## 3 Results and Discussion
### Boundary Layer Flow in the Stalled Passage
Figure 6 shows the distribution of the instantaneous skin friction coefficient on the blade surface at the axial cross section near the hub of the impeller (\(Z/b_{2}=0.15\)). The curves are shown from the leading to trailing edges of the blade at several moments similar to that described above. It is seen from the figure, the variation amplitude of the skin friction coefficient is higher on the pressure surface and the magnitude is also generally higher. Consequently, we can anticipate that the velocity of the near-wall flow is higher on the pressure surface, which induces more energy loss. From the position of \(S/S_{0}=0.5\), the curves start to increase sharply and to reach the local peak at around \(S/S_{0}=0.7\). The increased wall-tangent velocity is mainly attributed to the acceleration from APG on the surface as the primary vortex circulates in the passage, as seen in Figure 4. The magnitude of velocity then decreases until \(S/S_{0}=0.95\) as the passage expands in the area in which the velocity generally decreases close to the outlet of the impeller, and the jump right at the trailing edge results  
Figure 5: Distribution of pressure coefficient on the blade surface for the \(Z/b_{2}=0.15\) cross section: (a) pressure surface and (b) suction surface. Figure 6: Distribution of skin friction coefficient on the blade surface for the \(Z/b_{2}=0.15\) cross section: (a) pressure surface and (b) suction surface. from the local favorable pressure gradient (FPG) field, and the interaction of the passage flow and that outside of the passage, where the flow is not severely restricted by the blade surfaces and has a higher velocity. Different distribution patterns of the skin friction coefficient occur for the suction surface; more increasing or decreasing is observed. The local minimum at around \(S/S_{0}=0.55\) corresponds to the junction of the two vortices on the suction surface in which the local flow approximately perpendicularly impinges on the surface. It is worth emphasizing here that, for the boundary layer flow of both pressure and suction surfaces, there is almost no temporal variation of the velocity for the whole surface, as reflected by the overlapping curves in both subfigures. This indicates that, although the structural evolution of the separated vortices is clearly seen in Figure 4, the boundary layer flow is hardly affected in terms of velocity magnitude during the rotation but is almost constant. The analysis and discussions presented above demonstrate the characteristics of boundary layer flow of the blade under the constraint of the hub surface. To investigate the characteristics of flow in the central  
In order to evaluate the effect of the hub and shroud surfaces on the internal flow of the impeller, Figure 9 gives the distribution of the skin friction coefficient on the pressure and suction surfaces of passage I on cross sections at various axial positions, i.e., from cross section close to the hub surface (\(Z/b_{2}=0.15\)) to the shroud surface (\(Z/b_{2}=0.84\)) with a distance between two neighboring cross sections of 1 mm. Since we have already confirmed in Figures 6 and 8 that the skin friction coefficient almost does not vary in time, the difference among the curves in Figure 9 undoubtedly reflects the three-dimensionality of the flow, which is seldom considered in previous works in literatures. In general, the three-dimensionality of the flow is the most pronounced on the downstream half of the pressure surface roughly in the region \(S/S_{0}=0.50\)-\(1.00\); the flow is generally decelerated close to the hub surface but accelerated in the central region of the passage. The curves also reveal that, due to the different geometries of the curved shroud and flat hub, the boundary layer flow is remarkably affected as revealed by the difference between the curves of \(Z/b_{2}=0.15\) and \(Z/b_{2}=0.84\), and of \(Z/b_{2}=0.32\) and \(Z/b_{2}=0.67\). The boundary layer flow moves at a relatively higher velocity on the cross section close to the shroud.