## 3 Measurement Technique and Data Reduction Methodologies
### Evaluation of Skin Friction Drag Reduction
Constants for the riblet (\(C_{r\_{riblet}}\)) and smooth wall (\(C_{\_{smooth}}\)) cases may be different so they are expressed as \(A\) and \(B\), respectively:  
\[\frac{\Delta c_{f}}{c_{f,\_{smooth}}}=\frac{A\cdot(Re_{riblet})^{-\frac{1}{4}}- B\cdot(Re_{smooth})^{-\frac{1}{4}}}{B\cdot(Re_{smooth})^{-\frac{1}{4}}}= \frac{\frac{A}{B}\cdot\left(\sigma_{u,\_{riblet}}\cdot L_{0,\_{riblet}}\right) ^{-\frac{1}{4}}-\left(\sigma_{u,\_{smooth}}\cdot L_{0,\_{smooth}}\right)^{- \frac{1}{4}}}{\left(\sigma_{u,\_{smooth}}\cdot L_{0,\_{smooth}}\right)^{- \frac{1}{4}}}. \tag{16}\]  
From this equation, only the first term in the numerator on the right side is affected by riblets. Since the coefficient (\(A\)/\(B\)) also represents the effect of riblets on the change of the skin friction, the first term in the numerator on the right side represents the effect of riblets. This coefficient (\(A\)/\(B\)) has to be quantified properly to compute the reduction of skin friction drag. In order to evaluate the unknown coefficient (\(A\)/\(B\)), a simple method to estimate the value is proposed here. The basic procedure is as follows. The relationships among all variables and assumptions are presented in Figure 8 along with the procedure. (i) For a smooth wall, the local skin friction coefficient can be calculated by assuming a smooth flat plate and therefore by using Equation (7) [27]:  
\[c_{f,\_{smooth}}=0.059\cdot Re_{x}^{-\frac{1}{8}}, \tag{17}\]  
where \(x=600\) mm and \(U=41.7\) m/s. (ii) On the other hand, the local skin friction coefficient is also calculated by the following, which is equivalent to a term appeared in Equation (16):  
\[c_{f,\_{smooth}}=B\cdot(Re_{smooth})^{-\frac{1}{4}}. \tag{18}\]  
By comparing those two equations, the unknown coefficient \(B\) is obtained for the smooth wall case:  
\[B=0.059\cdot Re_{x}^{-\frac{1}{5}}\cdot(Re_{smooth})^{\frac{1}{4}}. \tag{19}\]  
(iii) Then, for the riblet wall case, since the local skin friction coefficient for this case cannot be calculated by Equation (17) because of the fact that the coefficient may be different for the riblet wall, the local skin friction coefficient is assumed to be calculated as a rough surface wall model by the [PERSON]'s method [30] as follows:  
\[c_{f,\_{riblet}}=\left(2.87+1.58 log_{10}\frac{x}{k_{s}}\right)^{-2.5}, \tag{20}\]  
where \(k_{s}\) is the surface roughness and it needs to be obtained. To this end, total skin friction is used. By comparing total skin friction coefficient (\(C_{F}\)) values obtained by the two different methods shown below, \(k_{s}\) is obtained, where \(l=900\) mm:  
\[C_{F,\_{riblet}}=\left(1.89+1.62 log_{10}\frac{l}{k_{s}}\right)^{-2.5} \tag{21}\]\[c_{f}=\frac{\tau_{0}}{\frac{1}{2}\rho\overline{U}^{2}}=C\left(\frac{\overline {U}\cdot\delta}{v}\right)^{-\frac{1}{4}}, \tag{14}\]0  
Equation (21) is the Prandtl-Schlichting method [30] and Equation (22) is based on the momentum thickness (\(\theta\)) given by Equation (2), which is calculated based on the pitot-rake data. In Equation (22), the difference of the momentum thickness was calculated between \(x=20\) mm where the boundary layer was tripped and hence considered to be the virtual origin and \(x=600\) mm where the boundary layer profile was measured. Consequently, the total skin friction coefficient, \(C_{F}\), and surface roughness, \(k_{s}\), are obtained. Then, the local skin friction coefficient for the riblet case is calculated by Equation (20). (iv) At the same time, like the smooth wall case, another method is used to obtain the local skin friction coefficient as expressed in Equation (23). By comparing those two local skin friction coefficients, the unknown value \(A\) is obtained as expressed by Equation (24):  
\[c_{f}=\frac{\tau_{0}}{\frac{1}{2}\rho\overline{U}^{2}}=C\left(\frac{\overline {U}\cdot\delta}{v}\right)^{-\frac{1}{4}}, \tag{14}\]1  
\[c_{f}=\frac{\tau_{0}}{\frac{1}{2}\rho\overline{U}^{2}}=C\left(\frac{\overline {U}\cdot\delta}{v}\right)^{-\frac{1}{4}}, \tag{14}\]2  
(v) Now, local skin friction coefficients for the two cases are obtained.