# Propulsion Performance of the Full-Active Flapping Foil in Time-Varying Freestream
## 3 Results and Discussions
### The Influence of the Time Varying Freestream
#### 3.2.2 The Effect of the Time-Varying Flow in Different Periods  
To observe the effect of the time-varying flow on the propulsion performance of the flapping foil in different periods, the variation of the mean propulsion power coefficient in different periods and oscillating frequency coefficient is illustrated in Figure 7. When \(f^{*}=0.34\), \(\theta_{0}=10^{\circ}\) and \(\gamma=0.1\), the numerical simulation obtains a stable periodic solution after 10 cycles. Therefore, we choose the data within 10-20 cycles for comparison. According to Figure 7, the oscillating frequency coefficient of the time-varying flow has an apparent regularity for the impact of the mean propulsion power coefficient of the flapping foil. When \(\tau=0.5\) and \(\tau=1.0\), with the variation of the oscillating frequency coefficient and simulation period, the mean propulsion power coefficient is almost unchanged. According to Equation (3), when the reduced frequency is an integer multiple of the oscillating frequency coefficient of the time-varying flow, the average flow velocity of the time-varying flow in one cycle is equal to the constant term \(U_{\infty}\). Therefore, the time-varying freestream has no significant effect on the hydrodynamic performance of the full-active flapping foil. When the relationship between the oscillating frequency  
Figure 6: The evolution of the mean propulsion power coefficient with different oscillating amplitude and oscillating frequency coefficient for one period. coefficient of the time-varying freestream and the reduced frequency is not an integer multiple, with the variation of the simulation period, the oscillating frequency coefficient directly affects the mean propulsion power coefficient of the full-active flapping foil. When \(\tau<0.5\), the mean propulsion power coefficient of the flapping foil fluctuates significantly, and when \(\tau>0.5\), the mean propulsion power coefficient is almost unchanged with the variation of the period, as shown in Figure 7. The influence of the oscillating frequency coefficient on the flapping foil can also be found in Figure 6. The reason of the above phenomenon is that the oscillating frequency coefficient of the time-varying flow affects the average flow velocity in one cycle. Especially when \(\tau<0.5\), the change of the average flow velocity in different periods is more prominent. The average flow velocity leads to differences in the hydrodynamic performance of the flapping foil in different periods. The effect of the time-varying flow on the propulsion performance of the flapping foil for \(f^{*}=0.34\) and \(\theta_{0}=10^{\lx@math@degree}\) is described above. Then, we analyze the influence of the time-varying flow on the flapping foil with the variation of the period under different motion parameters (reduced frequency and pitching amplitude). The evolution of the mean propulsion power coefficient for different reduced frequency and pitching amplitude at \(\tau=0.2\) and \(\tau=0.8\) is shown in Figure 8. We note that with the increase in the pitching amplitude of the flapping foil, the variation amplitude of the mean propulsion power coefficient of the flapping foil caused by the oscillating frequency coefficient increases gradually. When \(f^{*}=0.68\) and \(\tau=0.2\), the maximum change rates of the mean propulsion power coefficient caused by the time-varying freestream for \(\theta_{0}=10^{\lx@math@degree}\),\(20^{\lx@math@degree}\),\(30^{\lx@math@degree}\) and \(60^{\lx@math@degree}\) are 2.3%, 3.7%, 6.8% and 23.6%, respectively. When \(f^{*}=0.68\) and \(\tau=0.8\), the maximum change rates of the mean propulsion power coefficient caused by the time-varying freestream for \(\theta_{0}=10^{\lx@math@degree}\), \(20^{\lx@math@degree}\), \(30^{\lx@math@degree}\) and \(60^{\lx@math@degree}\) are 2%, 2.1%, 4.1% and 21.4%, respectively. Furthermore, we reveal that with the variation of the reduced frequency, the maximum change rates of the mean propulsion power coefficient caused by the time-varying freestream increase gradually.