# Propulsion Performance of the Full-Active Flapping Foil in Time-Varying Freestream
## 1 Introduction
In contrast, the influence of non-uniform flow velocity on the flapping foil's energy harvesting performance has been investigated systematically [26; 27; 28]. [PERSON] and [PERSON] [29] study the effect of small amplitude disturbances of freestream on the hydrodynamic performance of flapping foil and point out that the influence on the energy harvesting performance of the flapping foil is almost negligible. [PERSON] [26] and [PERSON] [27] analyze the effect of linear shear flow on the energy harvesting performance of two-dimensional flapping foil. They find that the flow field under a low shear rate can slightly improve the flapping foil's performance, while the energy harvesting efficiency of flapping foil reduces sharply when the shear rate is large enough. [PERSON] [30] examines the effect of time-varying flow velocity on the energy harvesting performance of two-dimensional flapping foil. The study also points out that flapping foil's hydrodynamic performance is greatly affected by the time-varying flow rate in one cycle. When calculating the mean performance in eight periods, the time-varying freestream velocity has little effect on the flapping foil's energy harvesting performance. In addition, other researchers have found that the free surface and waves also have an important impact on the hydrodynamic performance of the flapping foil. [PERSON] [31] investigates the possibility of extracting energy from gravity waves by a two-dimensional flapping foil. He points out that the marine propulsion of the flapping foil can absorb energy form sea waves and improve the propulsion performance when the design parameters are correctly maintained. [PERSON] and [PERSON] [32] analyze the hydrodynamic performance of the flapping foil under free surface and in waves by unsteady boundary element method. They note that significant efficiency of the flapping foil can be obtained under optimal wave conditions. [PERSON] [33] investigates the propulsion of a flapping foil in heading wave through the velocity potential theory and indicates that the frequency difference between the foil and waves has a significant effect on the hydrodynamic performance of the flapping foil. Referring to the non-uniform flow on the energy harvesting performance of flapping foil, we notice that both the small amplitude shear flow and the time-varying flow velocity have little effect. Therefore, we can infer that the influence of the non-uniform flow with a small amplitude on the propulsion performance of the flapping foil should also be weak. To verify the above inference, we analyze the effect of time-varying flow velocity on the flapping foil's propulsion performance through numerical simulation. In this paper, the kinematic model of full-active flapping foil under time-varying flow velocity is established. The finite volume method (FVM) is used to calculate the unsteady Reynolds averaged Navier-Stokes (URANS) by commercial CFD software Fluent version 19.1. We first investigate the propulsion performance of flapping foil in the parameter space of reduced frequency and pitching amplitude at a uniform flow velocity. Then, we study the influence of time-varying flow velocity on the propulsion performance of flapping foil. Moreover, the effect mechanism of time-varying flow velocity on the hydrodynamic performance of flapping foil is explained by analyzing the evolutions of vortex topology, flow velocity, and pressure distribution around the foil.