# Study on the Propulsion Characteristics of a Flapping Flat-Plate Pumping Device
## 2 Physical Model
### Hydrodynamic Performance Parameters  
To evaluate the performance of the flapping flat-plate hydrofoil, parameters such as thrust coefficient, lift coefficient, flapping hydrofoil efficiency, pumping efficiency, and head are quantified. During the study of its motion, the Strouhal number \(St\) is defined as \(St=2 fh_{max}/\overline{U}\), where \(\overline{U}\) is the mean flow velocity of the steady flow field and \(f\) is the flapping frequency [25].  
The instantaneous thrust coefficient \(C_{T}\) and instantaneous lift coefficient \(C_{L}\) of the flapping flat-plate hydrofoil are given as follows [26]:  
\[C_{T}=\frac{F(t)}{\frac{1}{2}\rho\overline{U}^{2}A}, \tag{3}\]  
\[C_{L}=\frac{L(t)}{\frac{1}{2}\rho\overline{U}^{2}A}, \tag{4}\]  
where \(F(t)\) denotes the instantaneous thrust generated by the flapping flat-plate hydrofoil; \(L(t)\) represents the instantaneous lift; \(\rho\) is the fluid density; and \(A\) is the projected area of the flapping flat-plate hydrofoil.  
The mean thrust coefficient \(C_{\overline{T}}\) and mean lift coefficient \(C_{\overline{L}}\) of the flapping flat-plate hydrofoil are defined as follows:  
\[C_{\overline{T}}=\frac{1}{\overline{T}}\int_{t}^{t+T}C_{\overline{T}}(t)dt, \tag{5}\]  
\[C_{\overline{L}}=\frac{1}{\overline{T}}\int_{t}^{t+T}C_{L}(t)dt, \tag{6}\]  
where \(T\) denotes the motion period.  
The formula for the average input power \(\overline{P_{a}}\) of the flapping flat-plate hydrofoil is as follows:  
\[\overline{P_{a}}=\frac{1}{\overline{T}}\left(\int_{0}^{T}L(t)y^{\prime}(t)dt+ \int_{0}^{T}M(t)\theta^{\prime}(t)dt\right), \tag{7}\]  
where \(y^{\prime}(t)\) represents the heaving velocity of the flapping flat-plate hydrofoil, \(\theta^{\prime}(t)\) denotes the pitching angular velocity, and \(M(t)\) is the instantaneous moment of the flapping flat-plate hydrofoil.  
Figure 5: Schematic diagram of flow channel outlet structure.  
To further investigate the propulsion performance of the flapping hydrofoil, parameters such as propulsion efficiency, flow rate, head, and pumping efficiency are introduced.  
The propulsion efficiency \(\eta\) of the flapping flat-plate hydrofoil is calculated as follows:  
\[\eta=\frac{\overline{F_{x}U}}{\overline{P_{a}}}, \tag{8}\]  
where \(\overline{F_{x}}\) denotes the mean thrust of the hydrofoil.  
The formula for the average outlet flow rate \(\overline{Q}\) after stabilization is as follows:  
\[\overline{Q}=\overline{U}WA, \tag{9}\]  
where \(W\) represents the channel width.  
The formula for the head \(H\) generated by the flapping flat-plate hydrofoil is as follows:  
\[H=\frac{\Delta\overline{P}}{\rho g}, \tag{10}\]  
where \(\Delta\overline{P}\) denotes the pressure difference between the average inlet pressure and average outlet pressure of the flow channel.  
To represent realistic ultra-low-head conditions typically found in plain river network areas, especially in narrow tributaries and shallow channels, the head values used in this simulation are maintained within the range of 0.03 m to 0.06 m. This range is consistent with documented field data and engineering references for low-lift pumping systems in similar hydrological contexts.  
All efficiency evaluations and percentage improvements reported in this study are based on this defined operating head range, ensuring their applicability to real-world riverine hydraulic conditions [5,6].  
The pumping efficiency \(\eta_{pe}\) of the flapping flat-plate hydrofoil is calculated as follows:  
\[\eta_{pe}=\frac{\Delta\overline{PQ}}{\overline{P_{a}}}, \tag{11}\]  
The paragraph has been revised as follows: The performance graphs, including flow velocity and vorticity contour plots, are based on numerical simulations conducted using ANSYS Fluent 2024R2. These simulations model the fluid dynamics around the flapping hydrofoil under different operating conditions.