The angle mθ represents the tangential direction of the trajectory, and the angle of attack α(t) is also an independent variable. The definition makes it easier to develop an optimization method for the motion trajectory or the angle of attack. In the nature, fish and birds adjust the angle of attack and the trajectory constantly to eliminate the unwanted forces and keep the speed and the efficiency at a high level.

1.3 Numerical method

Using the commercial software Fluent (ANSYS Inc., Canonsburg, USA), the governing equations are discretized by the finite volume method (FVM) with a first-order discretization in the time and a second order discretization in the space. The fluid motion is computed based on an unstructured, tetrahedral grid.The simulation is a dynamic process and the boundaries of the foil are changing constantly. Therefore,in this paper, the dynamic mesh method is adopted to deal with the moving boundary problem. The motion of the foil is realized through user-defined functions(UDFs) linked to the Fluent. In the calculation, the SIMPLE method of the pressure-speed amending method and the k-ε turbulence mode are adopted.

The simulations start from the static state of the foil, and then the foil moves according to the motion model. The cubic computational domain (2.5 m×1.2 m×1.2 m) is discretized into 8.1×105 grid , refined meshes are used near the boundaries of the foil. To verify the grid in dependency, the simulation with a mesh of smaller cell size is carried out and only a difference of 4% between the coarse grid and the fine grid is found.

To evaluate the hydrodynamic performance of the foil, the force and the torque acting on the foil are extracted and converted to dimensionless forms:

where ρ is the density of the fluid, Fy is the transverse force, -Fx is the thrust force, Mθ is the torque, and S is the projected area of the foil. The propulsive efficiency η is defined as the ratio of the output power oP to the input power eP, which can be defined as:

where T is the period of the motion. The chord Reynolds number and the Strouhal number St are defined as:

where ν is the fluid kinematic viscosity.

2. Experiment approach and setup

With the developments of computer technology and computational fluid dynamics (CFD), numerical methods are having more and more extensive applications. Compared with the traditional experimental method, the numerical method has the advantage of high repeatability, great maneuverability, and less input requirement. Moreover, it is easier to obtain the details of the flow field with the numerical method,thus, many difficulties of measurement in the experimental method can be avoided. Therefore, the numerical method becomes an important method in the mechanism research. However, due to the limitations of the theoretical model, the reliability of the numerical results still needs to be improved. In addition, the computation parameters of the numerical models need to be modified and confirmed according to the experimental results. So usually both the numerical method and the experimental method are adopted in studies.

To validate the reliability of the numerical method used in this paper, an experiment platform is designed and the verification experiments are carried out. Figure 3 shows the design schematic diagram of the experiment platform.

To realize the movements of three degrees of freedom, three servo motors equipped with reducers are used. The synchronous belt transmission is adopted in the towing direction and the gear-gear stripe transmission is adopted in the heave servo motors are controlled by computer.

Fig.3 (Color online) Design schematic diagram of the experiment platform

Fig.4 (Color online) Measuring module of the experiment platform

Fig.5 (Color online) Actual photo of the experiment platform

Except the power module and the transmission module, the measuring module is an important part of the system. As is shown in Fig.4, two slide rails are installed on the moving platform along the heave direction. The force sensors act as the transmission components to supply power. According to the mechanical analysis of the moving platform in the horizontal plane, there are the hydrodynamic force,the frictional force, the traction force and the inertia force. Because the coefficients of friction of the slide rails are very small, the frictional force can be inertia force can be calculated through the motion equations. The traction force can be provided by the force sensor. Through mathematical calculations, the hydrodynamic force acting on the foil can be 5 shows the actual photo of the experiment platform.

3. Results

Firstly, for comparing with the existing related studies[13, 14], the motion parameters of the foil are selected as follows: Re =, St=0.3, h=c=0.055 m, αm ax=25° and the stroke angle β=45°,90° and 135°. Figure 6 shows the force coefficients at different stroke angles.

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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