## 4 Results and Analysis
### Single Passage Model
#### 4.1.2 Meridional Plane  
The distributions of relative Mach number and pressure coefficient at the design point are shown in Figure 8. The pressure coefficient shown is defined as  
\[C_{p}=\frac{P}{P_{total,in}}, \tag{6}\]  
where, \(P\) is the static pressure and \(P_{total,in}\) the total pressure at inlet.  
In general, the uniform distribution of Mach number and pressure has a positive impact on efficiency and stall margin. But, it is found in Figure 8a that there is a region of high relative Mach number both in AL and LL impeller, which is very close to the turning point of the shroud line. In this region, the pressure is relatively lower, as shown in Figure 8b. So, this kind of nonuniformity of relative Mach number and pressure (shown in Figure 8) probably is one of the negative affects which would  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Impeller** & **Prototype** & **AL** & **LA** & **LL** \\ \hline SM & 0.17 & 0.27 & 0.15 & 0.29 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Stall distinction.  
Figure 8: Aerodynamic parameters distribution on the meridional plane at the design point.  
lead to lower efficiency of the AL and LL impeller. In contrast, distribution of Mach number and pressure of the prototype and LA impeller is more uniform, which is beneficial to efficiency.  
In another words, skin friction loss is another important negative effect to efficiency, which is generated by boundary layers. And the pressure loss coefficient which is resulted from skin friction of solid surfaces can be defined [18] as  
\[\omega_{SF}=4C_{f}{(\overline{W/W_{1}})}^{2}L_{B}/d_{h}, \tag{7}\]  
where, \(C_{f}\) is the skin friction coefficient, \(\overline{W}\) is the averaged relative velocity, \(\overline{W_{1}}\) is the averaged incidence velocity, \(L_{B}\) is the average length of hub and shroud, and \(d_{h}\) is the outlet hydraulic diameter. The averaged relative velocity of the four impellers is very close to each other as is shown in Figure 7a so that skin friction loss is approximately proportional to \(L_{B}\). Among these four impellers, \(L_{B}\) of LL impeller is shortest and then the skin friction loss of the LL impeller is smallest.  
Furthermore, hub to shroud loading loss, which is produced by secondary flow, can be expressed by the followed equation [18].  
\[\omega_{Hs}={(\overline{k}_{m}\overline{BW/W_{1}})}^{2}/6, \tag{8}\]  
where, \(\overline{k}_{m}\) is the blade curvature and \(\overline{B}\) the hub-to-shroud width. It is apparent that the hub-to-shroud width of LL impeller is smallest, which means weaker secondary flows and less loading loss.  
Base on above analysis, although the nonuniformity of Mach number and pressure ratio distribution can be observed in the LL impeller, its performance can be made up by the less skin friction loss and loading loss. So, as shown in Figure 8, there is a little performance difference among four impellers at the design point.  
In addition, the connection form with straight line of LL impeller arise two low Mach number regions can be found in the flow field near the break points because of the sharp change of slope, which needed to be concerned in this study. The more clearly pictures of flow situation of LL impeller can be seen in Figure 9. Nevertheless, this low Mach number region is small compared to the whole flow passage, and this phenomenon may be improved by other method of hub separation.  
As the machine is operated near the stall point, it is shown in Figure 10 that the meridional relative velocity distribution (Figure 10a) and pressure coefficient distribution (Figure 10b) of four impellers with 0.85 flow rate. It is apparent that there is a high relative velocity domain near the turning area of shroud line of LL impeller, which is beneficial to stall margin.  
Figure 9: Relative March number.