## 2 Method
### Description of Flow Solvers
#### 2.3.1 XFOil  
XFOil is a code for airfoil design and analysis consisting of inviscid, inverse and viscous formulations [75]. The viscous formulation can be used to calculate free or imposed forced flow transition to handle transitional separation bubbles, to calculate aerodynamic coefficients and to cope with moderately trailing edge separation. It uses an approximate e\({}^{\text{N}}\) envelope method to calculate the flow transition. The turbulence level settings are kept as default with free transition features. Both the boundary layer and the wake parameters  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline
**Parameter** & \(\mathbf{n_{c}}\) & \(\mathbf{\beta_{tr}}\) & \(\mathbf{x_{t}}\) & \(\mathbf{y_{t}}\) & \(\mathbf{dz_{tr}}\) & \(\mathbf{b_{b}}\) & \(\mathbf{b_{15}}\) & \(\mathbf{a_{tr}}\) & \(\mathbf{z_{tr}}\) & \(\mathbf{\gamma_{tr}}\) & \(\mathbf{x_{c}}\) & \(\mathbf{y_{c}}\) & \(\mathbf{b_{0}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{b_{17}}\) \\ \hline Lower & \(-0.036\) & \(0.0009\) & \(0.0009\) & \(0\) & \(0.045\) & \(0.18\) & \(0\) & \(0.135\) & \(0.045\) & \(0\) & \(0.009\) & \(0.09\) & \(0\) & \(0\) & \(0\) \\ Upper & \(-0.003\) & \(0.3\) & \(0.9\) & \(0.03\) & \(0.6\) & \(1.5\) & \(0.6\) & \(1.2\) & \(0.45\) & \(0.003\) & \(1.5\) & \(1.8\) & \(2.1\) & \(2.7\) & \(2.7\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Values of the upper and lower bounds of parameters used in the BP method.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Leading Edge** & \multicolumn{3}{c}{**Trailing Edge**} \\ \hline \(x_{0}=0\) & \(y_{0}=0\) & \(x_{0}=x_{t}\) & \(y_{0}=y_{t}\) \\ \(x_{1}=0\) & \(b_{1}=b_{8}\) & \((\tau_{x_{t}}+9\%_{b}^{2}/2\tau_{le})/4\) & \(y_{1}=y_{t}\) \\ \(x_{2}=\frac{-3\%_{t}^{2}}{2\tau_{x}}\) & \(y_{2}=y_{t}\) & \(x_{2}=3x_{t}+15\%_{b}^{2}/4\tau_{le}\) & \(y_{2}=(y_{t}+b_{8})/2\) \\ \(x_{3}=x_{t}\) & \(y_{3}=y_{t}\) & \(x_{3}=b_{15}\) & \(y_{3}=dz_{te}+(1-b_{15})\tan(\beta_{te})\) \\  & & \(x_{4}=1\) & \(y_{4}=dz_{te}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Thickness profile curve control points definition.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Leading Edge** & \multicolumn{3}{c}{**Trailing Edge**} \\ \hline \(x_{0}=0\) & \(y_{0}=0\) & \(x_{0}=x_{c}\) & \(y_{0}=y_{c}\) \\ \(x_{1}=b_{0}\) & \(y_{1}=b_{0}\tan(\gamma_{te})\) & \(x_{1}=\frac{1}{2}(3x_{c}-y_{c}cot(\gamma_{te}))\) & \(y_{1}=y_{c}\) \\ \(x_{2}=b_{2}\) & \(y_{2}=y_{c}\) &are calculated with a two-equation integral boundary layer formulation as shown in Equations (8) and (9), respectively [76].  
\[\frac{d\theta}{d\xi}+\left(2+H-M_{e}^{2}\right)\frac{\theta}{u_{e}}\frac{du_{e}} {d\xi}=\frac{C_{f}}{2}+\left\{\frac{v_{0}}{u_{e}}\right\} \tag{7}\]  
\[\theta\frac{dH^{*}}{d\xi}+(2H^{**}+H^{*}(1-H))\frac{\theta}{u_{e}}\frac{du_{e} }{d\xi}=2C_{D}-H^{*}\frac{C_{f}}{2}+\left\{(1-H^{*})\;\frac{v_{0}}{u_{e}}\right\} \tag{8}\]  
where shape factors \(H^{*}\) and \(H^{**}\) are defined in the following equations  
\[xH^{*}=\frac{\int_{0}^{\infty}\frac{u}{U}\left(1-\left(\frac{u}{U}\right)^{2} \right)dy}{\int_{0}^{\infty}\frac{u}{U}\left(1-\left(\frac{u}{U}\right)\right) dy} \tag{9}\]  
and  
\[H^{**}=\left(\frac{0.064}{H_{k}-0.8}+0.251\right)M_{e}^{2}\text{ with }H_{k}=\frac{H-0.290M_{e}^{2}}{1+0.113M_{e}^{2}}\]  
At high angles of attack where stall occurs, XFoil has difficulties in giving a converging solution for the airfoil analysis. To obtain the convergence of the solution, non-converged solutions are given a high penalty by setting the fitness function arbitrarily large via a penalty function, and thus they are eventually eliminated during the design process. The solver tolerance value used for this penalty function calculation is 0.002.  
Many researchers have employed XFoil software in morphing airfoil studies. In one of these studies, the results obtained using XFoil were compared with those obtained using the Reynolds-Averaged Navier-Stokes (RANS) solvers for camber morphing, and only a slight difference between the results was observed [77].