# Gradient-Based Aerodynamic Optimization of an Airfoil with Morphing Leading and Trailing Edges
## 3 Results and Discussion
### Mechanism of Drag Reduction  
Table 3 presents the drag breakdown, in which \(\delta(\cdot)\) is the change of the component with respect to that of baseline airfoil. By comparing the contribution of drag reduction of friction and pressure drag, we understand that the latter contributes the most drag reduction and the former has a slightly adverse effect. Note that, although the data in Table 3 are obtained based on original mesh, the finer meshes are used for illustrating the lambda shock pattern in Figure 14. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Name**} & \multirow{2}{*}{**Mesh**} & **AOA** & \multirow{2}{*}{**C\({}_{\text{d}}\) (Counts)**} & \multirow{2}{*}{**C\({}_{\text{df}}\) (Counts)**} & \multirow{2}{*}{**C\({}_{\text{dp}}\)** (Counts)**} \\  & & **(Degrees)** & & & \\ \hline Baseline & 192 \(\times\) 100 & 2.286 & 129.38 & 57.00 & 72.38 \\ Baseline & 300 \(\times\) 150 & 2.248 & 126.88 & 56.31 & 70.57 \\ Baseline & 400 \(\times\) 200 & 2.202 & 125.39 & 56.52 & 68.87 \\ \hline mLETE & 192 \(\times\) 100 & 2.162 & 105.97 & 57.69 & 48.29 \\ mLETE & 300 \(\times\) 150 & 2.193 & 105.50 & 54.35 & 51.15 \\ mLETE & 400 \(\times\) 200 & 2.148 & 103.70 & 54.88 & 48.82 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Result verification with finer meshes. Figure 13: Flow structure with original mesh (192 \(\times\) 100) and finer mesh (400 \(\times\) 200). (**a**) Baseline mesh 192 \(\times\) 100. (**b**) Baseline mesh 400 \(\times\) 200. (**c**) mLETE mesh 192 \(\times\) 100. (**d**) mLETE mesh 400 \(\times\) 200. Figure 14 demonstrates the region of the supersonic flow of baseline and optimized airfoils. There are two types of supersonic region shape: a single hump shape with a strong shock at around \(55\%\) chord that belongs to baseline and optimized mTE airfoil; a three-hump-like shape and lambda shock patterns for optimized mLE and mLETE. That phenomenon suggests the mechanisms behind the drag reduction of using mLE and mTE are different. The optimized mTE is more like an improvement of supercritical airfoil, with a relatively flat top and a larger positive camber on the rear 25 percent of the airfoil. A flatter upper skin leads to a sluggish acceleration of the flow speed, extending the chord-wise length of the supersonic region. Furthermore, the extent of the supersonic flow is closer to the surface, the local supersonic Mach number is lower, and the terminating shock wave is weaker, thereby creating less drag. Reference [39] also reported that mTE was able to improve supercritical airfoil. Similar trends can be seen by comparing the \(C_{p}\) distributions for the optimized airfoil with baseline airfoil in Figure 10.  
One interesting finding is the appearance of the lambda shock patterns, which indicates the interaction between shock and boundary layers. As illustrated in Figure 14, there  
Figure 14: Mach contour of baseline and optimized airfoils; please notice the lambda shock in the supersonic region and separation points. (**a**) Mach flow. (**b**) Leading edge. (**c**) Separation point 1. (**d**) Separation point 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Name** & \(\mathbf{C_{df}}\) & \(\mathbf{\Delta C_{df}}\) & \(\mathbf{C_{dp}}\) & \(\mathbf{\Delta C_{dp}}\) & \(\mathbf{C_{d}}\) & \(\mathbf{\Delta C_{d}}\) \\ \hline Baseline & \(57.00\) & & \(72.38\) & & \(129.38\) & \\ mLE & \(57.46\) & \(0.46\) & \(51.29\) & \(-20.63\) & \(108.74\) & \(-20.63\) \\ mTE & \(57.95\) & \(0.95\) & \(62.26\) & \(-9.16\) & \(120.21\) & \(-9.16\) \\ mLETE & \(57.69\) & \(0.69\) & \(48.29\) & \(-23.40\) & \(105.97\) & \(-23.40\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Drag breakdown (counts). are two lambda shocks and one normal shock at the end. The maximum Mach number is \(\mathit{Mach}=1.41\), which is observable on around 5% of the leading edge. The resulting first shock is much stronger than subsequent shocks, and produces a lower Mach number in the mainstream behind it.