## 3 Measurement Technique and Data Reduction Methodologies
### Integral Time and Length Scales  
This section describes the procedures for deriving the integral time and length scales, which represent the statistically largest-scale vortex structure containing most of the energy of the flow. The integral time and length scales were derived by the following procedure based on the autocorrelation for the fluctuating component of the \(U\) velocity. Figure 5: Schematic diagram of hot-wire anemometry (**a**) and freestream temperature change for a representative smooth wall case during wind tunnel operation (**b**). Figure 6: Illustration of pitot-rake assembly attached to the flat plate model surface. (i) Measured \(U\) velocity was decomposed into its mean component (\(\overline{U}\)) and its fluctuating component (\(u^{\prime}\)) by the use of the Reynolds decomposition expressed by Equation (8):  
\[U=\overline{U}+u^{\prime}. \tag{8}\]  
(ii) The autocorrelation coefficient (\(R\)) for the fluctuating component of the time-series output signal (\(u^{\prime}=U-\overline{U}\)) with data length of \(N\) was calculated using Equation (9). The autocorrelation coefficient can be plotted against the lag time (\(\Delta t\)) as illustrated in Figure 7:  
\[R(\Delta t)=\frac{\sum_{i=1}^{N-m}\bigl{(}U_{i}-\overline{U}\bigr{)}\bigl{(}U_ {i+m}-\overline{U}\bigr{)}}{\sum_{i=1}^{N}\bigl{(}U_{i}-\overline{U}\bigr{)}^ {2}};\ \Delta t=m/f. \tag{9}\]  
(iii) By taking the integral for the autocorrelation coefficient function and using Equation (10) [28] for \(0\sim\Delta t\) s, a certain \(\Delta t\) value, which becomes equal to the integrated autocorrelation function value, can be found. The resulting \(\Delta t\) is the integral time scale (\(t_{0}\)). This procedure for computing \(t_{0}\) is illustrated in Figure 7. Here, the integration from \(0\) to \(t_{0}\) (the first term in the right side in Equation (10)) was only considered to have a significant value and the integration from \(t_{0}\) to infinity (the second term in the right side in Equation (10)) was considered to have a negligible effect on the integration of the autocorrelation function. Thus, the latter was truncated and was considered to be an uncertainty. This also helps to reduce uncertainty when integrating to infinity since the upper limit of the integration depends on the acquired data length. The derived integral time scale corresponds to the largest time scale of turbulence and represents the largest scale of the flow structure that contains most of the energy in the energy cascade process where the Kolmogorov's -5/3 log-law can be applied. This definition is a general definition to identify the integral time scale, and therefore was used in this study. For calculating the integral, various methods, such as curve fitting for a complex integral profile, can be applied. In this analysis, the MATLAB interpolation function was used with a time interval of \(0.1\Delta t\) (= 5 \(\upmu\)s), so the uncertainty inherent in this integration was considered to be up to \(0.1\Delta t\). This uncertainty was also considered to include the uncertainty resulting from the aforementioned tiny amounts \(\Delta\) that appeared in Equation (10):  
\[t_{0}=\int_{0}^{\infty}R(\Delta t)d\Delta t=\int_{0}^{t_{0}}R(\Delta t)d \Delta t+\int_{t_{0}}^{\infty}R(\Delta t)d\Delta t=\int_{0}^{t_{0}}R(\Delta t )d\Delta t+\Delta. \tag{10}\]  
By multiplying the integral time scale (\(t_{0}\)) and mean velocity (\(\overline{U}\)) together under [PERSON]'s hypothesis of frozen turbulence (Equation (11)), the turbulent length scale (\(L_{0}\)) was obtained as expressed in Equation (12):  
\[\frac{\partial}{\partial t}=|U|\frac{\partial}{\partial x}, \tag{11}\]  
\[L_{0}=\overline{U}\cdot t_{0}. \tag{12}\]  
Here, [PERSON]'s hypothesis is generally applicable when the flow condition is \(u^{\prime}/\overline{U}\ll 1\); the value for the smooth wall, for example, was \(0.011\) at maximum, and therefore, it is plausible that [PERSON]'s hypothesis is applicable to the current measurement.