## 3 Results and Discussions
### Spectral Analysis  
Various studies regarding the prediction of noise sources in centrifugal compressors showed that the pressure fluctuations generated by the interactions between the impeller and diffuser blades and turbulent structures inside the compressor channels are the dominant factors, which affect the far-field acoustics [12]. Therefore, the unsteady static pressure at mid-span point of 1.08 \(R_{2}\) near the impeller-diffuser interface is monitored with respect to time. Figure 20 shows the pressure signal at the compressor design point, for which the time periodic of the pressure fluctuations confirms the convergence of the transient simulation. Figure 20 shows that the current URANS approach has predicted the pressure  
Figure 19: Distributions of time/space normalized static pressure at the mid-span cutting plane of the diffuser vane for various inclined leading edges: (**a**) original diffuser, (**b**) 30\({}^{\circ}\), (**c**) \(-\)30\({}^{\circ}\), (**d**) 75\({}^{\circ}\), (**e**) \(-\)75\({}^{\circ}\). fluctuations with a good accuracy in terms of magnitude and pattern compared to those calculated by LES in [43]. Figure 21a,b shows the spectra of the near-field pressure fluctuations computed by the current URANS approach for the original leading edge and various inclined leading edges. The sound pressure level (SPL) is evaluated using the fast Fourier transform approach on the pressure fluctuations. The SPL is used to investigate the different contributions of the transient pressure variations in the frequency domain. It should be noted that the SPL spectra of all the inclined leading edge cases were compared to the spectrum of the original diffuser vane, and the horizontal axis is normalized by BPF. As shown in Figure 21, there are clear fundamental peaks in the spectra, which corresponded to the BPF noise and its harmonics. The first and second harmonics of BPF noise are observed at 5.4 kHz and 10.8 kHz, respectively, which are indicated in Figure 21. Figure 21 clearly confirms that the total BPF noise and its harmonics, generated by the interaction between the impeller and diffuser, are the dominant noise sources in the current centrifugal compressor. Figure 21: Comparison of near-field SPL spectrum for various inclined leading edges with the original diffuser: (**a**) \(30^{\circ}\), \(-30^{\circ}\), (**b**) \(75^{\circ}\), \(-75^{\circ}\). Figure 20: Static pressure time history at mid-span point (1.08 \(R_{2}\)) for the design mass flow rate. Figure 21 shows that by applying 30\({}^{\circ}\) inclination angle to the leading edge of the diffuser vane, not only the tonal peak at the first BPF is decreased by 7.6 dB, but also the spectrum level is decreased. In addition, Figure 21a shows that the 30\({}^{\circ}\) inclined leading edge in the diffuser vane reduces the overall sound pressure level (OASPL) by 5.74 dB. While the 75\({}^{\circ}\) inclined leading edge reduces the first BPF tonal noise, the OASPL is increased, due to transient and separated flows inside the diffuser channels. For the original diffuser vanes with the straight leading edges, a spanwise uniformly distributed pressure profiles from the upstream flow are impinged on the leading edges of the diffuser vanes. Therefore, the pressure fluctuations are generated on the vane surfaces and then are propagated toward inside the diffuser passages, which result in separated flows inside the diffuser channels. However, by applying the hub-to-shroud inclination angle on the leading edge of the diffuser vanes, the incoming pressure profiles from upstream are not uniformly distributed in the spanwise direction compared to that of the straight leading edge. This is caused by the pressure variations close to the inclined leading edge region, which could lead to the tonal noise reduction. However, as shown in Figure 21, for the shroud-to-hub inclination angles (\(-30^{\circ}\) and \(-75^{\circ}\)), the tonal noise and OASPL are not reduced.