# Propulsion Performance of the Full-Active Flapping Foil in Time-Varying Freestream
## 3 Results and Discussions
### The Influence of the Time Varying Freestream
#### 3.2.1 The Effect of the Time-Varying Flow in One Period  
According to Figure 4, we choose different parameters of reduced frequency (\(f^{*}=0.34/0.68\)) and pitching amplitude (\(\theta_{0}=10^{\lx@math@degree}/30^{\lx@math@degree}/60^{\lx@math@degree}\)) to study the influence of oscillating amplitude \(\gamma\) and oscillating frequency coefficient \(\tau\) of the time-varying flow on the propulsion performance of the full-active flapping foil. The evolution of the mean propulsion power coefficient of the flapping foil with different oscillating amplitude and oscillating frequency coefficient is shown in Figure 6. We notice that the oscillating frequency coefficient of the time-varying flow has a significant effect on the mean propulsion power coefficient. In particular, the effect of the oscillating frequency coefficient on the flapping foil is related to the pitching amplitude and reduced frequency of the flapping foil. When the motion parameters are large, the time-varying flow has a more significant effect on the flapping performance of the flapping foil. When \(f^{*}=0.68\), the maximum change rate of the mean propulsion power coefficient caused by the oscillating frequency coefficient \(\tau\) for \(\theta_{0}=10^{\lx@math@degree}\) and \(\theta_{0}=60^{\lx@math@degree}\) is 9.8% and 13%, respectively. When \(f^{*}=0.34\), the maximum change rate of the mean propulsion power coefficient caused by the oscillating frequency coefficient \(\tau\) for \(\theta_{0}=10^{\lx@math@degree}\) and \(\theta_{0}=30^{\lx@math@degree}\) is 4.3% and 5.3%, respectively. Besides, we also found that when the vibration coefficient, the impact of the oscillating frequency coefficient on the propulsion performance of active flapping is significantly reduced. We further notice that when the  
Figure 5: The propulsion efficiency contour on the parametric space of reduced frequency and pitching amplitude.  
oscillating frequency coefficient \(\tau>0.5\), the impact of the time-varying flow on the mean propulsion performance of the flapping foil is significantly reduced, as shown in Figure 6. [PERSON] [30] and [PERSON] [35] have obtained similar conclusions when studying the effect of time-varying freestream on the energy harvesting performance of the full-active flapping foil.  
Referring to Figure 6, we also notice that the oscillating amplitude of the time-varying flow also affects the hydrodynamic performance of the full-active flapping foil. With the increase in the oscillating amplitude, the variation amplitude of the mean propulsion power coefficient of the flapping foil caused by the oscillating frequency coefficient increases proportionally, but the variation tendency does not change. When \(f^{*}=0.68\) and \(\theta_{0}=10^{\circ}\), the maximum change rate of the mean propulsion power coefficient caused by the oscillating frequency coefficient \(\tau\) for \(\gamma=0.05\) and \(\gamma=0.15\) are 1.5% and 5.3%, respectively. Since the oscillating amplitude does not affect the tendency of the time-varying flow on the propulsion performance of the full-active flapping foil, in order to simplify the simulation calculation, the oscillating amplitudes in the subsequent calculations are all fixed values, \(\gamma=0.1\).