## 6 Prestall Disturbances
### Prestall Propagating Disturbances; Numerical Results  
The frequency analysis was then applied to the numerically acquired pressure fluctuations sampled at 200 monitor points evenly distributed around the circumference. The axial positions of the pressure transducers in the experiment and the monitor points in the CFD model were the same. The simulated period of time was 0.26 s covering nearly 24 rotor revolutions with a resolution of 16,683 steps per second. The main purpose of this analysis procedure is to prove that the computational setup is suitable for modelling RI. The average Fourier transform of all 200 series demonstrates that the short sequence of pressure data is sufficient to verify the existence of RI based on its characteristic signature in the frequency spectrum. In comparison to the experimental results, the side-by-side peaks arise in the same frequency range--they have approximately the same frequency spacing in-between (see Figure 8). Furthermore, the peaks within the signature do not represent lower harmonics of the BPF. In order to enable an explicit comparison between the experimental and numerical data, coherence and phase spectra have been evaluated for two selected monitor points with a circumferential offset of 8 deg. In general, increased coherence values higher than 0.5 can be found in the entire frequency range, but two global maxima standing out at 0.4% BPF and 0.8% BPF coincide with RI (RI and first higher  
Figure 8: Frequency analysis - comparison between experiment and simulation. harmonic). The decreased slope of the phase response indicates that in the numerical simulations the propagation velocity of prestall disturbances is slightly higher. Even if there are minor differences between the experimental and numerical results regarding the amplitudes of pressure fluctuations, the computational setup is able to capture the predominating characteristics of RI. The primary achievement of the computation was to simulate the propagation of modal flow disturbances as they could be proved in the experiments. By applying a mode decomposition technique to the pressure signal of all monitor points, it is possible to assign a circumferential mode to each peak in the frequency spectrum. The analysis procedure has been precisely described by [16; 17]. Using 200 monitor points, an azimuthal mode analysis is feasible up to mode orders \(m=\pm 99\). Generally, the circumferential mode is equal to the number of flow disturbances being evenly distributed around the casing. Figure 9 shows that the peaks are of subsequent numbered modes with a dominant mode order of \(m=11\). Corresponding to the number of blades, the BPF is linked to the mode order \(m=14\). The interaction between rotor blades and propagating disturbances is responsible for the modal pattern symmetrically arranged around the BPF at about 800 Hz and 1800 Hz (BPF \(-\) RI; BPF + RI, circled in red). It is assumed that associated modes are cut-on. Correspondingly, in the experiments at this very operating point, a distinct sound with a certain howling quality was noted. The special modal signature in Figure 9, also shown in the experimental results of [PERSON] et al. (2015) [18], represents the most important indication that the computational setup used in this work is able to capture the main properties of RI. To understand which particular flow mechanism can be assigned to the disturbances, an isosurface of the \(\lambda_{2}\) vortex criterion has been evaluated at a specific time (see Figure 9, right). Eleven discrete disturbances are represented by vortex tubes being concentrated at the blade tips. The vortices appear as tornado-like tubes with a highly deflected trajectory where the casing ends of the tubes propagate circumferentially along a path which is located upstream of the leading edges. Meanwhile, the number of vortices varies slightly in time (Video S1). In the context of RI, existence of radial vortex structures has been already suggested in the past [19].