# Mechanisms of Sweep on the Performance of Transonic Centrifugal Compressor Impellers
## Case Description
### Sweep Definition  
In axial cases, sweep can be defined as moving the shroud section parallel along the axial or chord direction and keeping the chord length unchanged [19]. However, the definition of sweep in centrifugal impellers cannot be directly inherited from that of the axial cases. Otherwise, the trailing edge radius and the work input of the centrifugal compressor will be greatly influenced. There are  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbols** & **Values** \\ \hline Number of blades & Z & \(13+13+13\) \\ Normalized leading edge hub radius & r\({}_{\text{1h}}\)/r\({}_{2}\) & 0.41 \\ Normalized leading edge tip radius & r\({}_{\text{1t}}\)/r\({}_{2}\) & 0.56 \\ Normalized exit blade height & b\({}_{2}\)/r\({}_{2}\) & 0.05 \\ Pressure ratio & \(\pi\) & 3.0 \\ Flow coefficient & \(\phi\) & 0.05 \\ Pressure coefficient & \(\psi\) & 0.68 \\ Specific speed & N\({}_{\text{s}}\) & 0.60 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main specifications of datum impeller.  
Figure 1: Datum impeller.  
two types of sweep definition in centrifugal impellers. The first one is to cut or to extend the tip of the leading edge along the chord direction while maintaining the rest part unchanged [25], as showed in Figure 2a. The second definition is to adjust the tip of the leading edge by scaling up or down the shroud chord while keeping the blade angle and thickness distribution along the relative chord length unchanged, as illustrated in Figure 2b. Because the second definition allows the sweep feature to be independent from blade angle and thickness distributions, it is more practical when coupling with optimization techniques. Therefore, the second definition is adopted in this study.  
To quantitatively define the extent of sweep, the sweep angle between the leading edge trace and the radial direction is adopted, as illustrated in Figure 3a. A positive value of sweep angle represents forward sweep, and the opposite is aft sweep. During the parametric study, the sweep definition of twin splitters is inherited from the main blade, where the camber line of the main blade is truncated to define the splitters with respect to the unchanged relative meridional position of the splitter leading edge. In other words, when the sweep angle of the main blade changes, both splitters will change in similar fashion, as presented in Figure 3b. Through this simplified approach, sweep features of all blades can be defined by only one parameter, while sweep effects on impeller performance and flowfields can still be well estimated and captured. For a real impeller optimal design process, it is necessary to decouple the sweep design of different blades. The investigated variation range of the sweep angle of main blade has been chosen from \(-25^{\circ}\) to \(+25^{\circ}\), which covers most of the swept impeller designs in earlier researches.  
Admittedly, the current definition of sweep will introduce extra lean. It is shown in Figure 3b that an extra lean will be introduced from the mid-chord to the leading edge of the main blade, while the  
Figure 3: Swept geometry: (**a**) meridional view and (**b**) shroud sections of all blades.  
Figure 2: Sweep definition: (**a**) maintaining rest blade part and (**b**) maintaining relative blade angle and thickness distribution.  
exit lean angle (or rake angle) is remained the same. The ratio of the change in lean angle to that in sweep angle has been analyzed to be only \(-13\%\) in this case (Appendix A). Because such extent of blade lean at axial part of the impeller has limited effect on impeller efficiency [28] and flowfields [29], it is neglected except for the choke analysis.