# Comparative Aerodynamic Performance Analysis of Camber Morphing and Conventional Airfoils
## 3 Methodology  
### Computational Method and Verification  
The fluid flow around the 2D airfoils was simulated using CFD software ANSYS Fluent to predict the aerodynamic forces: \(L\) and \(D\). The adopted numerical technique is a finite volume discretization method, with a pressure-based solver to solve the Reynolds-averaged version of these equations, coupled with a one-equation Spalart-Allmaras (S-A) turbulence model [42] to compute the Reynolds stresses in the Reynolds-averaged Navier-Stokes (RANS) equations. The calculations depend on computing the S-A working variable, which obeys the following transport equation:\[\frac{\mathrm{D}\widetilde{\mathrm{V}}}{\mathrm{D}t}=\mathrm{c}_{\mathrm{b12}}[1- \mathrm{f}_{\mathrm{f2}}]\widetilde{\mathrm{S}}\widetilde{\mathrm{V}}+\frac{1}{ \sigma}\left[\
abla\cdot\left(\left(\mathrm{v}+\widetilde{\mathrm{v}}\right) \
abla\widetilde{\mathrm{v}}\right)+\mathrm{c}_{\mathrm{b2}}\left(\
abla \widetilde{\mathrm{v}}\right)^{2}\right]-\left[\mathrm{c}_{\mathrm{w1}} \mathrm{f}_{\mathrm{w}}-\frac{\mathrm{c}_{\mathrm{b1}}}{\kappa^{2}}\right] \left[\frac{\left(\widetilde{\mathrm{v}}\right)}{\mathrm{d}}\right]^{2}+ \mathrm{f}_{\mathrm{f1}}\Delta\mathrm{U}^{2} \tag{1}\]  
A C-type unstructured mesh was used in the fluid domain, with a \(C\) segment radius of 7.5 m and a rectangular block measuring 16.5 m by 15 m. These dimensions were set based on the length recommended for simulating the flow around an airfoil of chord length 1 m [43]. Then, 25 inflation layers were generated around the airfoil, with a growth rate of 1.1 to resolve the viscous sublayer in the near-wall region and the first layer thickness, y, computed from the given equation [44]:  
\[\mathrm{y}=\frac{\mathrm{y}^{+}\mu}{\rho\sqrt{\frac{1}{2}\mathrm{C}_{\mathrm{f }}\mathrm{v}^{2}}} \tag{2}\]  
where:  
\[\mathrm{C}_{\mathrm{f}}=0.058\ Re^{-0.2} \tag{3}\]  
The values for the \(\rho\) and \(\mathrm{v}\) used in the equation correspond to the \(Re\) the flow was being simulated for, and the first layer thickness calculated to maintain \(\mathrm{y}^{+}\) values of less than 1 at the wall. It was confirmed that the \(\mathrm{y}^{+}\) values at the airfoil boundary in all our setups were in a range of 1 to less than 1, as shown in Figure 3, which also shows the computational domain, and the grid from one of our setups. A mesh independence study was conducted as well. A Richardson extrapolation has been adopted to calculate the grid convergence index, and it was confirmed that an index of less than 3% was obtained for the various grids tested. Computational time also plays a role in deciding the mesh element size. The grid comp  
Figure 3: (**A**) Computational domain; (**B**) mesh near the airfoil; (**C**) inflation layers around the airfoil; (**D**) histogram of airfoil Wall Yplus (\(\mathrm{y}^{+}\)). (element size: 0.01 m) was the most refined mesh tested, but as shown in Figure 4, varying the element size and thus the number of cells, shows only minor changes in the values of \(C_{D}\) obtained from the simulations. Therefore, the element size was adjusted to 0.05 m for the rest of the study, which would generate around 350,000 elements for the geometric configurations tested. The values for the flow properties such as the freestream velocity and the fluid density, and parameters of viscosity such as \(\mu\), input in the setup, were set to correspond to the \(Re\) the flow was being simulated for. A \(Re\) of 76,630 was chosen for this study since we compare the performance of a camber morphing and a conventional airfoil with a plain flap, at actual flight conditions, of a small UAV or MAV. A velocity-based inlet and a pressure-based outlet were used as the boundary conditions with a no-slip condition imposed on the airfoil surface. In the solution methods, a coupled scheme was found to be more accurate and compatible with our unstructured mesh, and a convergence criterion of \(10^{-6}\) was set for all computed residuals.