## 3 Measurement Technique and Data Reduction Methodologies
### Quantitative Interpretation for Output Signals from Hot-Wire Anemometry  
The output signal from hot-wire anemometry is a function of both the freestream velocity and freestream temperature. Therefore, it is necessary to convert the measured output signal to a quantitative velocity by applying an appropriate temperature correction method. If the temperature variation throughout the wind tunnel run is relatively small, the measured signal (i.e., voltage) can be regarded as a function of the flow velocity (i.e., \(E_{output}=f(U);f\) denotes a functional dependence), and [PERSON]'s law [31], which is presented as Equation (25), can be applied:  
\[E^{2}=a+bU^{n}:n=0.45, \tag{25}\]  
Figure 8: Procedure for deriving unknown coefficients \(A\) and \(B\) for evaluating the skin friction coefficient. where constants \(a\) and \(b\) are determined by the calibration wind tunnel experiment to relate the voltage value and the freestream velocity. However, if the temperature variation throughout the wind tunnel experiment is relatively large (as seen in Figure 5b), the temperature dependence can no longer be ignored and the measured signal must be regarded as a function of both the velocity and temperature (i.e., \(E_{output}=f(U,T)\)). In this case, a correction that considers temperature variations, such as the one expressed in Equation (26) [32], is useful:  
\[U=v\cdot f\left(\frac{E^{2}}{k\Delta T}\right)=v\cdot f\left(\frac{E^{2}}{k(T_ {w}-T_{a})}\right), \tag{26}\]  
where \(v\) is the kinematic viscosity, and the coefficient \(k\), which is the thermal conductivity of air, is determined by Equation (27) [33] using freestream temperature, \(T_{a}\):  
\[k=418.4\left(5.75\times 10^{-5}\left(1+3.17\times 10^{-3}T_{a}-2.1\times 10^{-6}T_ {a}^{2}\right)\right). \tag{27}\]  
Since the wire temperature, \(T_{w}\), was unknown here and the flow temperature was measured at each probe height, \(T_{w}\) in Equation (26) was parametrically changed and its value that makes the difference of the \(U\) velocity profiles between the hot-wire anemometry data with Equation (26) and the pitot-rake data was sought to become zero or minimum. This modified temperature correction method with a best-fit approach was taken in this study for quantitative analysis. The resulting \(T_{w}\) values for each case are presented in Table 1 and this in-situ temperature correction was applied. Note that the derived \(T_{w}\) value includes not only the true wire temperature itself but also the correction effect of the output signal for a time decay as mentioned above. Those factors were considered to be the cause of the relatively large difference in \(T_{w}\) between smooth and riblet wall cases. It should also be noted that other temperature correction methods proposed in previous studies may be used, depending on the available dataset. A representative profile of the raw output signal, which qualitatively corresponds to the \(U\) velocity, for the smooth wall is shown in Figure 9a. The measurement was carried out by fixing the probe position in the \(x\)- and \(y\)-directions, and traversed in the \(z\)-direction (height from the wall surface) as described in Figure 5a. The sequential numbers on the figure show the order of measurement, which are seen from Figure 5b. Each plotted point is a 15-s average, which corresponds to 300,000 data points. As stated above, the output signal strongly depends on the freestream temperature variation. Figure 9b plots the \(U\) velocity profiles after applying the temperature correction to the raw output signal of the hot-wire anemometry data. The profile obtained by the pitot-rake measurement is also plotted for comparison. The \(U\) velocity profile after temperature correction (HW) agrees with that determined by the pitot-rake measurement (Pitot). The maximum discrepancy between those two profiles around the boundary layer outer edge (9 mm) was 2.2%.