# Propulsion Performance of the Full-Active Flapping Foil in Time-Varying Freestream
## 4 Conclusions  
In this paper, the propulsion performance and hydrodynamic characteristics of the full-active flapping foil in the time-varying freestream are systematically investigated via two-dimensional URANS and the finite volume method. We define the time-varying freestream as a superposition of steady flow and sinusoidal pulsating flow. Through numerical simulation results, we summarize several conclusions as follows: We first analyze the propulsion performance of the flapping foil at uniform freestream, and note that when the reduced frequency is small, the smaller pitching amplitude can obtain the higher propulsion performance of the full-active flapping foil. When the reduced frequency is large, the mean propulsion power coefficient increases first and then decreases with the variation of the pitching amplitude. We also reveal that with the increase in the pitching amplitude, the propulsion efficiency of the foil decreases monotonically. Then, we examine the influence of the time-varying freestream on the propulsion performance of the flapping foil for various oscillating amplitude and oscillating frequency coefficient, and find that the influence of the time-varying flow is time dependent. For one period, we notice that the oscillating amplitude does not affect the tendency of the time-varying flow on the propulsion performance of the flapping foil. With the increase in the oscillating amplitude, the variation amplitude of the mean propulsion power coefficient caused by the time-varying flow increases proportionally. We reveal that the effect of the time-varying flow on the flapping foil depends on the range of the oscillating frequency coefficient. 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. In particular, when the oscillation frequency of the time-varying flow is an integer multiple of the reduced frequency, the time-varying flow has almost no effect on the flapping foil. Otherwise, we also note that the effect of the time-varying flow is related to the motion parameters of the flapping foil. The larger the motion parameters, the more significant the impact of propulsion performance of the flapping foil. For multiple periods, we note that the time-varying freestream has little effect on the propulsion performance of the full-active flapping foil at different pitching amplitude and reduced frequency. In conclusion, we reveal that the time-varying incoming flow has little effect on the flapping propulsion performance for multiple periods. Therefore, we can simplify the time-varying flow to a steady flow field to a certain extent for numerical simulation. Meanwhile, we also realize that only the small oscillating amplitude of the time-varying flow is considered in this paper. The impact of large oscillating amplitude has not been discussed. Otherwise, the hydrological conditions in the marine environment are more complicated; their impact on the propulsion performance of the full-active flapping foil cannot be predicted accurately. Figure 14: The pressure distribution of the flapping foil at different moments for \(f^{*}=0.68\), \(\theta_{0}=60^{\circ}\), \(\tau=0.8\). **Author Contributions:** Conceptualization, Z.Q. ; formal analysis, Z.Q. and L.J. ; funding acquisition, L.J. and J.Z; investigation, Z.Q., Y.Q. and J.S. ; methodology, Z.Q. ; validation, Z.Q. and L.J. ; writing--original draft, Z.Q. ; writing--review and editing, J.Z. All authors have read and agreed to the published version of the manuscript. **Funding:** This study was funded by the Natural Science Foundation of Tianjin (grant number 18 JCYBJC24000), the Key Laboratory of Submarine Geosciences, MNR (grant number KLSG1907) and the National Key Research and Development Project (grant number 2017 YFC0305900). **Acknowledgments:** The authors would like to thank the anonymous reviewers for their constructive comments and suggestions.