Fig.8 Performance curves against αmax with β=45°

Fig.9 Performance curves against αmax with β=90°

Fig.10 Performance curves against αmax with β=135°

3.2 Influence of the stroke angle

The performance curves against the stroke angle are shown in Fig.11. The dotted lines in Fig.11 represent the values in the case of β=90°. It is shown that the stroke angle greatly affects the hydrodynamic performance of the foil. By choosing a large stroke angle, the foil can obtain a large thrust force with a small transverse force. In contrast, with a small stroke angle, the foil can obtain both a large transverse force and a large thrust force, but the propulsive efficiency is relatively low. In addition, a stroke angle around 90° can achieve a smaller thrust force with a high propulsive efficiency. However, in the symmetrical mode, a higher efficiency can be achieved as com-pared with the asymmetrical mode. A reasonable stroke angle should be adopted in different engineering applications.

Fig.11 Performance curves against stroke angles

3.3 Influence of the heave motion

Here, the case of β=90° is chosen as a typical example to show the influence of the heave motion on the hydrodynamic performance. Figure 12 shows the force coefficients and the propulsive efficiency versus the amplitude of the heave motion. The Strouhal number is chosen to be 0.3 to keep a high is shown that the impact of the amplitude on the mean thrust force is small. Note that the propulsive efficiency reaches the peak value around h/ c ≈ 0.8.As is known, a large heave motion obviously will lead to a large thrust force but a larger heave motion will lead to a lower frequency to keep the Strouhal number constant. So the change of the mean thrust is insignificant when the amplitude of the heave motion h increases.

Fig.12 Performance curves against amplitudes of heave motion

3.4 Influence of Strouhal number

The Strouhal number is an important parameter in the biological swimming. Furthermore, most fish are shown to swim near a “universal” optimal value St =0.3 because of the high propulsive efficiency[19, 20].Figure 13 shows the performance curves against the Strouhal number when β=90°. Here, the frequency is chosen to vary to obtain different Strouhal numbers. It is shown that the propulsive efficiency keeps a relatively large value within the range of 0.2-0.4, as is consistent with the range of fish in the nature. Furthermore, a large Strouhal number can lead to a large thrust force, but the transverse force increases correspondingly.

Figure 14 shows the vortex pattern in the wake for different values of St when β=90°. At a low value of St, the 3-D structure of the wake is a single and continuous vortex ring. At a high value of St, a more complex and laterally diverging structure is observed in the wake. Smaller-scale vertical structures attaching to the hairpin vortices are observed in the wake. The energy dissipation is greater in this case, so the propulsive efficiency is lower at a high value of St.In addition, the more complex and laterally diverging structure impacts the transverse force and a considerable net transverse force is produced. The results of asymmetric motion models are similar to those of symmetric motion models.

Fig.13 Performance curves against Strouhal numbers

Fig.14 (Color online) Performance curves against amplitudes of heave motions

3.5 Experimental results

To validate the reliability of the numerical method used in this paper, verification experiments are carried out. Table 1 shows the numerical results and experimental results with same parameters. Figure 15 shows the photo of the experiment site. The selected motion parameters of the foil are as follows:Re =, St=0.3, h =0.05 m and αm ax=25°.Through a comparison, it is found that the present results compare well with those in the previous , the adopted numerical method is reliable.

Table 1 Comparison of results

Fig.15 (Color online) Photo of experiment site

4. Conclusions

In this paper, a universal kinematic model with three degrees of freedom is adopted. The motion parallel to the flow direction considered in the model can produce trajectories of different prototypes. Based on this kinematic model, numerical simulations of the flapping foil with different trajectories are certain parameter ranges, the mean thrust forces of different motion modes in a cycle could have positive values, that is, the foil can obtain a thrust force. And the propulsive efficiency of the symmetric motion model is significantly greater than those of the asymmetric motion models. In addition, the influences of the motion parameters are analyzed. It is found that the motion parameters play important roles in the hydrodynamic performance of the flapping , to validate the reliability of the numerical method, an experiment platform is designed and verification experiments are carried out. The consistence between the numerical results and the experimental results indicates that the numerical method of the flapping foil is reasonable.

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