# Numerical Study on Hydrodynamic Performance of Bionic Caudal Fin
## 2 Materials and Methods
### The Numerical Method  
The Navier-Stokes equations for viscous and incompressible flows can be described by the equations:  
\[\rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\
abla\mathbf{u}\right)=- \
abla p+\upmu\
abla^{2}\mathbf{u} \tag{13}\]  
\[\
abla\cdot\mathbf{u}=0 \tag{14}\]  
where \(\mathbf{u}\) is the fluid velocity, \(\rho\) is the density, \(p\) is the pressure, and \(\upmu\) is the dynamic viscosity. Using the commercial software Fluent (ANSYS Inc., Canonsburg, PA, USA), the governing equations are discretized by the Finite Volume Method (FVM) with first-order discretization in time and second-order in space. The fluid field is computed on an unstructured, tetrahedral mesh. In order to improve calculation efficiency, a fine mesh is used near the moving surface and coarse mesh is used far away from the moving surface. The surface of the moving fin consists of triangle elements. The grids of the caudal fin are shown in Figure 3. The fluid field is divided into \(8.9\times 10^{5}\) elements. To test the grid independency, the calculation with smaller cell size is carried out, but little difference between the coarse grid and the fine grid is found. The impact from the caudal fin to the fluid is carried out by imposing wall boundary conditions on the moving surface. The boundary conditions of the overall computation domain are set as symmetry boundary condition. In the numerical simulations, the caudal fin starts to move from rest. Due to the movements of the fin, the grids around the moving surface need to be updated. Using the Fluent software, a dynamic mesh is applied to solve the problem. Furthermore, layering and local remeshing methods are used in the dynamic mesh to guarantee the stability of the calculation. The mesh is set to be updated in each time step. In the paper, the motion of the moving surface is specified in user-defined functions (UDF) linked to Fluent in the dynamical simulation. As is described by the motion equations, the caudal fin is towed forward at constant speed \(U\) and is set to carry out the heave motion and pitch motion. Without taking into account the flexibility, the caudal fin can be treated as a rigid body. In the paper, the DEFINE_CG_MOTION function is adopted to simulate the motion of the rigid fin. Using the DEFINE_CG_MOTION function, the translational and rotational motion of the rigid body can be defined in each time step. Taking into account the flexibility, the caudal fin can not be treated as a rigid body. Here the DEFINE_GRID_MOTION function is adopted to simulate the motion of the flexible fin. Using the DEFINE_GRID_MOTION function, the deformation velocity of each grid point can be defined in each time step. The deformation velocity of each grid point can be calculated through motion equations. Each motion period is divided into 200 time steps and the maximum number of iteration step in a time step is set to 50. In addition, the SIMPLE method is adopted to solve the pressure-velocity coupling problem. After several motion periods, the performance curves settle to a periodic behaviour. The calculation procedure takes 6-8 h per cycle on the personal computer. In order to eliminate the influences of the walls, enough space is kept around the caudal fin. The size of computational domain is shown in Figure 4.  
Based on the existing experimental results [8], numerical simulations of flapping foil are conducted to verify the accuracy of the basic algorithm. The motion parameters of the foil are selected as follows: \(Re=11000\), \(St=0.3\), \(h=L=0.055\) m, \(B=0.3575\) m, and \(\alpha_{max}=25^{\circ}\). Figure 5 shows the time histories of the force coefficients and torque coefficient of the fin. It is shown that the force coefficients and torque coefficient settle to a periodic behavior. There are two peaks in a period which are caused by the motions of downstroke and upstroke, respectively. Furthermore, the mean \(C_{x}\) is 0.58, the mean \(C_{y}\) is 0.0065, and the thrust efficiency is about 47.0%.