## 3 Results and Discussion
### Effects of Reynolds Number and Surface Roughness on Skin Friction Coefficient  
Figure 7 presents the skin friction coefficient \(C_{\mathrm{f}}\) versus \(Re\) for the geometry of Case I when \(Ra\) ranged from 0 (smooth wall) to 6.3 \(\upmu\)m, and the corresponding values are listed in Table 3. It was found that \(C_{\mathrm{f}}\) decreased as \(Re\) increased for \(Re\) of 10\({}^{2}\)-10\({}^{7}\). The rate of decrease was almost constant when \(Re\) was less than 10\({}^{3}\), but it gradually decreased with \(Re>10^{3}\), which resulted in \(C_{\mathrm{f}}\) being independent of \(Re\) for values larger than 10\({}^{5}\). The conclusion was validated in the literature [5; 25], according to [PERSON]'s and [PERSON]'s studies. It was found that the \(C_{\mathrm{f}}\) of smooth walls was close to that of rough walls when \(Re<4\times 10^{2}\), but the difference between them was obvious when \(Re\geq 4\times 10^{2}\). The reason will be analyzed in Figure 8. In addition, the change trend of \(C_{\mathrm{f}}\) was different at \(Ra=0.8\)-6.3 \(\upmu\)m in two ranges of \(Re\), 10\({}^{2}\)-10\({}^{5}\) and 10\({}^{5}\)-10\({}^{7}\). When \(Re\leq 10^{5}\), the values of \(C_{\mathrm{f}}\) were considerably close, indicating that \(Ra\) does not influence \(C_{\mathrm{f}}\) at low \(Re\). When \(Re>10^{5}\), it was noted that \(C_{\mathrm{f}}\) gradually increased, but the rate of increase decreased when \(Ra\) increased from 0.8 to 6.3 \(\upmu\)m; thus, the effects of \(Ra\) on \(C_{\mathrm{f}}\) could not be ignored at high \(Re\). This is because the flow at high \(Re\) was more easily influenced by rough walls than at low \(Re\), resulting in larger \(C_{\mathrm{f}}\). In order to estimate the effect of \(Ra\) on \(C_{\mathrm{f}}\), the relative deviation \(\Delta\) between the skin friction coefficients of smooth and rough walls is defined as follows:  
\[\Delta=\frac{C_{\mathrm{f,\,r}}-C_{\mathrm{f,\,s}}}{C_{\mathrm{f,\,s}}}\times 1 00\% \tag{13}\]  
where \(C_{\mathrm{f,\,s}}\) denotes \(C_{\mathrm{f}}\) of rough walls and \(C_{\mathrm{f,\,s}}\) denotes \(C_{\mathrm{f}}\) of smooth walls. \begin{table}
\begin{tabular}{c c c} \hline \hline \(Re\) & \(Ra/\mathbf{mm}\) & \(C_{\mathrm{f}}\) \\ \hline  & 0 & \(2.69\times 10^{-2}\) \\  & 0.8 & \(2.70\times 10^{-2}\) \\ \(10^{2}\) & 3.2 & \(2.70\times 10^{-2}\) \\  & 6.3 & \(2.69\times 10^{-2}\) \\ \hline  & 0 & \(6.92\times 10^{-3}\) \\ \(10^{3}\) & 0.8 & \(7.96\times 10^{-3}\) \\  & 3.2 & \(7.88\times 10^{-3}\) \\  & 6.3 & \(7.88\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of \(C_{\mathrm{f}}\) at different \(Re\) when \(Ra\) ranges from 0 (smooth wall) to 6.3 \(\upmu\)m for Case I.  
Figure 7: Skin friction coefficient \(C_{\mathrm{f}}\) versus \(Re\) for the geometry of Case I when \(Ra\) ranges from 0 (smooth wall) to 6.3 \(\upmu\)m.  
Figure 8 presents the relative deviation versus \(Re\) under different \(Ra\) of 0.8\(\sim\)6.3 \(\upmu\)m conditions. It was found that the relative deviations of 0.3%-0.6% for \(Re=10^{2}\) were far smaller than those when \(Re\) ranged from \(10^{3}\) to \(10^{7}\). This indicated that the \(C_{\rm f}\) of rough walls was the same as that of smooth walls for extremely low \(Re\), which was consistent with the conclusion drawn in Figure 7. However, when \(Re>10^{2}\), the relative deviations rapidly increased with increasing \(Re\) and \(Ra\), attaining a maximum of 203% at \(Re=10^{7}\) and \(Ra=6.3\)\(\upmu\)m, indicating that \(C_{\rm f}\) was influenced by both flow and wall roughness. This is because at low \(Re\), the flow in the gap was laminar and the boundary layer was relatively thick, which means the grain of the rough walls was completely covered by the viscous sublayer for \(Ra=0.8\)-6.3 \(\upmu\)m; thus, \(C_{\rm f}\) and flow were little influenced by rough walls.