# Propulsion Performance of the Full-Active Flapping Foil in Time-Varying Freestream
## 3 Results and Discussions
### The Mechanism of the Time Varying Freestream  
In the previous section, we analyze the influence of the time-varying flow on the flapping foil. Then, we study the effects of the time-varying flow on the evolution of instantaneous thrust coefficients, vortex topology, velocity field, and the pressure distribution near the foil, and further explain the effect mechanism of the time-varying flow. Referring to Figure 8, we notice that when \(f^{*}=0.68\) and \(\theta_{0}=60^{\lx@math@degree}\), the oscillating frequency coefficient of the time-varying flow has a significant effect on the mean propulsion performance of the flapping. Therefore, we analyze the evolution of the instantaneous thrust coefficient under these motion parameters. The evolution of the instantaneous thrust coefficient and freestream velocity over three periods is shown in Figure 10. We notice that the instantaneous thrust coefficient has two peaks and troughs within one period. The peaks and troughs are the maximum instantaneous resistance and thrust of the flapping foil. Comparing the evolution of instantaneous thrust coefficient and time-varying flow velocity at different oscillating frequencies of the time-varying flow, we find that there is a correlation between the instantaneous velocity and the variation of the instantaneous thrust coefficient of the flapping foil. When \(t=15.15T\), the instantaneous thrust coefficients for \(\tau=0\), 0.2 and 0.8 are 1.9, 2.15 and 2.25, respectively. The corresponding instantaneous  
Figure 9: The evolution of the mean propulsion power coefficient with the variation of the oscillating frequency coefficient. velocity are 2.5 m/s, 2.65 m/s and 2.87 m/s, respectively. When \(t=17.65T\), the instantaneous thrust coefficients for \(\tau=0\), 0.2 and 0.8 are 1.9, 2.3 and 2.4, respectively. The corresponding instantaneous velocities are 2.5 m/s, 2.65 m/s and 2.87 m/s, respectively. In the vicinity of the peak, the instantaneous resistance of the flapping foil gradually increases with the increase in the instantaneous velocity. When \(\tau~{}=~{}0.2\), the instantaneous thrust coefficients for \(t=15.35T\), \(16.35T\) and \(17.35T\) are -5.41, -5.85 and -5.75, respectively. The corresponding instantaneous velocities are 2.79 m/s, 2.41 m/s and 2.39 m/s, respectively. We can also note that the instantaneous thrust of the flapping foil gradually increases with the increase in the instantaneous velocity in the vicinity of the trough. In order to further explain the mechanism of the influence of time-varying flow on the flapping foil, the three snapshots of the vorticity contours and the corresponding velocity contours for different oscillating frequency coefficient (\(\tau=0\), 0.2 and 0.8) are shown in Figure 11. According to Figure 10, we choose the moment when the instantaneous thrust coefficient reaches the peak. We notice that the vortexes near the foil for the three oscillating frequency coefficient is basically the same, which shows that the time-varying flow does not significantly change the vortex topology of the flapping foil. However, we find differences between the velocity distribution near the foil for different oscillating frequencies. Referring to Figure 10, we note that with the variation of the oscillating frequency coefficient, the instantaneous velocities increases. Comparing Figure 11a-c, we find that as the instantaneous velocity increases, the velocity close to the lower surface of the foil gradually increases. According to the Bernoulli equation, the pressure gradient drops as the velocity increases. Therefore, comparing \(\tau=0.8\) with \(\tau=0\) and \(\tau=0.2\), a more significant negative pressure region on the lower surface of the foil is generated, which leads to an increase in the resistance and reduces the propulsion performance of the flapping foil.