## 4 Results and Discussion
### Uncertainty Analysis  
As mentioned in the introduction, reducing skin friction drag by applying riblets to a surface would be on the order of several percent. Therefore, it is vital to quantify the measurement uncertainty to demonstrate the effectiveness of applying the riblet surface to reduce skin friction drag. The possible sources of uncertainty in deriving the turbulent scale were found to be due to the wind tunnel freestream, measurement uncertainty by hot-wire anemometry, and the cumulative uncertainty in the postprocessing. Those sources and their values are listed in Table 2. Note that the uncertainty resulted in the temperature correction (\(q_{1}\)) for the hot-wire anemometry data was considered at \(z=1.0\) mm as the measured \(U\) velocity at this height will be used to evaluate the skin friction drag reduction in a later section as mentioned before. The random uncertainty (\(q_{2}\)) inherent in the hot-wire anemometry was represented by the temperature-corrected velocity variance in the freestream region (i.e., \(z=12\)-\(20\) mm: Five measurement points) for the smooth wall case for a duration of \(5\) s. The uncertainty that may be caused by the velocity fluctuations in the other components than the streamwise component was not obtained since the velocity measurement was conducted by a single-wire probe and therefore no exact three-dimensional velocity components were known. Therefore, an assumption was made to estimate the uncertainty accounted for the other components of velocity fluctuations. Assuming that velocity variances exist in the streamwise direction (\(u\)), spanwise direction (\(v\)), and vertical direction (\(w\)), the effective velocity (\(U_{\textit{eff}}\)), which is the velocity information detected by the wire affected by \(u^{\prime}\), \(v^{\prime}\), and \(w^{\prime}\), can be expressed as follows. Here, the mean streamwise velocity that corresponds to the true velocity is \(\overline{U}\). The overbar indicates the time average. It should be noted that the flow direction to the wire sensor is assumed to be less than \(30^{\circ}\)[34] since the hot-wire probe was placed normal to the wall and the flow:  
\[U_{\textit{eff}}=\sqrt{\left(\overline{U}+u^{\prime}\right)^{2}+C_{H}\cdot v^{ \prime 2}+C_{K}\cdot w^{\prime 2}}, \tag{28}\]  
where \(C_{H}\) and \(C_{K}\) are directional sensitivity factors for each component of the flow [34]. By taking a time average and the Taylor series expansion, Equation (28) reduces to Equation (29):  
Figure 9: \(U\)-velocity profiles for the smooth wall case. \[\frac{U_{eff}}{\overline{U}}=1+\frac{1}{2}C_{H}^{2}\frac{\overline{\
u^{2}}}{ \overline{U^{2}}}+\frac{1}{2}C_{K}^{2}\frac{\overline{w^{\prime 2}}}{ \overline{U^{2}}}+\mathcal{O}\bigg{[}\frac{\overline{u^{\prime 4}}}{\overline{U^{4}}} \bigg{]}. \tag{29}\]  
The last term in the right side in Equation (29) stands for the order of magnitude of remaining terms in the Taylor series expansion. Further rewriting Equation (29) by assuming that the velocity variances are small (i.e., \(\left|u^{\prime}/\overline{U}\right|\ll 1\), \(\left|v^{\prime}/\overline{U}\right|\ll 1\), and \(\left|w^{\prime}/\overline{U}\right|\ll 1\)), the uncertainty caused by the directional effect can be expressed as Equation (30):  
\[\frac{U_{eff}-\overline{U}}{\overline{U}}\approx\frac{1}{2}Tu\big{(}C_{H}^{2} +C_{K}^{2}\big{)}\text{:}Tu=\frac{\sqrt{\frac{1}{3}\big{(}\overline{u^{\prime 2 }}+\overline{v^{\prime 2}}+\overline{w^{\prime 2}}\big{)}}}{\overline{U}}. \tag{30}\]  
_Tu_ is physically equivalent to the turbulent kinetic energy. Since exact values of \(v^{\prime}\) and \(w^{\prime}\) were unknown, it was roughly assumed that \(u^{\prime}=v^{\prime}=w^{\prime}\) for this analysis. Then, _Tu_ can roughly be estimated as 0.1 near the wall (at \(z=2\) mm for the smooth wall; and the details will be presented in a later section). For the current probe geometry, \(C_{H}\) and \(C_{K}\) can be estimated as approximately 1.05 and 0.10, respectively [34].