## 2 Measurement Approach
### Laminar-Turbulent Flow Transition Localization  
The basic principle that enables a localization of the laminar-turbulent flow transition in thermographic flow visualization images is the different heat transfer coefficients in the different flow regimes. An initial temperature difference between fluid and surface leads to different surface temperatures in the regions of the different flow regimes that can be evaluated as the normalized intensity in the thermographic image. Figure 1a shows an example flow visualization with a flow direction along the \(x\)-axis from left to right and a laminar (\(0~{}px\leq x\leq 145~{}px\)) and turbulent (\(146~{}px\leq x\leq 250~{}px\)) flow regime colored in green and red, respectively. Figure 1b shows the intensity profile \(I\) (thin black line) along the center row in the flow visualization image marked with a white dashed line. Two distinct intensity plateaus representing either the laminar (green) or turbulent (red) flow regime are clearly distinguishable. Between the two flow regimes, a distinct, steep change of the intensity with a high gradient exists. The corresponding gradient \(\text{d}I/\text{d}x\) of the intensity profile is shown as a thin dotted black line. One approach for localizing the position of the flow transition center between both flow regimes is an evaluation of the intensity gradient profile. The aim is to localize the position of the maximal gradient in flow direction by an approximation of the gradient \(\text{d}I/\text{d}x\) with a Gaussian function  
\[\phi(a_{1},a_{2},a_{3},a_{4})=a_{3}\cdot\exp\left(-\frac{(x-a_{1})^{2}}{2\,a_ {2}^{2}}\right)+a_{4}. \tag{3}\]  
The position of the maximum of \(\phi\) after the least-squares approximation is given by the parameter \(\hat{a}_{1}\) and can directly be used as a measure of the flow transition position \(x_{\text{tr}}\) in the \(x\)-direction. Note that the optimal value \(\hat{a}_{1}\) is obtained with a least-squares estimation, so that \(x_{\text{tr}}\) is specified with sub-pixel accuracy. The curve-fitting result for the example in Figure 1b is shown as a thick dotted line. A second approach for localizing the laminar-turbulent flow transition with an even lower measurement uncertainty, compared to the approximation of the intensity gradient profile, can be carried out by an approximation of the intensity profile itself [22]. By fitting the Gaussian error function  
\[\Phi(b_{1},b_{2},b_{3},b_{4})=b_{3}\cdot\text{erf}\left(\frac{x-b_{1}}{\sqrt{ 2}\,b_{2}}\right)+b_{4} \tag{4}\]  
to the intensity profile \(I\), the flow transition can be determined directly by the position of the function's saddle point that is the parameter \(\hat{b}_{1}\). The approximation of the intensity profile with the Gaussian error function in the example in Figure 1b is shown with a thick black line. The systematic error of the flow transition localization depends on the capability of the approximation function to auto-center at the position of the flow transition. In order to achieve the auto-centering, the measurement data have to obey the approximation modelfunction over the entire considered profile region. For the Gaussian function this means a zero intensity gradient away from the flow transition, while for the Gaussian error function two distinct intensity plateaus with constant amplitudes are needed. The random error of the flow transition localizations depends on the absolute value of the intensity gradient at the position of the flow transition and is inversely proportional to the intensity difference between the laminar and turbulent flow regime for a given transition width [22]. An estimation of the random error \(\sigma_{\text{rand}}\) and the systematic error \(\sigma_{\text{syst}}\) of the flow transition localization can be conducted by evaluating the locations \(x_{\text{tr},i}\) for the intensity profiles at different y-positions \(y_{i}\).