# Unsteady Flow Characteristics of an Oscillating Piezoelectric Fan Blade at High Reynolds Numbers
## 3 Results
### Vortex Evolution  
The characteristics of the vortex generated by the vibration of the PZT fan blade is another key issue that has been widely discussed in previous studies. In order to obtain the vortex structure, an ensemble-averaged method for eight different phases \(\tau\) was adopted  
Figure 8: Time averaged velocity profiles extracted at different streamwise lines: (**a**) \(U_{x}\) profile along the midspan line (\(x=0\), \(0<z<40\) mm), (**b**) \(U_{z}\) profile along the midspan line (\(x=0\), \(0<z<40\) mm), (**c**) \(U_{x}\) profile along the line distancing 15 mm away from the midspan line (\(x=15\), \(0<z<40\) mm) and (**d**) \(U_{z}\) profile along the line distancing 15 mm away from the midspan line (\(x=15\), \(0<z<40\) mm). Figure 7: Time-averaged velocity magnitude (\(U_{\rm mag,xz}\)) field in the transverse plane (\(x\)-\(z\) plane) at different Reynolds numbers. here. Here, \(\tau=t/T\) where t represents time and \(T\) is the time of an oscillation period. The position of the PZT fan blade at these phases are shown in Figure 9. For every single phase, 200 pairs of images are used for the ensemble averaging. Figure 10 shows the ensemble-averaged result of \(x\)-component of vorticity (\(\omega_{x}\)) generated in one cycle at \(Re=550\). The PZT fan blade tip moves from the rightmost position to the leftmost position in the first half period, then moves in the opposite way in the rest of the cycle. A clockwise (CW) vortex and a counterclockwise (CCW) vortex are generated in a single oscillation cycle. Starting from phase \(\tau=0\), the blade starts to move, and a CW vortex starts to generate at the blade tip. The remnant of the CCW vortex from the previous cycle can be observed at this phase. As the blade accelerates and reaches the maximum velocity at phase \(\tau=0.25\), the CW vortex is developed and the strength of the vortex core is intensified. Then the fan blade decelerates and starts to move in the opposite direction (phase \(\tau=0.375\), \(\tau=0.5\)), resulting in the detachment and shedding of the CW vortex from the blade tip. Comparing the vortex generated under different Reynolds numbers, it can be found that the vortex develops in a similar procedure. As for the small Reynolds number case, the vortex basically dies out as the following vortex starts to generate and does not detach from the blade tip. As the Reynolds number increases, the vortex does not die out immediately when the movement of the blade changes direction. Instead, the vortex leaves the blade tip and propagates. In the case shown in Figure 10 with a relatively large Reynolds number (\(Re=550\)), the vortex splits at the point when the movement of the tip starts to change direction (\(\tau=0\), \(\tau=0.5\)), then the remnant vortex propagates to the downstream with the flow. During an oscillation cycle, the intensity of the CW and CCW vortex changes. Figure 11 shows the variation of the maximum \(y\)-component CW and CCW vorticity during a cycle. For different cases, the tendency of the maximum \(\omega_{y}\) is basically the same. The CW vortex reaches the maximum intensity around phase \(\tau=0.25\) and the CCW vortex reaches the maximum intensity around phase \(\tau=0.75\) for all cases. With the increase in the Reynolds number, the maximum intensity for both the CW and CCW vortexes are enhanced at different phases. In order to visualize the evolution of the vorticity, the 3D vortex structure generated by the vibrating blade is illustrated. We adopted a similar strategy used in previous studies [15; 16; 17] to construct the vortex structure. The concept is described in detail in Section 2 of this paper. The 2D PIV transverse slice at different positions is combined. In Figure 12, the iso-surface of \(y\)-component of vorticity is used to visualize the vortex structure generated during a full cycle of the fan blade vibration at \(Re=550\). Because of the similarity of the CW and CCW vortex generated in a full vibration cycle, here we focus on the temporary evolution of the CW vortex structure.