# Unsteady Flow Characteristics of an Oscillating Piezoelectric Fan Blade at High Reynolds Numbers
## 3 Results  
### Time  
In the current study, the time-averaged flow field is obtained from the PIV measurements in both longitudinal (\(y\)-\(z\)) and transverse (\(x\)-\(z\)) planes. In order to obtain the time-averaged flow field, 2000 pairs of PIV images were captured under the recording frequency, which is equivalent to 20 cycles of PZT fan oscillation. Figure 4 shows the contour of the time-averaged velocity magnitude (\(U_{\text{mag,yz}}\)) in \(y\)-\(z\) plane ranging from \(Re=140\) to 550, which is achieved by increasing the oscillation amplitude of the PZT fan. A slight asymmetricity about the midspan is observed, which might occur due to the material flaw of the fan blade and other disturbances in the testing environment. At a low Reynolds number (\(Re=140\)), the amplitude of the PZT blade is relatively small. The flow field can be characterized into three main regions with large velocity magnitude. They are two regions near side edges and the region at the trailing edge of the PZT fan blade. As the Reynolds number increases (\(Re=241\) or \(349\)), the time-averaged flow pattern changes. The high velocity regions expand and intensify, and the regions near the two side edges of the blade gradually converge with the high velocity region at the trailing edge. As the Reynolds number further increases, the three regions of high velocity observed in previous cases cannot be distinguished, and the airflow driven by the side edges and the trailing edge form a single high velocity region and propagate toward the streamwise direction along the midspan of the fan blade. Based on the observation from Figure 4, the high velocity region is mainly generated at the side edges at the low Reynolds number and at trailing edge at the high Reynolds number. Here, the average flow velocity magnitude at the trailing edge is denoted as \(U_{\text{main}}\) and the average flow velocity magnitude at the two side edges of the blade is denoted as \(U_{\text{side}}\), as shown in Figure 5. The velocity ratio can then be defined as \(R_{\text{U}}=U_{\text{main}}/U_{\text{side}}\). As shown in Figure 4b, the velocity ratio \(R_{\text{U}}\) increases linearly with the Reynolds number. This means the averaged velocity at the trailing edge increases faster than the averaged velocity at the side edges as oscillation amplitude increases. When the \(R_{\text{U}}\) increases to a high level, the flow generated at the side edges is entrained by the flow at the trailing edge,  
Figure 3: PIV measurement slices along the \(y\) axis with 1 mm apart in the transverse plane (\(x\)-\(z\) plane). forming a single high velocity region. An empirical correlation of \(R_{\rm U}\) as a function of \(Re\) can be obtained as:  
\[R_{\rm U}=0.0013\;Re^{1.1} \tag{3}\]  
Figure 6 shows the time-averaged velocity profile along the line of \(y=0\) and \(z=0\) on the longitudinal plane. Along the line \(y=0\) towards the \(+z\) direction (Figure 6b), at higher Reynolds number conditions (\(Re=550\) and \(427\)), the magnitude of velocity decreases monotonically when moving away from the trailing edge. For the case of \(Re=349\), the velocity magnitude decreases rapidly in the first \(10\) mm. Then the slope became flat as the flow from the side of the blade joined the mainstream. Along the \(+y\) direction (Figure 6c), for higher Reynolds number cases (\(Re=550\) and \(427\)), the velocity magnitude decreases monotonically and a turning point can be identified at approximately \(y=12.5\) mm. For the rest of the cases, the velocity decreases rapidly before the corner of the blade is reached and then the velocity increases again as the flow generated at the side edges of the blade comes to effect. From the experiment, the flow field in the longitudinal plane is affected jointly by the side edges and the trailing edge of the PZT fan blade. Figure 4: Time-averaged velocity magnitude (\(U_{\rm magyz}\)) field of the \(y\)-\(z\) plane at different Reynolds numbers.