## 2 Study on an Airfoil-Type Oscillator  
An airfoil shape would naturally be thought of as a logical shape for generating a large \(Cl\). However, it does not oscillate unless it is exposed to an appropriate turbulent flow. Furthermore, based on the authors' efforts to mechanically emulate the propulsion mechanism that can be observed in a regularly shaped fish as it swims at very high speeds, it was considered to be a natural selection for the oscillator [10; 11]. On the assumption that \(Cl\) would increase if there were some elasticity in the oscillating wing, a numerical experiment to find the optimum elastic modulus on a symmetric foil was carried out.  
As shown in Figure 2, an NACA0015 model foil, with a shaft located at 0.1 m from the front leading edge is used (the shaft is not considered in the calculation). A reciprocating displacement of \(y\) = 0.001 sin(2\(\pi\)t) m is given to the shaft, and the elastic modulus \(E\) of the foil is varied stepwise from \(E\) = 10\({}^{8}\) to 5 \(\times\) 10\({}^{4}\) Pa to investigate the change of \(Cl\), details of the simulation conditions are provided in Table 1.  
Figure 2: NACA0015 foil simulation model.  
This two way FSI problem was solved using ANSYS Mechanical and ANSYS FLUENT (14.5, Ansys inc., Canonsburg, PA, USA) [12] controlled by System Coupling. The results are plotted in Figure 3, from which significant variation of \(Cl\) due to variation in \(E\) can be clearly seen. When the foil's \(E=10^{8}\) Pa, there is almost no deformation; however, when \(E=10^{6}\) Pa, the maximum \(Cl\) is obtained, where \(Cl\) is approximately three times greater. However, the \(Cl\) will also drop when the value of \(E\) is decreased. Therefore, it can be concluded that the optimum elastic modulus is \(E=10^{6}\) Pa for this shape of foil, in the case of different cambers the optimum value of \(E\) will most likely vary.  
The overall deformation of the foil for \(E=5\times 10^{4}\), \(10^{6}\) Pa, and \(10^{8}\) Pa has been examined, providing an optimal \(Cl\) value for \(E=10^{6}\) Pa. The scale in the \(y\) direction is scaled up as the deformation is very small. \(E=10^{8}\) is not shown here, as the deformation is insignificant. Instantaneous deformations of the foil for \(E=5\times 10^{4}\) and \(E=10^{6}\) are shown in Figures 4 and 5, respectively. It is recognized that, compared to the case of \(E=5\times 10^{4}\), in which the body curves with more than one center of curvature, in the case of \(E=10^{6}\), the foil is singularly concave or convex, providing an asymmetric foil. This forms an attack angle for the foil, and is the reason for the increase of \(Cl\).  
Clearly, the lift force of a foil will change according to its attack angle. The lift increases as the attack angle increases within the range of \(\pm 15^{\circ}\). Facilitation of this lift angle must be considered when manufacturing an elastic oscillator-based power generator; however, this study only investigated the influence of the rigidity. In conclusion, this study has shown that the optimum value of the elastic modulus is \(E=10^{6}\) Pa; the lift will decrease with any increase or reduction of the value of \(E\).  
A reciprocating displacement of \(y=0.002\) sin(2\(\pi\)t) m occurs when \(E=10^{6}\) Pa. The \(Cl\) is three times that of the previous \(y=0.001\) sin(2\(\pi\)t) m. However, the calculation failed due to the occurrence of negative meshes when a larger displacement was used. The authors will investigate the reason for this in future research.  
\begin{table}
\begin{tabular}{c c} \hline Time Step Size for Flow Calc. (s) & 0.005 \\ Scale of flow domain (m) & 6 \(\times\) 3 \(\times\) 0.1 \\ Type of mesh & prism \\ Min. size of mesh (m) & 0.007 \\ Density of structure (kg/m\({}^{3}\)) & 1000 \\ Length of structure (m) & 1 \\ Depth of structure (m) & 0.1 \\ \hline \end{tabular}
\end{table}
Table 1: Simulation conditions.  
Figure 3: Variation of coefficient of lift (\(Cl\)) versus rigidity.