## 3 Results and Discussion
### Effect of the Flapping Motion on the Q-Criterion and Velocity Distributions
The existence of the separation bubble caused fluctuated pressure forces on the upper surface, reflecting the unstable pressure distribution over the airfoil model, as shown in Figure 6. Higher flapping frequencies tend to increase the velocity over the top surface of the airfoil, and, as a consequence, it results in a decrease in the aerodynamic pressure. Subsequently, the pressure drop could lead to an increase in the lift coefficient, and a decrease in drag coefficient. It is indicated that the pressure acting on the top surface tends to be small when the flapping fringe tip is at the bottom position, in contrast to that when the fringe tip is at the top position. Figure 2: The simulated instantaneous (time = 1 s) and time-averaged Q-criterion and velocity distributions with streamline of the bare airfoil model. Figure 4: Instantaneous Q-criterion distributions in the wake of the airfoil model with the fringe length of 8% chord at different flapping frequencies at t = 1.5 s.  
Figure 3: Q-criterion distributions around the airfoil model with the fringe length of 8% chord at different flapping frequencies. Left column: instantaneous Q-criterion distributions at the fringe tip moved to the top; Middle column: instantaneous Q-criterion distributions at the fringe tip moved to the bottom; Right column: time-averaged Q-criterion distribution. Figure 5: The velocity distribution around the airfoil model with the fringe length of 8% chord at different flapping frequencies. **Left column**: instantaneous velocity distributions when the fringe tip moved to the top; **Middle column**: instantaneous Q-criterion distributions when the fringe tip moved to the bottom; **Right column**: time-averaged Q-criterion distribution. Figure 2.0 Pressure coefficient distribution over the airfoil model with the fringe length of 8% chord at four flapping frequencies. The pressure coefficient distributions over the airfoil model with the fringe length of 8%C at four flapping frequencies are compared, as shown in Figure 2.1 The instantaneous pressure coefficient distribution shows a significant fluctuation corresponding to the shedding vortex over the upper surface as compared to the averaged pressure distribution over the baseline airfoil model. An obvious phase difference of the pressure oscillations can be observed between the case with the fringe tip at the bottom position and that with the fringe tip at the top position. The phase difference is more complicated in the region of the separation bubble compared to the region downstream of the separation bubble. By comparing the time-averaged pressure distributions, one can observe that higher flapping frequencies tend to move the separation bubble forward (towards the leading ledge). The distance from the leading edge to the take-off point of the separation bubble decreases from 33%C to 24%C as the flapping frequency increases from 80 Hz to 170 Hz. The separation bubble length is about 22%C, 25%C, 20%C, and 20%C for the case of \(f_{e}=80\) Hz, \(f_{e}=110\) Hz, \(f_{e}=140\) Hz, and \(f_{e}=170\) Hz, respectively. Overall, higher flapping frequencies tend to generate a lower time-averaged pressure coefficient over the upper surface, which would result in an increase in the lift coefficient. Figures 7-10 present the instantaneous Q-criterion distribution over the airfoil, the Q-criterion distribution in the wake, the velocity, and the pressure coefficient distributions over the airfoil model, respectively, with the fringe length of 10% at different flapping frequencies. Figures 11-14 present the instantaneous Q-criterion distribution over the airfoil, the Q-criterion in the wake, the velocity, and the pressure coefficient distributions over the airfoil model, respectively, with the fringe length of 12% chord at various flapping frequencies. Similarly, the vortex shedding characteristics are similar to the airfoil model with the fringe length of 8% chord.