# Study on the Hydrodynamic Performance of Typical Underwater Bionic Foils with Spanwise Flexibility
## Materials and Methods
### Numerical Solution  
The URANS (Unsteady Reynolds Averaged Navier Stokes) model is adopted in the numerical simulations. Using this model, the Navier-Stokes equations used to describe the hydrodynamics of fluids are restructured as:  
\[\frac{\partial U_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(}U_{i} U_{j}\big{)}=-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\
u\frac{\partial^{2} U_{i}}{\partial x_{i}\partial x_{j}}-\frac{\partial\overline{u_{i}^{\prime}u_{j}^{ \prime}}}{\partial x_{j}} \tag{11}\]  
\[\frac{\partial U_{i}}{\partial x_{i}}=0 \tag{12}\]  
where \(\rho\) is the density, \(\
u\) is the kinematic viscosity and \(p\) is the pressure. The subscript \(i\) and \(j\) represent the three coordinate directions. Here, the dependent variables are not only a function of the space coordinates but also a function of time. Therefore, the simulated velocity in URANS model can be split to the time-averaged part \(U\) and the fluctuations \(u\). To close the equations, the Reynolds stresses term, the last term in Equation (11), can be expressed with the mean velocity gradient via the eddy viscosity. The eddy viscosity can be represented by \(k\) (turbulent kinetic energy) and \(\varepsilon\) (the velocity and length scale of turbulent motion), that is the \(k\)-\(\varepsilon\) turbulence model. The numerical simulations are carried out by Fluent (ANSYS Inc., Canonsburg, PA, USA) software. The Finite Volume Method (FVM) is adopted  
Figure 5: Shape description of the flexible NACA foil. Figure 6: Shape description of the flexible crescent-shaped foil: (**a**) \(\gamma=20^{\circ}\); (**b**) \(\gamma=-20^{\circ}\). to discretize the Navier-Stokes equations into expressions that are first-order in time and second-order in space. The tetrahedral grid is used to divide the computational domain. The simulation is a dynamic process and the boundaries of the foil are changing constantly. The dynamic mesh method reconstructs the mesh at each time step and User-Defined Functions (UDF) linked to Fluent define the motion of the moving boundaries. For a single crescent-shaped foil separated from the fish body or NACA foil, the simulations are started from the static state and then the foil is towed forward and moves according to the motion model defined above. For the thunniform bio-inspired model, the self-propulsion model is adopted. In other words, the thunniform bio-inspired model attempts to generate thrust by swinging the caudal fins and waving the body. As a result, the hydrodynamic forces acting on the body and caudal fin achieve forward propulsion of the thunniform bio-inspired model. At each time step, the hydrodynamic forces are extracted and then the position and attitude of the thunniform bio-inspired model are calculated according to the hydrodynamic forces. Without taking into account the flexibility, the bionic foils can be treated as rigid bodies. In these cases, the translational and rotational motion of the bionic foils can be defined by DEFINE_CG_MOTION function. In addition, DEFINE_GRID_MOTION function is adopted to define the motion of the flexible foils and thunniform bio-inspired model. Using this function, the deformation velocity of each grid point can be defined in each time step. During the calculation, the SIMPLE algorithm of the pressure-speed amending method is adopted and the k-\(\varepsilon\) turbulence model is adopted. To ensure the accuracy of the calculations, the maximum number of iteration steps is 50 and every motion period contains 200 time steps. The wall boundary condition is set at the solid boundaries and symmetry boundary conditions are imposed on the boundaries of the fluid domain. To exclude the influence of the wall effect, the computational domain is set sufficiently large. To verify the grid independency, choosing thrust velocity as the assessment index, three numerical simulations with different grid size are carried out and GCI (grid convergence index) was calculated.