## 3 Results and Discussion
### Effect of the Flapping Motion on the Q-Criterion and Velocity Distributions
The flapping motion with a relatively lower \(f_{e}\) (<140 Hz) can break the large-scale vortex into small-scale vortices at the trailing edge, and, subsequently, alter the coherent structure of the vortex shedding. The irregularly shed vortices and the increased gap between each pair of vortices for cases with a lower \(f_{e}\) accelerated the vortex decay compared to those with a higher \(f_{e}\). As the flapping frequency goes up to 170 Hz, the shedding frequency of the vortex is dominated by the flapping motion, which results in a regulated pattern of the shedding vortices. The time-averaged Q-criterion distributions show that the resultant time-averaged vortex strength has been elevated by the flapping fringe. It is indicated that the extended flapping fringe did not always facilitate the reduction of vorticity; conversely, it could promote the vortex domination in the wake, its interaction with the blunt trailing edge, and, possibly, the consequent noise emission from the trailing edge of the airfoil. With the increased fringe length, the scale of the vortex became smaller, as shown in Figures 7 and 11. Correspondingly, the overall Q-criterion distribution over the top surface of the airfoil is decreased, as presented in Figures 8 and 12. These alterations in the flow field around the airfoil led to a decrease in the pressure coefficient over the upper surface of the airfoil model, as shown in Figures 10 and 14. Compared with the flapping frequency, the effect of the fringe length is less significant in terms of slight changes in both the vortex strength and the pressure distribution. the swirling strength of vortices shedding in the wake. For this case, the flapping fringe serves more like a propeller tail, such as the tail of a fish, and thus, reduces the drag, and improves the lift. The drag coefficients of the models with \(f_{\epsilon}\) = 80 Hz are slightly lower than that of the baseline model, whereas the lift coefficients are elevated dramatically. An \(f_{\epsilon}\) = 140 Hz, which is the natural shedding frequency for the bare airfoil, did not generate drastically different results. The difference in the lift and drag generation due to the fringe length is the minimum for this case. Figure 2.2 Instantaneous Q-criterion distributions in the wake of the airfoil model with the fringe length of 10% chord at different flapping frequencies at t = 1.5 s.  
Figure 2.3 Q-criterion distributions around the airfoil model with the fringe length of 10% chord at different flapping frequencies. Figure 2.4 Pressure coefficient distribution over the airfoil model with the fringe length of 10% chord at different flapping frequencies. Figure 2.5 The velocity distribution around the airfoil model with the fringe length of 10% chord at different flapping frequencies. Figure 2.6 Q-criterion distributions around the airfoil model with the fringe length of 12% chord at different flapping frequencies. Figure 2.7 Instantaneous Q-criterion distributions in the wake of the airfoil model with the fringe length of 12% chord at different flapping frequencies at t = 1.5 s.  
Figure 2.8 Pressure coefficient distribution over the airfoil model with the fringe length of 12% chord at different flapping frequencies. Figure 2.9 The velocity distribution around the airfoil model with the fringe length of 12% chord at different flapping frequencies. In summary, the effects of the fringe length and flapping frequency on the vortex wake and aerodynamic performance are investigated numerically. It is found that a favorable \(f_{e}\) should be determined according to the nature shedding frequency, \(f\), of the bare airfoil model. The structure of the shedding vortices or their strength would be either maintained or elevated by the vibration of the fringe at a higher \(f_{e}\) (\(>\)\(\prime\)). A higher flapping frequency (\(f_{e}\)) tends to serve as a booster, rather than a disturber, to increase the local velocity at the trailing edge, resulting in the elevation of the swirling strength, and a shrinking of the dimension of vortices.