## 4 Results
### Application Scenarios and Suitable Priors  
Previous knowledge can come from theory, simulations or previous experiments. In Sections 4.1 and 4.2, it was shown that the quality of the previous knowledge decides whether it lowers the uncertainty for determining the laminar-turbulent transition position on rotor blades of wind turbines. The present section evaluates different scenarios on how previous knowledge may be gained and in what way it can or cannot improve the measurement of the laminar-turbulent transition position on rotor blades of wind turbines. A proposed field of application for the Bayesian inference is to smoothen the spatial course of the flow transition position along the radial rotor blade position \(r\), which is the so-called transition line. The previous knowledge is obtained by combining all detected transition positions in a single image with a polynomial regression. In detail, the previous knowledge is defined to be normally distributed where the mean \(\mu_{A}=\text{polynomial}(r)\) corresponds to a specific point on a polynomial curve along the radial axis and has a constant standard deviation \(\sigma_{\text{prior}}\). The polynomial is fitted onto the detected transition positions via least squares regression and the mean squared error of this regression is taken as the standard deviation \(\sigma_{\text{prior}}\) of the prior. \[\sigma_{\text{prior}}^{2}\ =\frac{1}{c}\sum_{r=0}^{c}\left(\text{polynomial}(r)- \hat{A}(r)\right)^{2} \tag{18}\]  
By using the approximated course of the flow transition determined from the whole image as previous knowledge for the determination of the laminar-turbulent transition position, the resulting visualization of the transition line is expected to be less noisy. Each graph in Figure 9 shows the determined transition positions without previous knowledge in blue and with previous knowledge in orange. It further shows the mean \(\mu_{\text{prior}}\) of the prior as a solid line and the standard deviation \(\sigma_{\text{prior}}\) of the prior as transparent interval in green. Figure 9 applies this method to a second-order polynomial (top row) and a third-order polynomial (bottom row). The left column of Figure 9 uses the measured thermogram (see Figure 2c) and the right column uses a thermogram with noise (similar to Figure 2d). Figure 8: Normalized standard deviation \(\sigma_{\hat{A}}\) for determining the transition position on thermographic images superposed with Gaussian noise and comparison with the theoretical expectation from estimation theory [33], while considering different standard deviations \(\sigma_{\text{prior}}\) of the prior (also normalized). Through individually calculating the laminar-turbulent transition position for each column without sacrificing the spatial resolution, the application of the Bayesian method results in a smoother visualization of the transition line. Additionally, the Markov chain is an implicit uncertainty evaluation of the Bayesian result. However, in the example, the measured data has little influence on the result because the posterior result sticks very close to the chosen prior. The determined transition positions with prior information that follows a second order polynomial leave the course of the non-Bayesian measurement, which presumable introduces a systematic error (although the true value is not known here). The explanation is that the information within the prior is stronger than the measurement causing a prior-dominated result. Thus, the Bayesian inference can be used to visualize the transition line with less noise, but the result is dominated by the information within the prior so that it resembles a polynomial regression. Whereas the last result generated knowledge through averaging spatial position information from a single image, another idea is to use temporally related results from former experiments to form previous knowledge on the temporal course of the flow transition position.