# Propulsion Performance of the Full-Active Flapping Foil in Time-Varying Freestream
## 3 Results and Discussions
### The Propulsion Performance of Flapping Foil at Uniform Freestream  
The amplitude and frequency of the heaving and pitching motions are the main factors affecting the hydrodynamic performance of the full-active flapping foil at uniform freestream [9; 29; 37]. We first illustrate the contour of the mean propulsion power coefficient of the flapping foil in the parameter space of reduced frequency and pitching amplitude, as shown in Figure 4. We note that the reduced frequency and pitching amplitude have a significant effect on the mean propulsion power coefficient. Further, we observe that the effect of the pitching amplitude on the propulsion performance is related to the reduced frequency. When \(f^{*}>0.136\), the mean propulsion power coefficient of the flapping foil increases first and then decreases with the variation of the pitching amplitude. The optimal pitching amplitude appears at the middle of the pitching amplitude range, where the mean propulsion power coefficient of the flapping foil reaches the maximum value. When \(f^{*}=0.34\) and \(f^{*}=0.68\) \begin{table}
\begin{tabular}{c c} \hline \hline Parameters & Range \\ \hline Chord length, \(c\) & 0.17 m \\ Hydrofoil shape & 2D NACA0012 \\ Hydrofoil thickness, \(d\) & 0.02 m \\ Constant flow velocity, \(U_{\infty}\) & 0.25 m/s \\ Heaving amplitude, \(y_{0}\) & 1\(c\) \\ Pitching amplitude, \(\theta_{0}\) & 10–80\({}^{\circ}\) \\ Reduced frequency, \(f^{*}\) & 0.1–0.68 \\ Oscillating amplitude,\(\gamma\) & 0–0.15 \\ Oscillating frequency coefficient, \(\tau\) & 0–1 \\ Simulation Period, \(t\) & 10–20T \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main characteristics for computations. Figure 3: Comparison of results by the present study and [PERSON] and Dumas for \(y_{0}=1c\), \(\theta_{0}=70^{\circ}\), \(f^{*}=0.14\), (**a**) thrust coefficient and (**b**) lift coefficient. the maximum values of the mean propulsion power coefficients are 0.51 and 2.32, respectively, and the corresponding optimal pitching amplitudes are both \(30^{\lx@math@degree}\). According to Figure 4, we further notice that with the reduction in the reduced frequency, the optimal pitching amplitude of the flapping foil is decreased gradually. When \(f^{*}=0.23\), the optimal pitching amplitude is \(20^{\lx@math@degree}\), and when \(f^{*}=0.11\), the optimal pitching amplitude is reduced to \(10^{\lx@math@degree}\). Furthermore, we realize that when \(f^{*}\leq 0.11\), the mean propulsion power coefficient of the flapping foil decreases monotonically with the variation of the pitching amplitude. When \(f^{*}=0.068\), the mean propulsion power coefficients corresponding to \(\theta_{0}=10^{\lx@math@degree}\) and \(\theta_{0}=80^{\lx@math@degree}\) are 0.027 and 0.32, respectively. The contour of the propulsion efficiency of the full-active flapping foil on the parametric space of reduced frequency and pitching amplitude is illustrated in Figure 5. The dark blue region of the Figure 5 indicates that the mean propulsion power coefficient and propulsion efficiency in this range are less than zero. According to Figure 5, we reveal that with the variation of the pitching amplitude, the propulsion efficiency decreases monotonically. When \(f^{*}=0.68\), the propulsion efficiencies of the flapping foil corresponding to \(\theta_{0}=10^{\lx@math@degree}\) and \(\theta_{0}=60^{\lx@math@degree}\) are 48.1% and 14.1%, respectively. When \(f^{*}=0.23\), the propulsion efficiencies of the flapping foil corresponding to \(\theta_{0}=10^{\lx@math@degree}\) and \(\theta_{0}=50^{\lx@math@degree}\) are 4.7% and 0.5%, respectively. According to Figure 5, we also note that with the variation of the reduced frequency, the propulsion efficiency increases monotonically. When \(\theta_{0}=10^{\lx@math@degree}\), the propulsion efficiencies of the flapping foil corresponding to \(f^{*}=0.27\) and \(f^{*}=0.45\) are 6.4% and 19.3%, respectively.