## 4 Results and Discussion
### Wavenumber-Frequency Analysis
The acoustic power spectrum of the internal aerodynamic noise obtained in this way is shown in Figure 15. It shows the power spectrum of three components: the incompressible pressure fluctuation part due to turbulence fluctuation, the compressible pressure fluctuation part due to acoustic waves, and the total pressure fluctuation including the incompressible and compressible parts. From the spectrum and overall values of all parts, the power level in Region 1, which is the region more upstream, is greater than in Region 2. The influence of the incompressible part due to flow is dominant in the low frequency region below 1000 Hz, but the influence of the compressible part due to the acoustic wave is dominant in the region of 1000 Hz to 2000 Hz. However, in the higher frequency range, it is rapidly dissipated due to poor time and spatial resolution. It can be seen that the incompressible part becomes weaker from Region 1 to Region 2 because the vortex rapidly dissipates as shown in Figure 11b. For the incompressible part related to the turbulence fluctuation, during the convection of vortical waves downstream, its energy is transferred to smaller-scale eddies oscillating with the high frequency, that is so-called energy cascade phenomena occurring in the inertial subrange. The decay rate of its spectrum is known to follow the \(-\)5/3 power law. The even smaller eddy scale, known as the Kolmogorov Scale, belongs to the dissipation range where energy is dissipated due to heat or sound, and is known to follow the \(-\)7 power law [17]. It can be found from Figure 15 that, in both Region 1 and Region 2, the inertial subrange exists in the region between 100 Hz and 1000 Hz, and a dissipation range exists in the frequency region higher than 1000 Hz. Especially in Region 1, where larger coherent turbulent fluctuations exist and are followed by its energy cascade phenomena, the inertial subrange and dissipation range are more prominent and distinct. Note that the decay rate of the spectrum in the frequency range higher than around 2000 Hz to 3000 Hz is due to numerical damping. The good agreement of the decay rate of the predicted spectrum confirm the validity of the LES simulation and the wavenumber-frequency analysis. It is noteworthy that the magnitude of compressible components is higher than that of incompressible ones in the frequency range belonging to the dissipation rage. This fact ensures that the energy of turbulent fluctuations is transferred to the compressible acoustic waves in the dissipation range. It can be also seen from the overall power level that both incompressible and compressible components are attenuated downstream, but the incompressible part is further reduced compared to the compressible part due to more rapid dissipation of the vortex. Overall, these results show that the current numerical methods predict the acoustic power due to the valve flow in a pipe more precisely than the existing ones relying on the quasi-empirical equations by [PERSON] and [PERSON], considering more complex geometry and conditions. Figure 15: Overall acoustic power level (OAPWL); (**left**) Region 1, (**right**) Region 2.