# Study on the Propulsion Characteristics of a Flapping Flat-Plate Pumping Device
## 1 Introduction  
With accelerating urbanization, many small channels in plain river networks have suffered from eutrophication and poor water quality due to flat terrain and insufficient hydrodynamic conditions [1; 2; 3]. To address this issue, water conservancy infrastructures such as pumping stations and shuice gates are commonly used to enhance hydrodynamics and self-purification [4]. However, in narrow tributaries [5] and under low-lift conditions, this approach is costly and often yields limited effectiveness [6], failing to fully resolve water quality degradation in small rivers. Bionic pumping devices [7; 8] inspired by fish tail flapping offer high efficiency, low noise, and ecological compatibility under low-lift conditions, providing an effective means of enhancing flow in small rivers--especially under shallow water conditions where they demonstrate distinct advantages [9]. This advantage is closely related to the body structure and swimming mechanisms of fish, which enable low-energy, high-efficiency propulsion and have found applications in fields such as communication engineering, tidal energy harvesting, and underwater rescue [10; 11; 12; 13]. In recent years, with the advancement of hydrofoil fluid dynamics research, the kinematic parameters and optimization of flapping hydrofoils have emerged as key research topics in this field. [PERSON] et al. [14] conducted a numerical study on a hydrofoil with an aspect ratio of two under low-Reynolds-number conditions. They found that the thrust coefficient reached its maximum--an increase of 137.12%--when the phase difference between spanwise bending and pitching was 150\({}^{\circ}\), while propulsion efficiency peaked at a phase difference of 270\({}^{\circ}\), exceeding that of a rigid hydrofoil by up to 110.87%. In the study by [PERSON] et al. [15], the bidirectional fluid--structure interaction method was investigated for optimizing the stiffness and flapping modes of flexible hydrofoils. The study employed numerical simulations to analyze the effects of varying stiffness and flapping modes on the hydrodynamic performance of hydrofoils. This provides an optimization approach for ecological water purification under low-head conditions. [PERSON] et al. [16] performed numerical simulations of unsteady hydrodynamic characteristics during flapping hydrofoil propulsion using the Realizable k-\(\varepsilon\) turbulence model combined with dynamic mesh techniques. The study focused on analyzing how kinematic parameters such as flapping frequency, heave amplitude, and pitch amplitude affect thrust and propulsion efficiency. [PERSON] et al. [17] developed a two-degree-of-freedom model of a two-dimensional rigid flapping hydrofoil to compare four typical flapping modes. Their analysis of average thrust, lift, and flow field structure revealed that the mode with the largest initial pitch angle and synchronized heave and pitch motions achieved the best propulsion performance. [PERSON] et al. [18], using the RANS method and fluid-solid interaction modules, investigated the effects of various non-sinusoidal flapping modes on hydrofoil thrust and efficiency. They found that when the maximum effective angle of attack remained below 20\({}^{\circ}\), both average thrust coefficient and propulsion efficiency improved, with maximum efficiency gains reaching up to 40%. [PERSON] et al. [19] built a bidirectional fluid--structure interaction model to study how different flapping modes and structural flexibility affect the propulsion performance of three-dimensional bionic hydrofoils. The results showed that asymmetric flapping achieved higher peak thrust under similar power input, while moderate flexibility enhanced both propulsion efficiency and thrust output. [PERSON] et al. [19] aimed to improve flow conditions in small river channels and conducted numerical and experimental studies on three different flapping modes.