## 3 Results and Discussions  
### Aerodynamic Performance and Stall Margin  
In a previous study by the authors [11], the inclined leading edges were found to suppress the separated flows near the throats of the diffuser, which can enhance the aerodynamic efficiency of the transonic centrifugal compressor. The pressure ratio and efficiency of the compressor at the design point condition for various inclined leading edges are shown in Figure 8a,b, respectively. While decreasing the magnitude of the inclination angle for the shroud-to-hub inclined vanes increases the pressure ratio, the values of area are always lower than those obtained for the original diffuser van. In the case of hub-to-shroud inclined leading edges in the range of \(0<\alpha<45\), the pressure ratio is larger in comparison with the original diffuser vane; but for \(\alpha>45\), the pressure ratio is significantly dropped. A similar trend is also observed for the compressor performance in the case of shroud-to-hub (\(-\alpha\)) inclination angles, and \(30^{\circ}\) inclined leading edge showed the best efficiency compared to the other cases. In addition, the efficiency curve is also dramatically decreased for inclination angles bigger than 45. Turbulence kinetic energy is one of the most important variables, which measures the intensity of the turbulence and velocity fluctuations in turbulent flows. Figure 9 shows the time-averaged non-dimensional turbulence kinetic energy distributions near impeller exit for the original centrifugal compressor. The turbulent kinetic energy is normalized by the tip velocity. As shown in the figure, there is a high magnitude region of turbulence kinetic energy near the impeller shroud, between the pressure side of the impeller main blade and splitter suction side (indicated by with circles in Figure 9). This high magnitude region indicates that there are more turbulent flows and more velocity fluctuations near  
Figure 7: Diffuser vanes with inclined leading edge in comparison with the original diffuser vane. the impeller shroud, which later these high-turbulence kinetic energy region enters into the impeller-diffuser vaneless gap. Therefore, by applying the inclination angles on the leading edge from hub-to-shroud, the vanless gap between the impeller and diffuser close to the shroud is increased, which leads to provide a more uniform flow within the vaneless region and diffuser throats. In order to evaluate the effect of the inclined leading edge on the stall margin, the total pressure ratio and total-to-total efficiency curves were calculated for two different inclined leading edges (\(-30^{\circ}\) and \(30^{\circ}\)), and compared with the reference case as shown in Figure 10a,b, respectively. In the case of the \(-30^{\circ}\) case, the efficiency near stall and the stall margin are dramatically reduced in comparison to those of the original vaned diffuser. However, the \(30^{\circ}\) inclination angle can effectively increase the stall margin, compared to the original diffuser, and simultaneously improve the efficiency of compressor. The stall margin improvement can be calculated by Equation (7) for a test case compared to the reference case, which was suggested by [PERSON] [38]. \[\Delta SM(\%)=\left[\left(1-\frac{PR_{peak}}{PR_{stall}}\times\frac{\dot{m}_{ stall}}{\dot{m}_{peak}}\right)_{test}-\left(1-\frac{PR_{peak}}{PR_{stall}}\times \frac{\dot{m}_{stall}}{\dot{m}_{peak}}\right)_{ref.}\right]\times 100 \tag{7}\]  
Figure 8: Prior study of different inclined leading edges in the diffuser vanes compared to original diffuser vane [11]: (a) total pressure ratio, (b) total-to-total efficiency at the design point. Figure 9: Time-averaged non-dimensional turbulence kinetic energy close to the impeller exit (main blades and splitter blades are visualized by yellow and red color, respectively). where \(PR_{peak}\) and \(\dot{m}_{peak}\) represent the pressure ratio and mass flow rate at the peak efficiency point, respectively.