# Study on the Hydrodynamic Performance of Typical Underwater Bionic Foils with Spanwise Flexibility
## Materials and Methods  
### Geometric Model  
The thunniform mode features high efficiency and high speed. Many fish species that have a singular ability to swim adopt this mode, such as sharks, swordfish and tuna. The fish species utilizing the thunniform mode mainly generate thrust force by swinging their caudal fins. Specifically, the caudal fins achieve thrust force through a motion that contains translational motion following the tail and rotation motion around the tail. Except for the motion of the caudal fin, the motion of the fish body exists mainly in the rear part and front part of the fish body is relatively rigid. Moreover, a crescent-shaped caudal fin is the most common shape among thunniform swimmers. For example, a crescent-shaped caudal fin can provide ninety percent of the thrust force for a tuna, meanwhile pectoral and pelvic fins play roles only in controlling the attitude. A crescent-shaped caudal fin with a high aspect ratio is conducive to the shedding of the vortex rings, which in turn reduces the energy consumption during the shedding process and achieves high propulsive efficiency. In this section, as shown in Figure 1, three-dimensional geometric models are established with the thunniform group as the bionic prototype. Fins that work as auxiliary organs, such as pectoral and pelvic fins, are not taken into account. The geometric model contains three parts: head, trunk and caudal fin. The lengths of these three parts are 0.28 m, 0.4 m and 0.12 m, respectively.  
The cross-sectional shape of the geometric model is shown in Figure 2 and the curve equations are defined as follows:  
\[\begin{array}{rl}z_{1}=&\frac{12}{35}\sqrt{0.28^{2}-x^{2}}\\ z_{2}=&-29.24x^{5}+30.62x^{4}-9.424x^{3}+0.3853x^{2}\\ &-0.06764x^{2}+0.096\\ z_{3}=&-55.36x^{3}+70.42x^{2}-28.35x^{2}+3.64\\ z_{4}=&426.8x^{3}-643.5x^{2}+325.1x^{2}-54.92\\ y_{1}=&\frac{6}{35}\sqrt{0.28^{2}-x^{2}}\\ y_{2}=&-14.62x^{5}+15.31x^{4}-4.712x^{3}+0.1927x^{2}\\ &-0.03382x+0.048\\ y_{3}=&-\frac{1}{20}(x-0.52)\end{array} \tag{1}\]  
Figure 1: Three-dimensional geometric model: (**a**) thunniform bio-inspired mode; (**b**) crescent-shaped foil.  
In the process of geometric modeling, the section shape perpendicular to incoming flow is defined as elliptical surface. In the process of geometric modeling of the crescent-shaped foil, firstly \(z_{3}\) function and \(y_{3}\) function in Equations (1) and (2) are used to construct the incomplete three-dimensional geometry of the foil, after that a curved surface defined by \(z_{4}\) function in Equation (1) is used to cut the geometric model. Finally, the three-dimensional geometric model of the crescent foil which is similar as the caudal fin of thunniform mode is completed.  
In nature, locomotive organs with different shapes exist in the animal kingdom. In particular, the animals that swim or fly in fluid media usually have locomotive organs with similar cross sections along the flow direction. The force acting on locomotive organs can be regarded as the resultant force of force components acting on cross sections. Here, a flapping foil model is adopted to represent the general form of these locomotive organs. The cross sections of the bionic foils adopted in this paper have same shape as the NACA (National Advisory Committee for Aeronautics) airfoil model. As shown in Figure 3, the NACA0013 airfoil is chosen as the cross section, where \(L\) is the chord length, \(B\) is the length along the spanwise direction and \(\mathcal{O}\) is the reference point. The reference point is set a quarter of the chordwise length away from leading edge.  
With this NACA foil as the research object, the results of the study would contribute to revealing the mechanism of biological propulsion. In addition, a propulsion system with a NACA foil can fully absorb the advantages of different bionic prototypes and can be applied to existing propulsion systems.