## 3 Numerical Simulations
### Grid Strategy  
Viscosity is a measure of the internal viscosity of a fluid and all fluid flows involve some degree of viscosity effect. Viscosity causes the fluid directly in contact with the airfoil wall to be stuck to the surface without slip, called the no-slip condition. Due to the no-slip condition, within a very small normal distance adjacent to the airfoil wall, the fluid velocity will decrease from a large value to the same as the speed of the wall, resulting in a considerable gradient of the normal velocity of the wall. This area where the viscous effect causes the velocity gradient is called the boundary layer. The boundary layer is a thin viscous layer near the airfoil's surface, containing all the viscous effects, and the flow outside the boundary layer is non-viscous. The calculation of the correlation coefficient is closely related to the viscous boundary layer. To make the numerical simulation results closer to the experimental data, the meshing of the airfoil adjacent to the wall is vitally crucial. Calculations for the height of the first layer cells of the boundary layer are begun by calculating the flow Reynolds number:  
\[R_{e}=\frac{\rho UL}{\mu} \tag{12}\]  
where \(L\) is the characteristic length of the airfoil, \(\mu\) is the dynamic viscosity of the air, \(\rho\) is the air density, and \(U\) is the freestream velocity. An empirical correlation formulation to estimate the skin friction coefficient (\(C_{f}\)) is adopted, which is for the fully developed turbulent flow over a flat plate:  
\[C_{f}=\left[2\log_{10}(R_{e})-0.65\right]^{-2.3} \tag{13}\]  
Having computed the skin friction coefficient, the wall shear stress (\(\tau_{W}\)) is calculated:  
\[\tau_{W}=\frac{1}{2}\rho U^{2}C_{f} \tag{14}\]  
After that, we can calculate the friction velocity (\(u_{\tau}\)) from the wall shear stress (\(\tau_{W}\)):  
\[u_{\tau}=(\frac{\tau_{W}}{\rho})^{0.5} \tag{15}\]The Equation for \(y^{+}\) is:  
\[y^{+}=\frac{\rho y_{P}\mu_{\tau}}{\mu} \tag{16}\]  
Therefore, the height of the wall adjacent cell centroid from the wall (\(y_{P}\)) is:  
\[y_{P}=\frac{y^{+}\mu}{u_{\tau}\rho} \tag{17}\]  
Finally, the first height of the boundary layer cell (\(y_{H}\)) is double \(y_{P}\):  
\[y_{H}=2y_{P} \tag{18}\]  
The result of the first height of the boundary layer cell is an estimate and would need to be updated using information from the CFD analysis. Ref. [24] suggests that the \(y^{+}\) value should be less than 1, which could guarantee that the viscosity-affected region of the boundary layer is adequately resolved, and the first layer cells of the boundary layer should be in the sub-viscosity layer during the simulation process. The turbulent viscosity ratio (TVR) is adopted to evaluate if the boundary layer profile has been captured well. Initially, \(y^{+}\) is 1 and \(y_{H}\) is 1.53 \(\times\) 10\({}^{-5}\) m according to Equations (12)-(18), while the test results show that the maximum \(y^{+}\) of airfoil surface exceeds 1 as the AOA increases during the simulation. The tests show that a \(y^{+}\) value of 0.5 guarantees that the maximum \(y^{+}\) of the airfoil surface stays lower than 1 during the entire simulation process, and the corresponding \(y_{H}\) is 4.4 \(\times\) 10\({}^{-6}\) m. Figure 3 shows the corresponding plots of the TVR for the mesh with \(y^{+}\) of 0.5; since the \(y^{+}\) value is 0.5, there is no turbulent viscosity within the laminar sub-layer. As the mesh moves through the buffer region and into the logarithmic region, the TVR value is increased and the maximum value occurs near the middle of the boundary layer. Then, the turbulence gradually dissipates as the mesh approaches the freestream condition, which is expected. Therefore, the \(y^{+}\) value is sufficient. In Figure 4a, the computational domain is shown. The bold black dot is the (0,0) point of the coordinate system.