## 4 Conclusions  
A compressible URANS method was performed to numerically predict the flow structures in a transonic centrifugal compressor with pressure ratio of 4. The aerodynamic performance of the original diffuser vane computed by the numerical simulation was validated by the measurements at various operating point conditions. Two types of the inclined leading edges with respect to the diffuser vane height, from hub-to-shroud and shroud-to-hub, were proposed. 1. The numerical results of various inclined leading edges showed that the inclination angles are effective tools for improving the efficiency of compressor and stall margin. With the optimal inclination angle of \(\alpha=30^{\circ}\) from hub-to-shroud, the performance at the design point increased by 2.6% and the stall margin by 3.88%, in comparison with the original diffuser. However, for accurate prediction of detailed flow physics near the stall point, transient approaches would be required. Detailed flow investigation inside the diffuser channels confirms that using the hub-to-shroud inclination angles can suppress the separated and secondary flows within the diffuser passage, especially on the vane pressure side. 2. The distribution of the static pressure and velocity contours in the diffuser channels demonstrated that the hub-to-shroud inclination angles reduced the flow unsteadiness. In addition, time-averaged streamlines on the surfaces of the diffuser vanes showed that the inclination angle in diffuser vane affects the boundary layer growth inside the diffuser channels, which leads to reduction of the recirculation zones, and thus the flow separations are postponed. It is evident that the hub-to-shroud inclined leading edges are very useful in mixing low and high-momentum flow regions. 3. The time/space pressure distribution at the midspan of the diffuser revealed that transient pressure waves mainly occur near the leading edges of the diffuser vanes, and decay very quickly along the diffuse passages. The number of these waves is the same as the number of impeller main and splitter blades, and their slope denotes the convective velocity. Instantaneous time/space distributions of the static pressure for various inclined leading edges confirmed that the 30\({}^{\circ}\) inclined leading edge had the maximum pressure level. Additionally, in the case of 30\({}^{\circ}\) inclination angle, in contrast with other cases, the pressure oscillations generated by the pseudo-periodic transient separation bubbles with various slopes were not observed on the time/space pressure distributions. 4. The spectrum of the near-field pressure fluctuation well predicted the fundamental frequencies corresponding to the first (5.4 kHz) and second (10.8 kHz) BPF harmonics. It is shown that the BPF tonal noise and its harmonics, generated by the interactions between the impeller and diffuser blades, are the dominant noise sources in the centrifugal compressors. The BPF noise and OASPL were decreased by 7.6 dB and 5.74 dB, respectively, by applying 30\({}^{\circ}\) inclination angle on the leading edges of the diffuser vanes. This confirms that the inclined leading edges are very effective technique, not only for the reduction of the tonal noise and OASPL, but also for the improvement of the aerodynamic performance. Conceptualization, methodology, software, writing--original draft preparation, A.Z. ; software, K.S.P. ; writing--review and editing, M.C. and J.T.C. All authors have read and agreed to the published version of the manuscript. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2020R11A1A01060578). Not applicable. Data Availability Statement: Data sharing is not applicable to this article. The authors declare no conflict of interest. The spectrum of the near-field pressure fluctuation well predicted the fundamental frequencies corresponding to the first (5.4 kHz) and second (10.8 kHz) BPF harmonics.