## 4 Results  
### Analysis on the Inherent Unsteady Flow Structure  
The natural type of unsteady separation flow structure was analyzed as a reference state in this section. The DES simulations have been performed at different mass-flow coefficients. Characteristic lines of the quasi-3D model were shown in Figure 5. The mass-flow coefficient was defined as the ratio of averaged axial velocity to averaged total velocity. The pressure coefficient \(\Delta\)Cp was defined as Equation (5):  
\[\Delta\text{Cp}=\Delta\text{p}/\left(\frac{1}{2}\rho\text{U}_{\text{m}}^{2}\right) \tag{5}\]  
where \(\Delta\)p was pressure raise, Um was the rotational speed at midspan, and \(\rho\) was the density. In continuum mechanics, the vorticity \(\overset{\rightarrow}{u}\) is a pseudovector field defined as the curl of the flow velocity \(\overset{\rightarrow}{u}\) vector. It describes the local spinning motion of a continuum near some point. The definition can be expressed by the vector analysis formula:  
\[\overset{\rightarrow}{\omega}\equiv\
abla\times\overset{\rightarrow}{u} \tag{6}\]  
where \(\
abla\) is the del operator. The vorticity of a two-dimensional flow is always perpendicular to the plane of the flow, and therefore can be considered a scalar field. In this paper the working point A near the stall was selected as the research working point to investigate unsteady separation flow structure. The instantaneous vorticity contour at the working point A was presented in Figure 6. As the graph showed, the separation on the suction side started near the middle length of chord, and then the separation vortex spread downstream and induced a shedding vortex at the trailing edge of blade. The vortex shedding phenomenon at the blade trailing edge, which was physically similar to the Karman vortex street, could be described as that separation vortices on the two sides of the blade combined with each other at the trailing edge with different vorticity directions, and then shed off in pairs when the balance of vorticity was obtained. Analyzed from this physical mechanism, three different types of unsteady flows existed in the flow field: Shedding vortex, suction-side separation  
Figure 5: Characteristic lines of the quasi-3D model. Figure 6: Instantaneous vorticity contour at working point A.  
vortex, and pressure-side separation vortex. To obtain the characteristic frequencies of the three inherently unsteady flows, the analysis on vorticity fluctuations on monitor points P1, P2, and P3 (marked in Figure 6) was conducted. The vorticity fluctuations in the time domain and corresponding spectrum analysis at the three monitor points at working point A are shown in Figure 7.  
A dominant frequency can be identified in all three locations, which was 36.5 times the rotation frequency (RF). The maximum amplitude at this frequency was located in the region of shedding vortices, which signified that it was the frequency of shedding vortex \(F_{\textit{shed}}\). The dominant frequency marked by the red square reached its maximum amplitude near the separation point on the suction side, which indicated the suction-side vortex separated at 7.12 times the rotation frequency (\(F_{\textit{ss}}\), frequency of suction-side vortex shedding). The dominant frequency at 27.6 times the rotation frequency (marked by the blue square) reached its maximum amplitude near the trailing edge in the pressure side. Based on the physical mechanism, this frequency was the inherent frequency of the pressure-side separation vortex named as \(F_{\textit{ps}}\). Variations of the frequencies to mass-flow coefficients are shown in Figure 8 for the three types of unsteady separation flows. It could be concluded that the vortex shedding frequency \(F_{\textit{shed}}\) varied more than 100% with mass-flow coefficients, while the frequencies \(F_{\textit{ps}}\) and \(F_{\textit{ss}}\) were insensitive to the variation of working conditions.