gap width conditions. However, the centrifugal force was larger for higher \(Re\); hence the impact of the end walls was not significant, which resulted in the vorticity near the end wall decreasing as \(\eta\) increased, as shown in the vorticity contours for \(Re\) of \(10^{4}\) and \(10^{7}\).