# Thermographic Stall Detection by Model-Inspired Evaluation of the Dynamic Temperature Behaviour
## 2 Principle of Measurement
### Thermodynamic Response around an Operating Point  
The signal processing approach around an operating point considers the surface temperature response to temporal oscillations in the temperature of the incoming flow. The aim is to identify boundary layer flow regions on the processed thermographic images by means of unambiguous relations between the features of the evaluated surface temperature oscillations and the flow-dependent heat transfer coefficient. Therefore, the surface temperature response behaviour to oscillating input signals is analysed.  
Small-scale oscillations in the input signal around an operating point induces thermodynamic oscillations in the output signal. Considering small-scale oscillations around an operating point, the system's input-output relation corresponds to that of a linear time-invariant system. For this reason the input-output relation indicated in Figure 2b is describable with a transfer function and can be evaluated with the powerful methods of linear system theory.  
One method for the evaluation of linear systems with harmonic input signals is the analysis of amplitude and phase information for selected frequency components of the output signal. With increasing heat transfer coefficients \(\alpha\), the intensity of the convective heat transport increases and the system's input-output relation has a reduced amplitude attenuation and phase shift [25]. As a result, the amplitude and phase information depends on the heat-transfer coefficient, and thus enables distinguishing between different boundary layer flow regions. Furthermore, the harmonic signal processing approach around an operating point provides two desirable features. First, assuming a negligible dependence of the heat transfer coefficient and the material properties on the surface temperature within the measurement range, the operating point has, on first approximation, no influence of the linear system behaviour around the operating point. Therefore, the amplitudes are independent of the initial temperature field \(T_{0}\). Second, due to the harmonic behaviour of the output signal, the influence of the heat conduction is reduced. Thus, spatial differences in amplitude and phase information can only result from different flow properties, i.e., from different heat-transfer coefficients \(\alpha\), or material inhomogeneities \(M\).  
Note that the amplitude and phase information depends on the amplitude and the frequency of the available harmonic components in the input signal. In particular, the amplitude response signal and, thus, also the signal-to-noise ratio decreases for increasing frequencies, due to the thermal inertia of the surface [25]. For this reason, the proposed  
Figure 2: Model function of the surface temperature with input and output quantities and cross-sensitivities, (**a**): Flow-dependent heat transfer coefficient \(\alpha\) as input quantity, (**b**): Analysis of the response behaviour for changing heat inputs, the flow properties in \(\alpha\) are considered as a parameter of the system’s input–output relation (cross-sensitivity).  
signal processing approach around an operating point is limited to low-frequency heat input oscillations.