# Study on the Hydrodynamic Performance of Typical Underwater Bionic Foils with Spanwise Flexibility
## 3 Results and Discussion
### Flexible Foil  
This section focuses on the influences of spanwise flexibility. Figure 12 shows the performance curves versus flexibility phase difference of the NACA foil. The case with \(\gamma=0^{\circ}\) stands for a rigid NACA foil. Additionally, an elliptic curve, a hyperbolic curve and a parabolic curve are used as curve functions respectively. The figure shows that spanwise flexibility can enhance the propulsive efficiency slightly within a certain range, but the mean thrust force coefficients can achieve higher values when the flexibility phase difference is positive. Moreover, the elliptic curve is more conducive to enhancement of the propulsive efficiency, but less conducive to enhancement of the mean thrust force when the phase difference is positive. And there is little difference between the performance of the hyperbolic curve and the parabolic curve. In addition, the NACA foil can achieve both higher mean thrust force and propulsive efficiency when the flexibility phase difference is between \(10^{\circ}\) and \(20^{\circ}\). Figure 13 shows the three-dimensional vortex pattern in the wake of the NACA foil for typical flexibility phase differences for the elliptic curve. The figure shows that the vortex rings are stretched and the volume is larger when \(\gamma=20^{\circ}\) compared to \(\gamma=-20^{\circ}\). A stronger jet stream can be generated between the vortex ranks when \(\gamma=20^{\circ}\) and, in turn, the reaction force acting on the foil is enhanced accordingly. As a result, the propulsion performance is enhanced. This result indicates that spanwise flexibility can influence the shedding of vortex rings, further influence the arrangements of vortex rings and eventually lead to a change of propulsion performance. Furthermore, positive values of flexibility phase difference are more conducive to enhancing the propulsion performance than are negative values. The geometric parameters and motion parameters of the NACA foil are assigned as follows: \(h=0.05\) m, \(L=0.1\) m, \(B=0.3\) m, \(\alpha_{max}=25^{\circ}\), \(St=0.3\) and \(Re=20000\).  
Figure 11: Three-dimensional vortex patterns in the wake: (**a**) NACA foil; (**b**) crescent-shaped foil.  
Figure 10: Forces and torque curves: (**a**) NACA foil; (**b**) crescent-shaped foil.  
Figure 14 shows the performance curves versus flexibility phase difference of the crescent-shaped foil. The elliptic curve is more conducive to enhancing the propulsive efficiency, which is similar to the result for the NACA foil and the curves of the mean thrust force coefficients have similar trends as those of the NACA foil. In contrast, the crescent-shaped foil can maintain a relatively high propulsive efficiency value while achieving a larger mean thrust force within a relatively large range of flexibility phase difference. Natural caudal fins of fish usually have a shape similar to that of the model when the flexibility phase difference is positive. The results indicate that reasonable spanwise flexibility can enhance the propulsion performance of a crescent-shaped foil significantly. As shown in Figure 15, the vortex rings in the wake of the crescent-shaped foil under typical flexibility phase differences have forms similar to those of the NACA foil, so further analysis is not described here. The geometric parameters and motion parameters of the crescent-shaped foil are assigned as follows: \(h\) = 0.06 m, \(L\) = 0.15 m, \(\alpha_{max}\) = 25\({}^{\circ}\), \(St\) = 0.3 and \(Re\) = 45,000.  
Figure 14: Performance curves versus phase difference of the crescent-shaped foil: (**a**) mean thrust force coefficients; (**b**) propulsive efficiency.  
Figure 12: Performance curves versus phase difference of the NACA foil: (**a**) mean thrust force coefficients; (**b**) propulsive efficiency.  
Figure 13: Three-dimensional vortex pattern in the wake of the NACA foil: (**a**) \(\gamma\) = 20\({}^{\circ}\); (**b**) \(\gamma\) = \(-\)20\({}^{\circ}\).