Introduction

The conventional screw propellers have the disadvantages of low efficiency, poor maneuvering performance, and large noise , which greatly limit their applications in narrow, complex and dynamic environments. However, after millions of years of evolution,creatures are capable of swimming or flying with high speed and efficiency. Therefore, the biomimetic propulsor attracts research attentions[1].

Most fast-swimming creatures undulate their fins to obtain thrust. Similarly, flying creatures undulate their wings to resist gravity and move forward. They have similar characters in shape and motion mode, in reducing resistance and obtaining valuable force. The flapping foil based on bionics is a sort of simplified models, which can imitate the motion of the wings or fins of fish and birds[2-6]. With different motion parameters, the hydrodynamic performance varies correspondingly. In addition, a simple imitation of motion is not enough, its mechanism is a more important hydrodynamic mechanism would provide a theoretical reference for the design of underwater vehicles based on the flapping propulsion.

In the movements of fish and birds, one sees the symmetric mode and the asymmetric mode. Most fish flap their caudal fins in the symmetric mode, with equal forces generated from both the upstroke and down stroke. On the other hand, with an oscillation parallel to the advance direction, the asymmetrical mode consists of a powerful down stroke generating large force and a weak upstroke with a weak force, as evidenced by the motion patterns of turtles and birds[7-11].

The symmetric mode and the asymmetric mode were usually studied separately, focusing on certain bionic prototypes, lack of generality. The kinematic parameters of living fish, birds and bionic robots were widely studied, but without enough attention to the hydrodynamic mechanism. In this paper, a united motion model containing the symmetric mode and the asymmetric mode is developed. By analyzing the flow field generated in the process of flapping and forcing of the foil, the propulsion mechanism of the bionic foil is studied. And then the influence of the motion parameters on the hydrodynamic performance of the bionic foil is analyzed. The simulation results are compared with the existing related studies to assess the reliability of the numerical method. In addition, an experimental method is adopted to further validate the numerical approach.

1. Materials and methods

1.1 Geometric model

Based on observations of fish and birds, the sections of fins or wings parallel to the flow direction take streamlined shapes, similar to the NACA the hydrodynamic force can be viewed as the resultant force of a series of forces acting on the streamlined sections. In this paper, the NACA0013 airfoil is chosen to construct the section. Figure 1 presents the geometry of the three-dimensional model,where c=0.1m is the chord length, L =0.3m is the length along the span direction, and O is the reference point. O is located 0.25c from the leading edge and the projected area (one-sided) S is 3.2×10-2 m2. The foil makes a periodic motion along the vertical direction and the horizontal direction,meanwhile, the foil makes a pitching movement around the reference axis.

Fig.1 (Color online) Geometric model of the foil

1.2 Motion model

In this study, the foil is allowed to move with three degrees of freedom. Specifically, the foil is towed forward at a constant speed U and there are three types of movements: the heave motion transverse to the direction of towing, the angular motion around a span wise axis, and the surge motion parallel to the direction of towing. The motion of the foil can be described by the following equations:

where f is the flapping frequency, t is the time,and β is the stroke angle[12]. In contrast to the traditional symmetric motion model, x( t) is adopted to describe the oscillatory motion parallel to the forward direction. With the stroke angle β, different trajectories can be formed to represent different motion models of different bionic prototypes. As shown in Fig.2, the case of β=45° describes the motion mode of birds. It can be seen that birds direct their wings forward during the down stroke, creating a highly asymmetric flap to generate a net lift force. The case of β=90° describes the motion mode of fish.In the symmetric mode, a symmetric trajectory is formed without the surge motion parallel to the direction of towing. The case of β=135° describes the motion mode of turtles. It can be seen that turtles direct their flippers backward during the down stroke,creating a highly asymmetric flap to generate a net trust force.

Fig.2 (Color online) Trajectories of different modes

In addition, the pitching angle θ is given by

where

The angle α(t) is the angle of attack. In the case of β =90°

and in the case of β≠90°