## 2 Numerical Modeling and Validation
### Numerical Methodology  
This paper employs the finite volume method and the well-established CFD solver Fluent to solve the two-dimensional, transient, and incompressible Navier-Stokes equations for the oscillating foil problem. The pressure-based coupled (SIMPLE) segregated algorithm is used to evaluate the pressure gradient in the momentum equation. For the stability and convergence, the convection terms and the diffusion term are discretized by a second-order accurate upwind scheme and a second-order central-differencing scheme, respectively.  
To simulate the coupled heaving and pitching motion of the oscillating foil with a large amplitude, the sliding mesh technique is applied by dividing the fluid domain into two sub-domains, as shown in Figure 2. The inner domain undergoes a rigid body motion, which means that the motion of the oscillating foil is prescribed and fixed during the simulation. The outer domain, on the other hand, needs to re-mesh to adapt to the oscillating foil's large-amplitude. The mesh slides and deforms according to the motion of the foil, enabling an accurate simulation of fluid-structure coupling interactions. A user-defined function (UDF) is used to customize the motion trajectory.  
We start with the baseline configuration NACA0015 airfoil under the condition of \(Re=1100\), \(H_{o}/c=1\), \(f^{*}=0.14\), \(\theta_{0}=76.3^{\circ}\), and \(\phi=90^{\circ}\), which is selected based on Kinsey [11]. Since there are no major qualitative changes in the cycles of power extraction except for slightly amplified extrema associated with amplified force coefficients, this paper does not study a higher \(Re\).  
The inlet velocity is calculated from \(Re=1100\) using the URANS approach based on the Spalart-Allmaras model. The boundary interface serves to link the inner and outer domains within the simulation. Symmetric no-slip boundary conditions are imposed on the upper and lower boundaries to ensure no flow through these boundaries. The entrance and exit boundary conditions are applied to the left and right boundaries, respectively, to simulate the flow entering and exiting the domain. The oscillating foil and cylinder are located in the inner domain, with the cylinder placed at the trailing edge. The outer domain measures \(75c\times 70c\) with the pitching axis of the oscillating foil located \(35c\) from the inlet boundary. A circular interface with a radius of \(5c\) connects the inner and outer domains. The outer domain is partitioned into a structured quadrilateral mesh using O-type partitioning to enhance mesh quality. As shown in Figure 2, the red box shows the grid area that needs further display, and along the direction of the red arrow, a partial enlargement of the grid drawing details is given. The inner domain is partitioned into a triangular mesh with boundary layers applied to the oscillating foil and cylinder surfaces. The initial layer mesh thickness of the boundary layer is set to \(10^{-5}c\), ensuring that the  
Figure 2: Mesh details with boundary conditions.  
value remains below 1 throughout the simulation. To reduce numerical calculation errors and flow field fluctuations caused by large variations in mesh size, denser mesh nodes are placed on the foil and cylinder surfaces due to the relatively small size of the trailing edge cylinder and its proximity to the foil.