## 2 Materials and Methods
### Aeroacoustic Measurements
The permeable trailing edge material promotes strong forced convection resulting in effective surface cooling at higher flow speeds. Especially towards the trailing edge tip, the surface temperature of the material approaches ambient conditions. Furthermore, the poor heat conduction, resulting from the high porosity of the material, and the lack of heating wires in the trailing edge tip region prevents homogeneous heating. For the maximum heating power input of \(P_{el}=210\) W, the wire surface temperature reaches approximately 270 \({}^{\circ}\)C which is much higher than the surface temperature. For quantification of the emitted TBL-TE noise from different trailing edge configurations, a phased microphone array is used to localize the trailing edge noise source and isolate it from interfering background noise sources. The acoustic array consists of 64 G.R.A.S. 40 PH free-field microphones with built-in preamplifiers. The microphones are distributed according to a variation of [PERSON]'s design [26] consisting of seven logarithmic spiral arms and one microphone in the center. The array is located 1.3 m away from the airfoil plane. In order to optimize the array resolution in vertical direction, the [PERSON] design is stretched. The microphones are spread across a 1 m wide and 2 m high plane and the array layout is illustrated in Figure 4.  
For each test configuration, the microphone recording time amounts to 60 s at a sampling frequency of 50 kHz. The raw signals are split into time windows of \(\Delta t=0.1\) s which results in a frequency resolution of \(\Delta f=10\) Hz. The data in every time window are weighted using a Hann function and a window overlap of 50 % is chosen. A Fourier transform is applied to the time domain sound pressure levels of every microphone in order to determine the complex pressure amplitudes \(p(f)\) as a function of frequency. For each of the 1199 windows, the spectral densities of the 64 microphones are linked by calculating the Cross Spectral Matrix \(CSM_{k}=0.5p(f_{k})p(f_{k})^{*}\) for a certain frequency \(f_{k}\) of interest. Results from the individual windows are averaged in order to obtain a single CSM for every frequency. For the given experimental setup, the range \(f_{\text{min}}=500\) Hz up to \(f_{\text{max}}=5000\) Hz is chosen during the post-processing. A conventional delay-and-sum beamforming method in the frequency domain [27] is employed to localize the noise source and its strength in the airfoil aligned scan plane of interest. The dimensions of the scan plane amount to 2 m in height and 1 m in width with a resolution of 0.01 m. The frequency-dependent resolution of the acoustic camera follows from [PERSON]'s limit which can be written as [28]  
\[\Delta x=l\tan\left(1.22\frac{c_{0}}{fD}\right)\, \tag{4}\]  
where \(\Delta x\) is the smallest resolvable length scale, \(l\) is the distance between the array and the scan plane, \(c_{0}\) is the speed of sound and \(D\) is the diameter of the circular array. For the highest frequency of interest \(f_{\text{max}}=5000\) Hz and under the assumption of \(D=2\) m in the vertical direction, a minimum value of \(\Delta x_{min}=0.05\) m results which justifies the chosen scan grid resolution. For every scan grid point \(j\), a frequency-specific steering vector is constructed which represents the received signal of microphone \(n\) emitted from a modeled monopole source (ideal point source) located at point \(j\) in the scan plane. The steering vector incorporates the distance between emitter and receiver as well as the resulting time delay of the signal. The corresponding amplitudes of the steering vectors are obtained in the form of the source autopowers by minimizing the difference between the recorded and modeled pressure data. In quiescent air, the signal time delay directly follows from the speed of sound \(c_{0}\) and the distance between scan point and receiver.