# Study on the Hydrodynamic Performance of Typical Underwater Bionic Foils with Spanwise Flexibility
## 3 Results and Discussion  
### Rigid Foil  
The geometric parameters and motion parameters of the thunniform bio-inspired model are assigned as follows: \(L_{f}\) = 0.16 m, \(\lambda\) = 1.0\(L_{f}\), \(\alpha_{max}\) = 20\({}^{\circ}\), \(h\) = 0.5\(L\) and \(f\) = 1.25 Hz. Here, \(L_{f}\) is the length of the thunniform bio-inspired model and \(L\) is the chord length of the caudal fin. The thunniform bio-inspired model moves from the static state. Figure 8 shows the velocity time-histories of the thunniform bio-inspired model. Here, \(u\)_-fish_ is the velocity along the propulsion direction and \(v\)_-fish_ is the velocity transverse to the propulsion direction. Initially, \(u\)_-fish_ is zero and then increases gradually, until it finally reaches dynamic stability. Furthermore, \(u\)_-fish_ and \(v\)_-fish_ are not constant, but oscillate around a constant value. The process for reaching dynamic force balance is gradual.  
Figure 9 shows the vortex pattern in the wake. There are vortex rings behind the thunniform bio-inspired model. The vortex ring consists of a train of inverted hairpin-like vortices braided together such that the legs of each vortex are attached to the head of the preceding vortex. Specifically, Figure 9b shows the vorticity magnitude and velocity vector in two-dimensional space. The vortices shed from the thunniform bio-inspired model are arranged in two ranks in the wake, which can generate a jet stream. As a result, the thunniform bio-inspired model obtains a corresponding reaction force. The arrangement and shape of vortex pattern are similar with those found in nature and agree well with other research [15]. Hence, the numerical model and solution approach adopted in this paper are reliable.  
The geometric parameters and motion parameters of the NACA foil are assigned as follows: \(h=L=0.055\) m, \(B=0.3575\) m, \(\alpha_{max}=25^{\circ}\), \(St=0.3\) and \(Re=11,\!000\). As shown in Figure 10a, the forces and torque acting on the NACA foil fluctuate periodically. In one cycle time, the resultant force of the transverse force in a motion period is zero and the thrust force is positive. That is, the NACA foil can obtain forward thrust force and eliminate transverse force. To verify validity and feasibility of the method, the geometry parameters and motion parameters are set to be consistent with those of other experimental study [19]. Furthermore, the mean \(C_{x}\) is 0.58, the mean \(C_{y}\) is 0.0065 and the thrust efficiency is about 47.0%. Comparing with the experimental study, the curves of force coefficients have similar trends and the performance parameters compare well with it (\(C_{x}=0.604\), \(C_{y}=0.0082\), \(\eta=49.1\%\)). Moreover, the forces and torque curves of the crescent-shaped foil are shown in Figure 10b. The curves are similar to those of the NACA foil. Additionally, three-dimensional vortex patterns in the wake of the NACA foil and crescent-shaped foil are shown in Figure 11. The vortex pattern has similar arrangement and shape with that of thunniform bio-inspired model, which also compare well with other researches [32; 33; 34]. Continuous vortex rings can be found in the wake, but the vortex rings behind the crescent-shaped foil are more complex because of the more complex geometry. 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 8: Velocity time-histories of the thunniform bio-inspired mode.  
Figure 9: Vortex pattern in the wake of the thunniform bio-inspired mode: (**a**) three-dimensional case; (**b**) two-dimensional case.