# Study on the Propulsion Characteristics of a Flapping Flat-Plate Pumping Device
## 3 Numerical Simulation Method
### Mesh Generation and Computational Setup  
The size of the computational domain and far-field boundary conditions significantly impact the results. To ensure full development of the wake behind the flapping wing, the total domain length is typically set to \(20c\). To efficiently utilize computational resources and avoid negative cell volumes, overset dynamic mesh technology is employed to handle the motion of the flapping flat-plate hydrofoil. The computational domain consists of a moving foreground mesh and a stationary background mesh, as shown in Figure 6. The foreground mesh is a hybrid mesh covering the hydrofoil motion region; the background mesh is a structured mesh with a maximum cell size of 30 mm, representing a 5 m \(\times\) 20 m rectangular river channel.  
Figure 6: Mesh generation schematic.  
The turbulence modeling in this study is based on the Realizable k-\(\varepsilon\) model, which is effective in capturing complex flow dynamics, particularly in regions involving vortex shedding and near-wall effects. The outer layer of the foreground mesh is sized consistently with the background mesh to further ensure computational accuracy. Different turbulence models and wall functions impose varying Y+ requirements [29; 30]. To accurately capture wall flow features and ensure computational stability, boundary layer meshes are deployed on the channel walls and hydrofoil surface. The first layer mesh height on the flapping flat-plate hydrofoil surface is set to 0.000046 mm, corresponding to a Y+ value of 1, which ensures that the boundary layer is well-resolved and the near-wall effects are captured accurately. This mesh configuration guarantees that the turbulence model effectively resolves the flow structures near the hydrofoil surface, which is critical for accurately simulating the flow characteristics in this study.  
In the numerical simulation, the pressure-based solver in FLUENT is used to solve the Navier-Stokes equations. The Coupled algorithm handles the pressure-velocity coupling. The turbulence kinetic energy and dissipation rate equations are discretized using first-order upwind schemes, while the momentum equations use second-order upwind schemes. Boundary conditions are set as follows: the inlet uses a pressure-inlet condition, the outlet uses a pressure-outlet condition, the outer boundary of the foreground mesh is set as overset, and the river channel, flow channel, and flapping flat-plate hydrofoil surfaces are set as wall boundaries. The overset mesh method is applied in overlapping regions. The motion of the flapping flat-plate hydrofoil is implemented via a User-Defined Function (UDF).  
In this study, a traditional mesh-based method (finite volume method) was employed for simulating the flapping hydrofoil motion. However, recent advancements have introduced mesh-free methods and collocation-based approaches, which can offer several advantages over traditional mesh-dependent methods, particularly in handling large deformations and dynamic boundaries. These methods do not rely on predefined grid structures and have shown promise in simulating complex fluid-structure interaction problems with improved flexibility and reduced computational cost [31; 32].  
While the present study uses the finite volume method with overset grids, we acknowledge the potential of mesh-free methods as a promising direction for future work. These methods could improve computational efficiency and accuracy in simulations involving highly dynamic systems, such as bionic propulsion devices.