Fig.6 Force coefficients at different stroke angles

The curves see a periodic behaviour, and the curves in a cycle are shown in Fig.6. In the case of β =90°, the curve of the thrust force coefficient has the same shape in the upstroke and the down stroke because of the symmetrical motion. Note that the mean value of Cx , which is equal to 0.58, is positive,while the mean value of Cy is almost zero. In other words, the foil obtains the thrust force without the net transverse force. In the case of β=45°, one sees small fluctuations at the beginning and the end of the upstroke when the foil rotates quickly. The mean thrust force coefficient is 0.21 and the mean transverse force coefficient is 2.12. In the case of β=135°, the mean thrust force coefficient in a cycle is 0.12. In the cases of β=45° and β=135°, the mean transverse forces are not equal to zero because of the asymmetrical motion. In the case of β=45°, it is the transverse force that increases significantly rather than the thrust force. The thrust force in the case of β =135° is relatively small because of the negative mean thrust force of the upstroke, and other experiments show similar phenomena during the upstroke,called “the memory effects” in the wake[13]. Besides,the propulsive efficiency of symmetrical mode is about 47.0% which is well within the range of fish.The propulsive efficiency in the case of β=45° is equal to 11.3%, which is low because the for wards biased flap increases the transverse force significantly rather than the thrust force. The propulsive efficiency in the case of β=135° case is equal to 31.0%,which is also relatively low because of the unwanted energy expenditure during the upstroke.

Figure 7 shows the vortex pattern in the wake. It can be seen that there are vortex rings in the wake,which is similar to the wake of fish and birds. The jet stream can be induced among the vortex rings and the foil obtains a reaction force correspondingly. Specifically, Figure 7(a) shows the 3-D vortex pattern in the case of β=90°. The vortex rings consist 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. The vortex structure is similar to those found in the nature and the simulations in other studies[15-18]. Figures 7(b), 7(c) show the comparisons of the vortex patterns in different motion modes viewed from z direction. It is shown that the vortex patterns of the asymmetrical mode are characterized by irregular and discontinuous structures. As a result,because of the asymmetrical motion , a net transverse force is produced and more useless energy is expended.

In the nature, birds direct their wings forward during the down stroke, creating a highly asymmetric flap that can generate a net lift force to support the weight. Turtles direct their flippers backward during the down stroke, creating a highly asymmetric flap that can reduce the oscillatory force. The fins of fish can form a symmetric trajectory without the surge motion parallel to the direction of towing and keep a high propulsive efficiency. In the following parts, the influence of the motion parameters on the hydrodynamic performance is analyzed. For comparing with the experimental results, the chord length is chosen to be 0.1 m and the length along the span direction is 0.3 m. The other parameters are selected as follows: αm ax=25°,h=0.05m, U=0.2 m/s , St=0.3. The force coefficients represent the mean value in a cycle.

Fig.7 (Color online) Vortex patterns for different stroke angles

3.1 Influence of the angle of attack

To study the influence of the angle of attack, its value is chosen to vary from 10° to 45°. The definition of the angle of attack is different in the symmetric and asymmetric modes, so the influence of the angle of attack for different stroke angles should be analyzed,separately. Figures 8-10 show the mean force coefficients and the propulsive efficiency of the foil with different stroke angles. In the symmetrical mode,the mean thrust force keeps a relatively large value when the angle of attack αmax varies from 20° to 30°,as is consistent with fish in the nature. Furthermore, as the angle of attack increases, the mean thrust force increases correspondingly and reaches the peak point,then starts to decrease. Besides, it is found that the propulsive efficiency sees a similar trend as compared with the mean thrust force. Larger or smaller αmax will bring a large transverse force and a reduced thrust force, and then decrease the propulsive efficiency. In the case of β=45°, the trajectory of the down stroke is more gradual than in other cases, so large αmax adopted in the powerful down stroke can increase the transverse force significantly and have little effect on the thrust force. As a result, the propulsive efficiency decreases significantly. In contrast, in the case of β= 135°, the trajectory of the down stroke is steeper,so large αmax is allowed to keep the thrust , the strong angular motion with a large αmax will increase the useless energy consumption, to reduce the propulsive efficiency.

文章来源：《水动力学研究与进展》 网址: http://www.sdlxyjyjzzz.cn/qikandaodu/2021/0114/465.html

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