# Study on the Propulsion Characteristics of a Flapping Flat-Plate Pumping Device
## 4 Results
### Impact of Flapping Mode on Water Body
#### 4.1.3 Effect of Motion Mode on Water-Pushing Performance  
To investigate the effect of different flapping modes on water-pushing performance at varying frequencies, the flapping frequency was gradually increased from 0.1 Hz to 1 Hz in 0.1 Hz increments, and then increased by 1 Hz intervals up to a maximum of 3 Hz.  
Figure 12 shows the pumping efficiency of the two flapping modes under different flapping frequencies. There is a significant difference between the two modes: the simple harmonic motion maintains a relatively high pumping efficiency overall. Between 0.1 and 2 Hz, its efficiency remains relatively stable, reaching a maximum of 21.5% at 0.7 Hz, followed by a notable decline between 2 Hz and 3 Hz. As shown in Figure 10, at lower frequencies, the reverse Karman vortex street generated by simple harmonic motion sheds periodically and consistently drives flow rearward. However, at higher flapping frequencies, the flapping flat-plate hydrofoil experiences greater acceleration, and the asymmetry of the instantaneous lift becomes more pronounced, resulting in an upward deflection of the trailing vortices.  
In contrast, the pumping efficiency of the quasi-harmonic motion fluctuates significantly across different frequencies. As shown in the Figure 12, it reaches a peak value of 17.8% at a flapping frequency of 0.6 Hz, followed by a sharp decline, and remains low beyond 1 Hz. At the peak pumping efficiency of the quasi-harmonic motion, the simple harmonic motion still demonstrates a 20.2% higher efficiency. Based on Figure 11, at lower frequencies, the trailing vortex structure under quasi-harmonic motion is relatively stable. Due to the quick-return characteristic of the crank-slider mechanism, the downward stroke of the flapping flat-plate hydrofoil is faster than the upward stroke, and stronger vortices form on the upper side, guiding the wake downward.  
At flapping frequencies between 0.1 Hz and 0.5 Hz, the trailing vortices are weak, resulting in low energy transfer efficiency in the wake, reduced fluid thrust, and consequently lower pumping efficiency. At 0.6 Hz, a stable wake structure is formed, and the  
Figure 12: Pumping efficiency curves for different flapping modes.  
reverse Karman vortex street exhibits minimal downward deflection. The wake momentum effectively drives the water, and the energy utilization of the trailing vortices reaches its optimum. Between 0.7 Hz and 1 Hz, the downward deflection of the trailing vortices becomes more pronounced, reducing wake stability and increasing momentum loss, which results in lower pumping efficiency. As the flapping frequency increases from 1 Hz to 3 Hz, the flapping velocity of the flat-plate hydrofoil rises, enhancing the asymmetry of vortex shedding. Consequently, the wake is directed more downward rather than rearward, leading to a further decrease in pumping efficiency.  
Figure 13 shows the variation of average flow rate and average head with flapping frequency for different motion modes.  
It can be observed that at low frequencies, the flow rate of the flapping flat-plate hydrofoil is approximately proportional to the flapping frequency, while the head is roughly proportional to the square of the frequency. At higher frequencies, both simple harmonic and quasi-harmonic motions are affected by vortex deflection, which in turn impacts the flow rate and head.