# Study on the Propulsion Characteristics of a Flapping Flat-Plate Pumping Device
## 4 Results
### Impact of Flapping Mode on Water Body
#### 4.1.2 Effect of Motion Mode on the Flow Field  
In the combined heaving and pitching motion of the flapping flat-plate hydrofoil, clockwise vortices are periodically generated and shed from the lower surface of the hydrofoil, while counterclockwise vortices are generated and shed from the upper surface. The alternating vortices are transported downward with the fluid, forming a stable, periodic  
Figure 10: Instantaneous lift coefficient curves for different flapping modes.  
vortex street structure. Due to the continuous motion of the flat-plate hydrofoil, these vortices align along the flow direction, exhibiting a pattern with clockwise vortices below and counterclockwise vortices above. This vortex street, which is opposite to the classic Karman vortex street arrangement, is referred to as a reverse Karman vortex street [35]. It generates a jet effect at the trailing edge of the flat-plate hydrofoil, thereby driving the fluid flow.  
To analyze the vortex structure within the flow field, the vorticity in the \(z\)-direction component is selected as the characteristic quantity, with its calculation formula given as in [36].  
\[\Omega_{\mathrm{z}}=\frac{\partial u_{y}}{\partial x}-\frac{\partial u_{x}}{ \partial y}, \tag{16}\]  
Figure 11 shows the vorticity contour plots for different flapping modes at various frequencies. It can be observed that in both flapping modes, the reverse Karman vortex street gradually forms towards the rear and extends backward within the flow field. As the frequency increases, the vortex street structure becomes more pronounced in both flapping modes, with an enhancement in vorticity and more intense variations.  
It is noteworthy that under simple harmonic motion at low frequency, the wake vortices are relatively symmetric, forming a reverse Karman vortex street aligned along the centerline of the flow field. However, at 3 Hz, the increased flapping speed of the flat-plate hydrofoil causes a more rapid change in the angle of attack, altering the direction of lift exerted by the fluid on the hydrofoil. The resulting upward average lift at the tail leads to an upward shift of the entire vortex street. Under quasi-harmonic motion, the trailing vortices of the flapping flat-plate hydrofoil exhibit a downward deflection at all  
Figure 11: Vorticity contour maps at different flapping frequencies for different flapping modes.  
frequencies, and this downward shift becomes more pronounced as the frequency increases. This is due to the fact that quasi-harmonic flapping is driven by a crank-slider mechanism, which inherently exhibits a quick-return characteristic. During the upward stroke, the flapping hydrofoil is in the working phase, moving more slowly and producing relatively uniform fluid disturbances. During the downward stroke, which serves as the return phase, the flapping flat-plate hydrofoil accelerates, producing greater instantaneous thrust. This increased thrust induces significant asymmetry in the fluid motion, thereby influencing the formation and evolution of the vortex street.