## 3 Results and Discussion
### Effect of the Flapping Motion on the Q-Criterion and Velocity Distributions  
The instantaneous (at t = 1 s) and time-averaged Q-criterion and velocity distributions with streamlines of the bare airfoil model (baseline) are presented in Figure 2. The quasi-steady-state of the simulation was reached by checking the periodicity of the flow field. The vortices were shed evenly in the wake, with an identical magnitude of the Q-criterion for each pair, indicating typical Karman vortex street characteristics. The time-averaged Q-criterion shows very weak vortices downstream in the wake. It implies that the rotational direction of two vortices for each pair was opposite, and thus, after averaging, the opposite vorticities counteracts the magnitude of the Q-criterion. A separation bubble was predicted around the middle section on the upper surface of the airfoil, as shown in the instantaneous velocity distribution panel in Figure 2. In the time-averaged flow field, the detachment and reattachment point for the separation bubble can be clearly identified by the streamlines surrounding the circulation region. The separation bubble has a length of about 25% of the chord. In fact, the shedding vortices downstream of the airfoil were initiated by the separation bubble. The Q-criterion (around the airfoil and in the wake) and velocity distributions for the airfoil model with the fringe length of 8% of the chord at four different \(f_{e}\) are presented in Figures 3-5. The large-scale vortices initiated over the upper surface of the airfoil were broken up into smaller scale vortices compared to the shed vortices of the bare airfoil model. The flapping motion at the flapping frequency of 80 Hz creates an irregular pattern of vortices compared to the cases with higher frequencies (110-170 Hz), as shown in Figure 3. The time-averaged Q-criterion distributions demonstrate an accelerated decay of vortices by the extended fringe with a low flapping frequency (80 Hz), resulting from the unstructured small-scale vortices being shed irregularly. This observation might be induced by the velocity \"offset effect\" between induced vortices and the shed vortices from upstream. With the increased flapping frequency, the vortices decayed slowly. This might be attributed to the fact that the contribution from the flapping fringe to the overall swirling strength becomes more dominant with the increased flapping frequency. In the downstream wake, the airfoil model with \(f_{e}\) = 110, 140, or 170 Hz shows evenly shed vortices, as shown in Figure 4. The increased flapping frequency seems to shorten the gap between each pair of vortices, which is directly correlated to the flapping frequency of the fringe. From the comparison of the four flapping frequencies, it is suggested that when the flapping frequency \(f_{e}\) is 60 Hz (40%) lower than \(f\) (natural vortex shedding frequency of the bare airfoil model, 140 Hz), the regular vortex shedding is interfered with, and results in an irregular vortex distribution with non-uniform scales and shorter gaps between each pair. As can be observed, the flapping trailing edge fringe with \(f_{e}\) = 80 and 100 Hz alters the coherent structure of the large-scale vortices in the wake, and at the same time, reduces the vorticity in the flow field (Figure 4). However, it is difficult to differentiate the influence of the fringe tip position (fringe tip at the top or bottom positions) on the distribution of the Q-criterion. Thus, the instantaneous and time-averaged flow fields around the airfoil were plotted to illustrate the effects of the flapping fringe on the flow field, as shown in Figure 5. The flapping motion induces movement of the separation bubble towards the leading edge of the airfoil for all cases (Figure 5).