## 3 Results and Discussions
### Flow-Field Analysis
Figure 15 compares the time-averaged normalized entropy production distribution on the streamwise plane close to the leading edge of the original diffuser vane and various inclined vanes. The entropy production is normalized by the entropy at the impeller exit. The distributions of entropy production for all the cases are almost similar, except for the near hub region. In the cases of the original and \(-30^{\circ}\) inclined vanes, the higher entropy production region close to the pressure side of the diffuser vane hub, indicated by the red circles, is observed, which is caused by the pressure side separations. However, for the \(30^{\circ}\) inclination angle, the high-entropy region near the hub is significantly reduced, which confirms that the separation flows are postponed inside the diffuser passages. The time-averaged streamline on the diffuser vane pressure sides for various inclined leading edges are shown in Figure 16. In the case of \(30^{\circ}\) inclination angle, no recirculation or separated regions are observed on the diffuser vane pressure side, which confirms the hub-to-shroud inclination angles applied in the leading edge of the diffuser vanes can  
Figure 14: (**a**) Time-averaged meridional velocity profiles in the spanwise direction near the leading edge of the diffuser vanes, (**b**) time-averaged pressure profiles at 50% span along the streamwise direction. Figure 15: Time-averaged normalized entropy production contours near the leading edge of diffuser vanes for different inclination angles. significantly affect the separated flow regions generated by the interaction between the upstream entering flow and the diffuser vane leading edges. It is clear that the hub-to-shroud inclination angle is effective at mixing the low and high momentum flow regions. Therefore, the separated flows are postponed, which leads to enhance the aerodynamic performance of the centrifugal compressor. In order to visualize the reattached and separation flow regions, time sequences of the \(Q_{criterion}\) iso-surface for different diffuser vanes during one impeller revolution are shown in Figure 17. The \(Q_{criterion}\) is defined as the second invariant of the velocity gradient tensor, which can be calculated by Equation (8) [39]:  
\[Q_{criterion}=\frac{1}{2}\Big{(}\|\Omega\|^{2}-\|S\|^{2}\Big{)} \tag{8}\]  
where \(\Omega\) and \(S\) are vorticity and the rate of deformation tensors, respectively. As indicated by circles in Figure 17a,c, for the original diffuser vane and \(-30^{\circ}\) inclination angle, separation bubbles close to the pressure side of the diffuser vanes, are formed by the impeller-diffuser interaction. These separation bubbles are developed, become larger, and are convected toward the downstream of the diffuser passages by the rotation of impeller with respect to time. However, as shown in Figure 17b for the \(30^{\circ}\) case, these separation bubbles are not observed within the diffuser passage during the impeller revolution, which confirms the effects of the hub-to-shroud inclined leading edge on postponing the separated flows. Figure 18a,b shows the time-averaged normalized entropy production and pressure ratio curves, respectively, with respect to the circumferential direction at 1.08 \(R_{2}\) at various spanwise locations for different inclination angles. In the case of the \(30^{\circ}\) inclination angle, the range between the minimum and maximum entropy production is decreased and the entropy profiles become more uniform in comparison to the original case. In addition, the values of the entropy production are lower with respect to the other cases, which leads to the reduction of the velocity fluctuations at the diffuser throats, and less separated flows inside the diffuser channels. In addition, as depicted in Figure 18b, increasing the inclination angle increases the static pressure at various locations in the spanwise direction.