## 4 Numerical Assessment of Operating Point \(OR=0.05\)/\(Pr=0.20\)
### Comparison of Numerical Findings to Experimental Data  
A comparison of the findings to test data is difficult as the experimental data are limited to the few wall pressure probes. Using raw numerical and experimental data, time histories, as shown in Figure 6, can be used for a first comparison of the experiment and numerical calculation. With the introduction of pure Fourier Transformations, frequency spectra similar to Figure 5 provide additional results and the occurrence of periodic phenomena can be revealed or confirmed.  
Seeking for a further validation of numerical findings and motivated by the similarity in Figure 5, the SPOD algorithm is used to correlate the experimental pressure signals.  
Using experimental data, SPOD modes were computed and resulting amplitudes assessed. Because the reconstruction of a full pressure field is not possible, the comparison focuses on the temporal evolution of amplitudes among the sensor positions. Using diffuser sensor Diffuser 3/1 as a reference, histograms of the phase lag between the sensors have been computed. To obtain a better resolution, the window size of the SPOD as well as the overlap were increased by a factor of 20 compared to the analysis in Figure 5b. In this case, the first mode captures roughly 55% of the overall energy and the phase lag was computed from this mode only.  
While the assessment of complex amplitudes from a pure Fourier Transformation results in a far less clearer view, the SPOD results in Figure 10 show clear correlations among the different sensors. Dominant phase lags of the four circumferentially distributed valve head pressure sensors are provided in the first row and phase lags of the four diffuser  
Figure 9: Reconstructed pressure evolution at 2700 Hz (CFD, \(M\) 1 + 2, f = (2700 \(\pm\) 200) Hz, T = 1/2700 s).  
wall pressure sensors downstream of valve Head Sensor 3, as indicated in Figure 4 in the second row.  
Focusing on the axial excitation at 600 Hz, a simultaneous pressure amplitude across the circumference is expected. Comparing the histograms of the phase lag, this numerical finding can be confirmed by the experiment. All valve head sensors as well as the two downstream diffuser sensors show a dominating phase lag of 0\({}^{\circ}\) at 600 Hz.  
Clearly visible in the diffuser, the phase lag increases almost linearly for higher frequencies (assuming constant fluid properties and thus a constant speed of sound \(a\), the phase lag \(da\) can be estimated from the distance \(dx\) and the frequency \(f\) as \(da=dx\times f/a\times 360^{\circ}\)). At 2700 Hz, the trend is interrupted and in alignment to the numerical SPOD analysis a phase lag of 0\({}^{\circ}\) is observed for the first three diffuser sensors. Because the three sensors are in phase even though based on the distance a phase lag is expected, the dominant pressure propagation at this frequency must evolve in lateral direction. The valve head sensors support this thesis as sensor Head 3 shows a similar yet slightly positive phase lag and Head 1 on the opposite side confirms the lateral phase lag of 180\({}^{\circ}\). The experiments therefore align with the numerical findings, and the results are considered plausible.