## 3 Results and Discussion
### Boundary Layer Flow in the Unstalled Passage  
As shown in Figure 4, the flow in the unstalled passages (passages II, IV, and VI) exhibit entirely different patterns compared with flow in the stalled passages. The flow moves through the passage without significant separation and formation of large-scale vortices. A small separated vortex is generated near the pressure surface during a portion of the vortex formation cycle which generally moves towards the outlet of the impeller with reduced size and intensity. Figure 10 gives the distribution of the pressure coefficient on both pressure and suction surfaces of the unstalled passage on the cross section at the axial position \(Z/b_{2}=0.15\). As shown in Figure 10a, the pressure coefficient of the pressure surface in the region \(S/S_{0}=0.50\)-\(0.80\) monotonically increases in time from \(t=1/12T\) to \(t=9/12T\). Under the adverse pressure gradient, the fluid in the boundary layer is subjected to higher pressure as it moves towards the outlet of the passage, while separation is not pronounced probably because of the relatively lower magnitude of adverse pressure gradient as compared with that of the stalled passage. The distribution of pressure coefficient on the suction surface is similar to that of the stalled passage. Figure 11 shows the distribution of skin friction coefficient on the pressure and suction surfaces of the unstalled passage on the cross-section at the axial position \(Z/b_{2}=0.15\). It is seen in the figure that the skin friction coefficients of both the pressure and suction surfaces remain basically unchanged during a vortex formation cycle. On the pressure surface, the coefficient keeps almost monotonically increasing from the leading to trailing edge of the surface without significant local variation due to the separated vortex as that in the stall passage. For the suction surface, however, the wall-tangent velocity of the boundary layer flow remarkably decreases as the flow moves towards the outlet of the impeller, and no obvious fluctuation is observed, reflecting the decreased kinetic energy of the flow as a result of the viscous drag from the wall and the adverse pressure gradient. Figure 10: Distribution of the pressure coefficient on the blade surface for the \(Z/b_{2}=0.15\) cross section: (**a**) pressure surface and (**b**) suction surface. We further explore the characteristics of boundary layer flow in the central region of the unstalled passage on the cross section at axial position \(Z/b_{2}=0.50\). The distribution of pressure coefficient along the pressure and suction surfaces of the passage is given in Figure 12. During a vortex evolution cycle, the pressure coefficient of the pressure surface continues to increase, ranging from \(-4\) to around 0. In the region \(S/S_{0}=0.6\)-1.0, the magnitude of the APG of the pressure surface decreases slightly in time. As the flow is decelerated by APG, the flow will be reversed, and the size of the separated vortex will increase to some extent and moves to the outlet of the impeller. Meanwhile, the separation of boundary layer flow on the middle section of the passage is inevitably affected by the neighboring fluid. The distribution of skin friction coefficient on the blade surfaces is quite similar to that on the cross section near the hub, as shown in Figure 13. In the region \(S/S_{0}=0.6\)-1.0, the skin friction coefficient of the pressure surface begins to increase significantly and, then, the velocity gradient of the boundary layer increases; the greater the shear stress, the greater the energy loss. The velocity gradient of the boundary layer on the suction surface is continuously reduced except for the region at the trailing edge of the blade where the boundary layer flow is notably affected by the outflow from other passages. Figure 11: Distribution of the skin friction coefficient on the blade surface for the \(Z/b_{2}=0.15\) cross section: (**a**) pressure surface and (**b**) suction surface.