## 4 Numerical Assessment of Operating Point \(OR=0.05\)/\(Pr=0.20\)
### Visualization of Dominant Flow Structures Using SPOD  
Following the experimental observations in Figure 2(b), numerical simulations were conducted and transient wall pressures on the valve head and diffuser as well as the pressure field on a cross section of the diffuser were recorded. Afterwards a SPOD analysis was performed for the transient surface data in case of the CFD and for the data from eight pressure transducers (sensor positions indicated in Figure 4: four sensors on valve head surface (circumferential distribution declared Heads 1-4; Head 1 bottom right/Head 3 bottom left) and four sensors in the lower half of the diffuser downstream of Head 3 (declared Diffuser 3/1-3/4; axial distribution in streamwise direction)) in case of the experiment. To be consistent, the 10 ms window size of the (time) segments and the overlap of 5 ms were used for both CFD and experimental datasets to obtain the same frequency resolution. As a result, each CFD segment covers 2000 samples (time step of \(5\times 10^{-6}\) s) while the experiment covers only 640 samples (sample rate of 64 kHz). In both cases, the default window function (Hamming-Window) was applied. The resulting spectra of SPOD eigenvalues are shown in Figure 5 as a fraction of the overall modal energy.  
Two peaks appear in the energy spectrum of the CFD data in Figure 4(a) for frequencies at 500 Hz to 700 Hz (declared as Peak\({}_{LF}\) for low frequency) and 2500 Hz to 2900 Hz (declared as Peak\({}_{HF}\) for high frequency). For the experimental analysis, only the second peak is clearly visible in Figure 4(b) while the first peak cannot be clearly gathered due to the limited sensor positions. At the discussed operating point, the position of the four head sensors Heads 1-4 is within the region of detachment (visible in Figure 4) leading  
Figure 5: Normalized eigenvalues of SPOD modes for \(OR=0.05/PR=0.2\): (a) CFD (approximately 40,000 grid points); and (b) experiment (eight pressure sensors).  
to partly non-periodic low frequency effects. Taking into account that the experimental SPOD uses only eight discrete positions while the transient data of the CFD are captured at about 40,000 positions, the energy spectra compare well. The first mode (\(M1\)) of the SPOD analysis of both CFD and experiment captures already about 40% and together with the second SPOD mode, which contributes a significant amount to the second peak in both cases, roughly 2/3 of the overall modal energy are represented. Concluding that a majority of the pressure oscillation can be represented by the first two modes, the sum of \(M1\) and \(M2\) is used for the following reconstructions.  
For a validation that the overall pressure characteristic is captured by the two frequency bands at Peak\({}_{LF}\) and Peak\({}_{HF}\), exemplary pressure signals corresponding to sensor Diffuser 3/3 are shown in Figure 6 for both CFD and experimental results. While the raw signal in both cases shows a rather chaotic trend, the application of bandpass filters according to the two peaks accurately captures the essential pressure trend. Throughout the illustrated time period in Figure 6a,b, a continuous oscillation at Peak\({}_{LF}\) and a superposition of amplitudes associated with Peak\({}_{HF}\) can be identified by the two bandpass filters. For both frequencies, temporal excitations and extinctions are observed. Based on the good agreement, the two dominating frequency bands are used to visualize the reoccurring pressure oscillations and their propagation in the flow field from the numerical SPOD analysis.