# Unsteady Flow Characteristics of an Oscillating Piezoelectric Fan Blade at High Reynolds Numbers
## 3 Results
### Turbulence Characteristics  
The results of instantaneous PIV measurement in previous research [16; 18] and the ensemble-averaged result in Section 3.2 suggested that a strong turbulence effect exists in the flow field produced by the vibrating PZT fan blade. However, the turbulence characteristics of the flow field are seldom discussed in previous studies. To quantify the turbulence effect, we calculated the turbulence intensity using the PIV data, and the \(TI_{xx}\) is defined as:  
\[TI_{xz}\ =\ \frac{\overline{u_{x}^{\prime\,2}}+\overline{u_{z}^{\prime\,2}} ^{2}}{\overline{u_{x}^{\prime\,2}}+\overline{u_{z}^{\prime\,2}}^{2}} \tag{4}\]  
where \(u_{x}^{\prime}\) and \(u_{z}^{\prime}\) are the fluctuating parts of the \(x\) and \(z\)-component of the velocity in the transverse plane respectively; \(\overline{u_{x}}\) and \(\overline{u_{z}}\) are the \(x\) and \(z\)-component of ensemble-averaged velocity at this phase. Similarly, for the longitudinal plane, \(TI_{yz}\) can be calculated as:  
\[TI_{yz}\ =\ \frac{\overline{u_{y}^{\prime\,2}}+\overline{u_{z}^{\prime\,2}} ^{2}}{\overline{u_{y}}^{2}+\overline{u_{z}}^{2}} \tag{5}\]  
Figure 15a-c shows the result of turbulence intensity at different phases in the longitudinal plane for different Reynolds numbers. In order to better visualize the turbulence intensity pattern at a lower Reynolds number, we apply different color scales for different cases. For the lowest Reynolds number \(Re\) = 140, the maximum \(TI_{xy}\) is lower than 0.1 for all phases, indicating that the effect of turbulence on the flow field at a low Reynolds number is insignificant and negligible. For the case of \(Re\) = 347, the turbulence intensity is enhanced around the area of the blade trailing edge and two side edges. The turbulence intensity around these areas for this case is in the range of \(0.1<TI_{yz}<0.5\). When the Reynolds number increases to \(Re\) = 550, the turbulence is greatly enhanced at the trailing edge along the streamwise direction. In the area where the unsteady vortex is generated and shed, the turbulence intensity is increased to \(TI_{yz}\) = 1 and even above. Figure 15d quantifies the maximum turbulence intensity for each phase at different Reynolds numbers. At \(Re\) = 140, the maximum turbulence intensity corresponding to each phase is relatively low. When the Reynolds number reaches 550, the large oscillation amplitude greatly disturbed the flow field. The turbulence intensity is relatively small at \(\tau\) = 0 and \(\tau\) = 0.5, where the blade tip velocity is zero. It gradually increases with the growth of the vortices generated by the blade oscillation. To evaluate the area affected by the turbulence, we calculated the area encompassed by the contour line of \(TI_{yz}\) = 0.05 at different phases as \(A_{\text{tur},yz}\). As can be seen in Figure 16, \(A_{\text{tur},yz}\) is close to 0 at a low Reynolds number (\(Re\) = 140). As the Reynolds number increases to \(Re\) = 349, the \(A_{\text{tur},y}\) is alleviated, ranging from 80 mm\({}^{2}\) to 170 mm\({}^{2}\). At \(Re\) = 550, the \(A_{\text{tur},y}\) is further increased by almost 2 to 3 times. Figure 17a illustrates the maximum turbulence intensity \(TI_{xz,\text{max}}\) at different phases in the transverse plane. Similar to the longitudinal plane, the maximum turbulence intensity is close to 0 for all phases when \(Re\) = 140. When the Reynolds number increases to 349, the \(TI_{xz,\text{max}}\) fluctuates between the 1 to 2, and the range is wider comparing to the longitudinal plane. At \(Re\) = 550, the \(TI_{xz,\text{max}}\) varies from 6 to 14. From Figure 17b, the area affected by turbulence at different phases extended as the Reynolds number increases. Comparing to the longitudinal plane, the \(A_{\text{tur},xz}\) is larger at the same Reynolds number, which is due to the stronger vorticity exhibited in the transverse plane.