## 3 Results and Discussion
### Boundary Layer Flow in the Stalled Passage  
In this section, we analyze and discuss the flow in the stalled passage I to demonstrate the evolution of the vortices. Since the near-wall flow is significant in determining the hydrodynamic performances of the impeller and even the pump, the characteristics of the flow are presented by the distributions of pressure coefficient and skin friction coefficient on the surfaces of the blade at various axial cross sections. The dimensionless pressure coefficient is defined as follows:  
\[C_{p}(\mathbf{x},t)=\frac{p(\mathbf{x},t)-p_{2}}{0.5\rho U_{1}^{2}} \tag{5}\]  
where \(p\) is the instantaneous static pressure at the blade surface; \(p_{2}\) is the static pressure at the outlet of the impeller; and \(U_{1}\) is the averaged inflow velocity. The skin friction coefficient is defined as follows:  
\[C_{f}(\mathbf{x},t)=\frac{\tau_{w}(\mathbf{x},t)}{0.5\rho U_{1}^{2}} \tag{6}\]  
where \(\tau_{w}\) is the local wall shear stress. The stalled passage I consists of the pressure surface of one blade (denoted as ps1) and suction surface of the other blade (denoted as ss1). In order to clarify the evolution of the separated vortices in a cycle, here, we present the instantaneous values of the quantities at six moments which are the same as in Figure 4, i.e., 1/12\(T\), 3/12\(T\), 5/12\(T\), 7/12\(T\), 9/12\(T\), and 11/12\(T\), that the impeller rotates for 20 degrees between two successive moments. Figure 5 gives the distribution of pressure coefficient on both surfaces of the blade for the cross section at the axial position \(Z/b_{2}\) = 0.15, which is close to the hub of the impeller. The local coordinate \(S\) along the curved surface is chosen as the abscissa and is normalized by the arc length \(S_{0}\) of respective surface; thus, \(S/S_{0}\) = 0.0 and 1.0 represent the leading and trailing edges, respectively. For the pressure surface shown in Figure 5a, the pressure of the fluid experiences an increase for the majority of the surface and is the most notable for the region \(S/S_{0}\) = 0.5-0.8, representing an adverse pressure gradient \(\partial p/\partial s>0\) (APG). Since the APG on the blade surface is generally aligned in the direction with of boundary layer flow, the flow is decelerated under the combined effect of the AFG and the viscous force and consequently separates from the blade surface. The separated boundary layer flow is lifted up away from the pressure surface and becomes part of the main flow in the passage; thus, a large-scale vortex is generated on the pressure surface, which occupies almost the entire passage, as shown in Figure 4. The flow downstream of \(S/S_{0}\) = 0.6, i.e., close to the outlet of the impeller, is accelerated by the favorable pressure gradient (FPG) roughly in the small region \(S/S_{0}\) = 0.8-1.0. For the distribution of pressure coefficient on the suction surface shown in Figure 5b, it is different from that of the pressure surface in that the pressure coefficient almost monotonically increases for the whole blade surface, resulting in a strong APG in the region \(S/S_{0}\) = 0.3-1.0. The pressure coefficient varies slightly for the inlet region of the impeller but is more pronounced close to the outlet of the impeller where the local flow is alternatively occupied by the large-scale separated vortex and reattached flow. As the boundary layer flow is separated from the blade surface, the fluid is injected into the external main flow, which may then reattach to the blade surface in the downstream region. The wall shear stress, as quantitatively represented by the skin friction coefficient, is a reflection of the magnitude of velocity of boundary layer flow on the blade surface and is affected by the local pressure gradient. Here, we analyze the distribution and evolution of the skin friction coefficient to demonstrate the intensity of the circulation of the separated vortices in the passage.