## 4 Results and Analysis
### Full Passage
#### 4.2.2 Unsteady Results  
All of the above studies are based on steady flow field, but, the impeller is exposed to downstream circumferential unsteady flow, which resulted in blade vibrations and auxiliary noise [20]. The acoustic Strouhal number is employed to characterize unsteady interaction of impeller and solute of compressible flows. This can be defined as follows:  
\[St=\frac{fL}{C}, \tag{9}\]  
where, \(f\) is the rotational frequency, \(C\) the speed of sound estimated with the inlet temperature. \(St\) represents temporal relationship between rotation speed of impeller and perturbation wave  
Figure 14: Blade to blade performance comparison at design point, 50% span wise. Figure 15: Performance of whole flow passage at design point. propagation of flow in the impeller. Unsteady effects can be neglected when \(St\ll 0.1\) and the relatively accurate results can be evaluated by steady calculations. While \(St>0.1\), more accurate results can be obtained through unsteady flow calculations [21]. In this case, \(St\) is about 0.12 at design point. It implies that the propagation of pressure wave is faster than the rotation speed of the impeller. Therefore, the unsteady numerical investigation, which is focused on the response of pressure distortions imposed by asymmetric geometry of volute, is necessary to be conducted. Since time-accurate simulation of the unsteady phenomenon, rotational speed and angular displacement between two successive computations should be considered into the numerical time step. The dual-time step (DTS) method was adopted to improve the time marching in unsteady calculation because the adopting of full passage mesh and the highly quality mesh in this study. The unsteady Rotor/Stator interface was set by domain scaling method [22]. The number of angular positions is set to be 20 in one single passage, so there are total of 240 angular positions. The time step \(\Delta t\) can be expressed as equation follows:  
\[\Delta t=\frac{2\pi}{\omega\cdot Nperiods\cdot NOFROT} \tag{10}\]  
where, \(\omega\) is the rotor rotation, \(Nperiods\) is the number of periodicities and \(NOFROT\) is the number of angular positions, so, in this study the time step was \(\Delta t=4.5\times 10^{-5}\)s, and the steady numerical simulation result is set to be the initial condition of unsteady numerical calculation in the first period. The positions of the seven probes are shown in Figure 16, of which six probes are located at the outlet of impeller and one at the tongue. It is shown in Figure 17 that shows the nondimensionalized static pressure variation of six probes at different locations, respectively. The horizontal ordinate is the physical time which means the rotor need unitary time step (\(\Delta t=4.5\times 10^{-5}\)s) to revolve one angular position, so there are total of 240\(\Delta t\) because of 240 angular positions. It can be seen that the flow field presented a periodic fluctuation and the shape of curves are different due to the asymmetric configuration of volute. Meanwhile, phase shift between different probe positions is also observed. It is due to the propagation of pressure perturbation from one blade channel to the next. The oscillation of nondimensionalized static pressure of probe 7 at tongue position in a period can be seen in Figure 18. The amplitude of pressure oscillation of tongue is smaller than those of impeller outlet positions, but the frequency of pressure oscillation is bigger. Since the amplitude of pressure oscillation is small, only about 1.5%, the steady numerical simulation result at the tongue position is accurate enough. Figure 16: Position of probe. It can be seen in Figure 19 that static pressure distribution at different time steps of 50% radial height near hub in one channel period. When the relative position between impeller and volute is the same as that of steady simulation, the \(t\) is adopted as 0. Transformation of static pressure distribution indicates unsteady characteristics, but the influence of which is not obviously.