# Wavenumber-Frequency Analysis of Internal Aerodynamic Noise in Constriction-Expansion Pipe
## 4 Numerical Results
### Wavenumber-Frequency Analysis  
The wavenumber-frequency analysis was performed using the LES results presented in Section 4.3. Figure 13 shows the wavenumber-frequency transform of the static pressure on a cylindrical inner duct wall, i.e., \(r=d/2\) for a length of 1 m from the pipe outlet. The frequency and time intervals are \(\Delta f=10\,\mathrm{Hz}\) and \(\Delta t=5\times 10^{-5}\) s, respectively. The spatial and wavenumber intervals are \(\Delta z=3.844\times 10^{-3}\) m, \(\Delta\theta=10^{\circ}\), and \(\Delta k_{z}=1.004\), respectively. The average speed of sound and mean flow Mach number over the target region are \(c_{0}=317\) m/s and \(M=0.296\), respectively. The speed of sound is considered to be the smallest value in the entire flow field.  
The black cut-off lines corresponding to the modes of (\(m\), \(n\)) = (0:4, 0:3) are also drawn to make it easier to understand the behavior of the pressure waves inside the pipe in the wavenumber-frequency domains. The \(m\)-mode is the circumferential mode, and the \(n\)-mode is the radial mode. The dashed line represents the incompressible pressure wave convecting at a constant mean flow speed of \(0.88U_{0}\). The strong compressible acoustic waves propagating at speeds of \(c+U_{0}\) (downstream propagating waves) and \(-c+U_{0}\) (upstream propagating waves) can be identified in the wavenumber-frequency diagram of the \(m\) = 0 and \(n\) = 0 modes. The condition for the higher-order modes is shown in Equation (10). From Equation (9), the variation in the \(z\)-direction wavenumber, \(k_{z}\), for the higher-order modes is described in Figure 13 in the parabola form, according to the \(n\)-modes [14]. It can be seen from  
Figure 11: Time histories of mass flow rates for large eddy simulation (LES).  
Figure 12: Flow fields for LES: (**a**) static pressure [Pa]; (**b**) turbulent viscosity [kg/ms]; and, (c) vorticity [1/s].  
Figure 13 that the \(n=0\) and \(n=1\) modes contributes dominantly, whereas the higher-order modes with \(n=2\) or higher are relatively negligible. [PERSON] mentioned that pipe failure due to acoustic waves occurs in the range of 500-2000 Hz [8; 15]. In this study, the higher-order modes above \(n=2\) occurred at frequencies greater than 2000 Hz. Thus, they did not belong to the pipeline failure region due to acoustic waves. The line of the \(n=0\) mode shows the velocity of the acoustic wave propagating upstream and downstream, where the area inside the line is the compressible pressure fluctuation component, and the area outside the line is the incompressible pressure fluctuation component. Based on the results of the wavenumber-frequency analysis, it can be seen that the compressible and incompressible pressure fluctuations can be separated from the entire flow field, as shown in Figure 14.  
Figure 15 compares the acoustic power level spectrum that was obtained from the inverse wavenumber-frequency transform of the compressible and total pressure fluctuation components. It shows that the acoustic power spectrum can be separated from that due to the total pressure fluctuation. Figure 16 shows the acoustic power spectrum of each circumferential mode. These were obtained from the inverse wavenumber-frequency transform of the wavenumber-frequency diagram, as shown in Figure 14. The contribution of each circumferential acoustic mode to the total acoustic power can be quantitatively assessed based on these results. This information can be utilized because  
Figure 14: Region of compressible pressure fluctuation component for \(m=0\) mode.  
Figure 13: Wavenumber-frequency diagram of surface static pressure on cylindrical plane \(r=d/2\).