## 2 Materials and Methods
### CFD Simulations  
In this work, three fringe lengths at four different flapping frequencies of the extended fringe were evaluated. These variables resulted in 12 simulation cases, and a bare airfoil model was also simulated as the baseline case. Simulations were performed using the CFD software Cradle SC/Tetra 2021 (Dayton, OH, USA). The computational software solves the N-S equations by using the finite volume method with hybrid computational grids, wherein hexahedral grids were generated as prism layers around the airfoil to capture the boundary layer behavior, and tetrahedral grids were generated in the rest of the region. A local mesh refinement technique (grids become coarser along the downstream direction) was adopted to resolve the flow around the flapping fringes installed at the trailing edge to save computational effort.  
The turbulence model LKE K-KL-\(\omega\) was adopted to evaluate the turbulence region over the airfoil. This model is one of the Reynolds-Averaged Navier-Stokes (RANS)-based approaches to predict the transitional flow [25; 27; 28]. In the LKE model, the energy of the disturbances in the pre-transitional region of a boundary layer is expressed as \"Laminar Kinetic Energy\", whereas the turbulence energy is defined as k, and the transport equation of KL is solved with two equations of the fully turbulent model. The flow over the airfoil is governed by the Navier-Stokes equation for two-dimensional, viscous, incompressible flow. The continuity equation and the Reynolds-averaged N-S equations can be expressed as:  
\[\frac{\partial(\rho\overline{u_{i}})}{\partial(x_{i})}=0 \tag{1}\]  
\[\rho\frac{D(\rho\overline{u_{i}})}{Dt}=\frac{\partial(\overline{P})}{\partial (x_{i})}+\frac{\partial}{\partial(x_{i})}(u\left(\frac{\partial(\overline{u_ {i}})}{\partial(x_{j})}+\frac{\partial(\overline{u_{i}})}{\partial(x_{i})} \right)-\rho\overline{u^{\prime}_{i}u^{\prime}_{j}} \tag{2}\]  
where \(i\) and \(j\) are coordinate indices; \(u\) is the velocity; \(x\) is the flow direction; \(\rho\) is the density of the fluid; \(\mu\) is the dynamic viscosity of the fluid.  
The dynamic motion of the fringe was fulfilled using the stretching mesh technique, as it enables the expansion and shrinking of the mesh elements to accommodate the movement of the solid boundary, which can avoid the pressure or velocity discontinuity/fluctuations at interfaces [25]. The CFD simulation, in terms of the turbulence model, grid sensitivity, and the dynamic moving mesh technique, has been validated statically by comparing  
Figure 1: The 2-dimensional computational domain and boundary conditions for the CFD simulation. C: chord length of the airfoil; \(u\): velocity in the \(x\)-direction; \(v\): velocity in the \(y\)-direction; \(\rho\): density of the fluid; \(\mu\): viscosity of the fluid; \(P\): pressure; \(\theta\): dynamic motion of the trailing edge fringe; \(A\): amplitude of the flapping angle and set at 5 degrees; \(f\): frequency of the flapping motion.  
static lift and drag coefficients with experiments, and dynamically, by comparing pressure contours for an oscillatory plunging motion of an airfoil with the literature reported in our previous study [25].  
The numerical simulations were performed on a computational cluster containing 3.4 GHz AMD Phenom II X4 965 Quad-cores and 8 GB RAMs. Each case was run with 32 processors. In the current study, a total physical time period of 1.5 s of the flow was simulated with a time step of \(5\times 10^{-5}\) s. The mean computational time for each simulation case was about 23 h.