## 4 Conclusions  
To investigate the influence of surface roughness on windage loss and flow in the shaft-type gap of generators, the skin friction coefficient \(C_{\text{f}}\) and flow characteristics versus Reynolds number \(Re\) were analyzed for different surface roughness \(Ra\), and the effects of radius ratio \(\eta\) were discussed under \(Ra\) of 0.8 \(\upmu\)m conditions. The conclusions of this paper will help develop a model for predicting \(C_{\text{f}}\) with rough walls, further improving the design of small supercritical carbon dioxide power devices, including turbine-alternator-compressor units. The main conclusions are as follows:  
1. \(C_{\text{f}}\) decreased as \(Re\) increased, and the rate of decrease was constant at low \(Re\) but it gradually decreased at high \(Re\). In addition, for \(Re=10^{2}\), the relative deviations between the skin friction coefficients of smooth and rough walls of 0.3-0.6% indicated that \(C_{\text{f}}\) was not influenced by rough walls. However, when \(Re>10^{2}\), the relative deviations increased with \(Re\) and \(Ra\), indicating that \(C_{\text{f}}\) was influenced by flow and rough walls because the grain was in blending or logarithmic law regions occurred, which impacted the flow.  
Figure 14: Pressure contours in the tangential direction and velocity distribution in the radial direction for \(Re=10^{4}\) (**a**) Case I; (**b**) Case II; (**c**) Case III; (**d**) Case IV.  
2. When \(Re=10^{2}\), the flow was laminar and similar for different \(Ra\), but when \(Re>10^{2}\), it transitioned to TC flow and periodic Taylor vortexes appeared, the number of which increased with \(Ra\). The velocity could be divided into three regions, and the speed-stable region increased with \(Ra\). The velocity gradient in speed-drop regions was larger than that in speed-stable regions. These results indicated that \(Ra\) influenced the thickness of the boundary layer and the flow at the center of the gap was more stable.
3. \(C_{\rm f}\) and the variation rate increased with \(\eta\), indicating the \(C_{\rm f}\) was seriously influenced by larger \(\eta\). Moreover, the relative deviations increased with \(\eta\) when \(Re\leq 10^{5}\), which indicated that \(C_{\rm f}\) was more easily influenced by rough walls for larger \(\eta\) under low \(Re\) conditions, since the interaction between rough walls and working fluid affected windage loss. However, the opposite was true when \(Re>10^{5}\) because the flow became the primary source of windage loss in a smaller gap width at high \(Re\).
4. The size of Taylor vortexes increased with \(\eta\), leading to fewer vortexes. In addition, the pressure gradually increased from the rotational to stationary walls but decreased with increasing \(\eta\) as a whole. Additionally, there were periodic low-pressure and high-pressure regions near the walls due to Taylor vortexes in the gap, but the periodic high-pressure regions near the stationary walls decreased at larger \(\eta\), since the number of Taylor vortexes decreased. Moreover, the velocity gradient near the stationary walls for Case IV was smaller than that of other Cases due to the significantly large \(\eta\).  
Methodology, L.H.; software, L.H.; data curation, Q.D. and J.L.; writing--original draft preparation, L.H.; writing--review and editing, Q.D. and Z.F.; visualization, Z.L.; supervision, Q.D.; funding acquisition, Q.D. All authors have read and agreed to the published version of the manuscript. This work was supported by the Joint Funds of the National Natural Science Foundation of China (U20A20303) and the National Key R&D Program of China (2017 YFB0601804). Not applicable. Not applicable. Data Availability Statement: Data will be made available on request. We acknowledge the support and internal funding from the Joint Funds of the National Natural Science Foundation of China (U20A20303) and the National Key R&D Program of China (2017 YFB0601804). The authors declared no conflict of interest.