## 3 Pressure Data Analysis
### Unsteady Pressure Analysis  
In this section, we investigate the Fourier spectra of the unsteady pressure signals acquired on the cone wall for all shutter area ratios. The fundamental frequency associated with the self-induced instability and the equivalent amplitude determined based on [PERSON]'s theorem [35] are obtained for each case. The precessing frequency is written in dimensionless form using the Strouhal number,  
\[\mathrm{Sh}=\mathrm{f}\frac{\mathrm{D_{t}}}{\mathrm{V_{t}}}, \tag{3}\]  
while the dimensionless amplitude of the pressure pulsation is given as follows,  
\[\mathrm{A}\equiv\frac{\sqrt{2}\mathrm{p_{RMS}}}{\varrho\mathrm{V_{t}^{2}}/2} \tag{4}\]  
The equivalent pressure pulsation amplitude is \(\sqrt{2}\mathrm{p_{RMS}}\) where \(\mathrm{p_{RMS}}\) is the root mean square of the unsteady part of the pressure signal according to [PERSON]'s theorem [26]. This value quantifies the level of the flow unsteadiness. The acquired pressure fluctuations (AS), the reconstructed signals (RS) and its Fourier spectra (FT) on all four levels of the cone wall are plotted in Figure 4 for the case with vortex rope (without diaphragm). The Strouhal number Sh\(\sim\)0.39 is associated with the dominant frequency of the helical vortex rope. Here, the maximum equivalent amplitude (EA\(\sim\)0.28) is obtained on L1 level being associated with the largest eccentricity of the vortex filament [36]. The experimental investigation of the unsteady pressure field on the cone wall is performed for all diaphragm openings corresponding to different shutter area ratios. Significant pressure pulsation mitigation is obtained for all cases, with case involving diaphragm in comparison with the case without the diaphragm. In this paper, the acquired pressure fluctuations (AS), the reconstructed signals (RS) and its Fourier spectra (FT) on all levels of the cone wall for two diaphragm shutter area ratios of 50% (d = 0.113 m) and 70% (d = 0.088 m) are plotted in Figures 5 and 6, respectively. One can observe that the maximum equivalent amplitudes of EA\(\sim\)0.17 were obtained. This value is diminished by over 45% in the case without the diaphragm. A synopsis view over dynamic performances of this new control method is provided in Figure 7. The evolution of equivalent amplitudes on the cone wall for all investigated cases is provided in Figure 7a. In addition, the Strouhal number associated with the self-induced instability for all investigated cases is plotted in Figure 7b. It can be observed that the frequency is changing in a limit of 10% for the case with the maximum shutter area ratio. The Strouhal number is practically unchanged when the shutter area is slightly increased while the maximum equivalent amplitude of the pressure pulsations is mitigated. On the other side, it significantly decreases the maximum equivalent amplitude on all levels, being around the same values for diaphragm cases with shutter area ratios from 50% to 70% (d = 0.113... 0.088 m). Figure 4: Pressure fluctuations on all levels without diaphragm: acquired signals (AS) vs. reconstructed signals (RS) (**left**) and it Fourier spectra (FT) vs. equivalent amplitudes (EA) (**right**). Figure 5: Pressure fluctuations on all levels with shutter area ratio of 50% (d = 0.113 m): acquired signals (AS) vs. reconstructed signals (RS) (**left**) and its Fourier spectra (FT) vs. equivalent amplitudes (EA) (**right**). Figure 6: Pressure fluctuation on all levels with shutter area ratio of 70% (d = 0.088 m): acquired signal (AS) vs. reconstructed signal (RS) (**left**) and its Fourier spectrum (FT) vs. equivalent amplitude (EA) (**right**). Two snapshots with flow structure captured on the test rig are presented in Figure 8 to support the results. To visualize the flow structures, we introduced air in the rig. One can observe a precessing helical vortex for a swirling flow with no control method in Figure 8a.