## 2 Simulation Basics
### Turbulence  
Turbulence is a state of fluid flow in which fluid moves irregularly. The NS equations describing fluid motion is a closed system of equations and direct numerical simulation requires enormous computational resources and is challenging to apply in practice. The Reynolds average (RANS) method is widely used to average the time term of turbulence fluctuation [25]. However, at the same time, a new unknown quantity lacking the governing equation is added, which leads to the NS equations not being closed. It is necessary to select an appropriate turbulence model to close the equations describing the turbulence. The primary challenge of numerical simulation is to choose a suitable model to describe turbulence. For most numerical simulation applications, Spalart-Allmaras (SA), \(k\)-epsilon, and SST \(k\)-omega are widely accepted, along with the relatively accurate RANS turbulence models [26; 27]. The Spalart-Allmaras model is explicitly designed for aeronautics and aerospace applications involving wall-bounded flows and has been shown to give good  
Figure 1: (a) Six NACA0012 airfoil models; (b) close view of different models. results for boundary layers subjected to adverse pressure gradients. The \(k\)-epsilon models are historically the most widely used turbulence models in industrial CFD. The draw-back of \(k\)-epsilon models is their insensitivity to adverse pressure gradients and boundary layer separation, which typically predict a delayed and reduced separation relative to observations. The \(k\)-epsilon model is therefore not widely used in external aerodynamics. The standard \(k\)-omega model in Ansys Fluent is based on a \(k\)-omega model proposed by [PERSON], the main problem of which is its strong sensitivity to freestream conditions. Therefore, the BSL \(k\)-omega model is developed to blend the robust and accurate formulation of the \(k\)-omega model in the near-wall region with the freestream independence of the \(k\)-epsilon model in the far-field. The SST \(k\)-omega model includes all the refinements of the BSL \(k\)-omega model and in addition, accounts for the transport of the turbulence shear stress in the definition of the turbulent viscosity. These features make the SST \(k\)-omega model more accurate and reliable for external flow fields of airfoils [24]. Therefore, we adopt the Spalart-Allmaras and the SST \(k\)-omega turbulence models in this paper. Figure 2: (a) Grid division of the blunt trailing edge; (b) grid division of the sharp trailing edge. Spalart-Allmaras is a one-equation RANS turbulence model, and the transport equation for the turbulent kinematic viscosity \(\widetilde{v}\) is  
\[\frac{\partial}{\partial t}(\rho\widetilde{v})+\frac{\partial}{\partial x_{i}}( \rho\widetilde{v}u_{i})=\frac{1}{\sigma_{\widetilde{v}}}\left\{\frac{\partial} {\partial x_{j}}\left[(\mu+\rho\widetilde{v})\frac{\partial\widetilde{v}}{ \partial x_{j}}\right]+C_{b2}\rho\left(\frac{\partial\widetilde{v}}{\partial x _{j}}\right)^{2}\right\}+G_{v}-Y_{v}+S_{\widetilde{v}} \tag{2}\]  
Then, the turbulent viscosity \(\mu_{t}\) is computed from  
\[\mu_{t}=\rho\widetilde{v}f_{v1} \tag{3}\]  
\(f_{v1}\) is the viscous damping function, which is given by  
\[f_{v1}=\frac{X^{3}}{X^{3}+C_{v1}^{3}} \tag{4}\]  
\[X\equiv\frac{\widetilde{v}}{v} \tag{5}\]  
The model constants have default values [27]. SST \(k\)-omega is a two-equation turbulence model and the turbulent viscosity \(\mu_{t}\) is computed as a function of the turbulence kinetic energy \(k\) and the specific dissipation rate \(\omega\).