Thermography-based localisation of laminar-turbulent transition on airfoils: a fundamental limit of measurability

Infrared thermography enables the contactless visualisation of laminar-turbulent transition on airfoils. Although the technique has already been used in many wind tunnel and field experiments, for example to study the flow behaviour around the rotor blades of helicopters and wind turbines, a description of the fundamentally achievable minimal measurement uncertainty concerning the position of the flow transition is pending. To this end, the Cramér–Rao bound for an approximate signal model is analytically derived and numerically verified. The signal model studied is a Gaussian error function superposed by additive white Gaussian noise. In addition, the effect of image pixelation on the Cramér–Rao bound is quantified by a numerical analysis, and the critical image resolution is determined for which the uncertainty limit due to pixelation becomes larger than the uncertainty limit due to the contrast-to-noise ratio. It is proven by Monte Carlo simulations that a classical nonlinear least-squares estimator attains the Cramér–Rao bound for a negligible pixelation or a sufficiently high signal-to-noise ratio, respectively. Thus, the minimal achievable measurement uncertainty for the thermographic measurement of laminar-turbulent transition position is clarified, and the validity of the findings is shown for an experiment on wind turbine rotor blades.