## 4 Results and Discussion
### Wavenumber-Frequency Analysis  
Based on the unsteady LES results, the wavenumber-frequency analysis was performed by using the Fourier transform in the cylindrical coordinate system as described in Section 2.4 with the in-house code, of which the validity was confirmed in the previous studies [8; 13]. The acoustic power caused by the sudden pressure variation inside the nozzle provides the source of the AIV of the pipe wall. Therefore, for more precise assessment of AIV problems, it is necessary to extract the acoustic wave components propagating in the downstream direction from the fluctuating pressure field in the pipe. Therefore, the pressure extraction area is selected to include the prescribed monitoring position as shown in Figure 13.  
The simulation time interval and the frequency interval used for the wavenumber-frequency analysis were \(5\times 10^{-5}\) s and 3.23 Hz, respectively, and the grid spacing was (\(\Delta\theta\), \(\Delta z\)) = (2.5\({}^{\circ}\), 0.0015 m). The average speed of sound and Mach number in each region were 343 m/s, 0.064 in Region 1 and 0.011 in Region 2. The results of the wavenumber-frequency analysis are shown in Figure 14.  
Figure 12: Total pressure level; (a) Sensor 1, (b) Sensor 2. In Figure 14, the black lines represent the modal cut-off lines in the pipe, and the circumferential direction mode is represented by \(m\) and the radial direction mode is represented by \(n\). The dashed line at the right bottom represents the wave components propagating at the constant speed V = 21.91 m/s in Region 1 and V = 3.87 m/s in Region 2, which is the incompressible pressure components convected at the mean flow velocity in the downstream direction and associated with turbulence fluctuation acting as an aerodynamic noise source. Since the acoustic wave generated by the aerodynamic noise sources propagates at the speed of sound, the compressible pressure fluctuation (plane wave components in the pipe) propagates in the downstream direction at the speed of c + u\({}_{0}\), which is the sum of the sound velocity and the flow velocity, and at the speed of c - u\({}_{0}\) in the upstream direction. In the case of other non-planar waves, the phase speed is higher than c - u\({}_{0}\) and c + u\({}_{0}\) in the upstream and downstream directions, respectively. Therefore, the pressure wave components located in the region between the two lines of which the phase speed is c - u\({}_{0}\) and c + u\({}_{0}\) denotes those propagating at a speed higher than the speed of sound and thus are compressible acoustic components. The cut-off line for each duct mode can be defined as Equation (10) and is depicted as a curve in Figure 14. In the case of the incompressible pressure fluctuation, the components propagating at the average mean flow velocity are not clearly visible, which is different from the results shown in the study by [PERSON] et al. [8]. The reason for this is because the vortices of various sizes are formed and propagate at various speeds due to an abrupt change in the cross-sectional area between the nozzle and the downstream pipe. In the case of the compressible pressure fluctuation component, it can be seen that the spectral density is prominently distributed in the vicinity of the cut-off line of each mode and becomes gradually weaker as the order becomes higher. As mentioned above, the predicted pressure level assures up to 2000 Hz, and high grid resolution is required for higher frequency range. Therefore, the high-frequency components above 2000 Hz are rapidly attenuated. The application of inverse wavenumber-frequency transform on the region of wavenumber-frequency diagram including only the compressible pressure fluctuations leads to the acoustic power spectral density, eliminating the incompressible pressure fluctuations. Figure 14: Wavenumber-frequency analysis result; (**a**) Region 1, (**b**) Region 2. \(n\) = the radial direction mode; \(m\) = the circumferential direction mode. Figure 13: Acoustic pressure region.