## 2 Methodology
The viscous-inviscid flow solver used for the prediction of \(\text{C}_{\text{L}}\) and \(\text{C}_{\text{D}}\) polars of the designed shapes during the optimization procedure is the in-house Foil2w code [9]. It is a low-cost viscous-inviscid interaction model, formulated on the basis of a standard panelmethod combined with a vortex blob approximation of the wake, in which the \"equivalent inviscid flow (EIF)\" velocity \(\overrightarrow{\mathbf{u}}_{\mathbf{e}}\) is expressed in the form:  
\[\overrightarrow{\mathbf{u}}_{\mathbf{e}}=\overrightarrow{\mathbf{u}}^{0}+ \overrightarrow{\mathbf{u}}^{*} \tag{1}\]  
where \(\overrightarrow{\mathbf{u}}^{0}\) is the velocity of the purely potential flow and \(\overrightarrow{\mathbf{u}}^{*}\) is the correction term that needs to be added in order to account for viscous effects. The potential flow part \(\overrightarrow{\mathbf{u}}^{0}\) is simulated on the basis of a standard panel formulation by distributing singularities along the airfoil geometry (piecewise constant sources and constant vorticity), while the wake is represented by free vortex particles, which are convected with the local flow velocity. The modeling of the viscous correction part \(\overrightarrow{\mathbf{u}}^{*}\) is also based on an integral representation (integration over the airfoil \(S_{\text{B}}\) and the wake \(S_{\text{W}}\)):  
\[\overrightarrow{\mathbf{u}}^{*}(\overrightarrow{x};\mathbf{t})=\frac{1}{2\pi }\!\!\int\limits_{S_{\text{B}}}\!\left(\overrightarrow{\mathbf{u}}_{\mathbf{ n}}^{*}(\overrightarrow{s};\mathbf{t})-\overrightarrow{\mathbf{u}}_{\mathbf{ \tau}}^{*}(\overrightarrow{s};\mathbf{t})\right.\overrightarrow{\mathbf{k}} \times\right)\frac{\overrightarrow{\mathbf{\tau}}}{\mathbf{\tau}^{2}}\, \text{ds}(\overrightarrow{s})+\frac{1}{2\pi}\!\!\int\limits_{S_{\text{W}}} \!\left[[\overrightarrow{\mathbf{u}}_{\mathbf{n}}^{*}]]_{\text{w}}( \overrightarrow{s};\mathbf{t})\frac{\overrightarrow{\mathbf{\tau}}}{\mathbf{ \tau}^{2}}\,\text{ds}(\overrightarrow{s}) \tag{2}\]  
where \(\overrightarrow{\mathbf{u}}_{\mathbf{n}}^{*}\) is the normal correction velocity on \(S_{\text{B}}\) and \([[\overrightarrow{\mathbf{u}}_{\mathbf{n}}^{*}]]\) the normal correction velocity jump on \(S_{\text{W}}\), while \(\overrightarrow{\mathbf{u}}_{\mathbf{\tau}}^{*}\) is the tangential correction velocity on \(S_{\text{B}}\) (note that tangential correction velocity jump in the wake is omitted). The viscous flow solution (the integral boundary layer quantities, displacement thickness, and momentum thickness) is obtained by solving the unsteady integral boundary layer equations as formulated in steady-state form by [PERSON] and [PERSON] [17] and re-casted in their unsteady form by [PERSON] and [PERSON] [9]. The viscous-inviscid interaction coupling is achieved through the normal to the wall and the wake shear layer transpiration velocity distribution that represents the mass flow difference between the real viscous flow and the equivalent inviscid flow within the boundary layer, given by:  
\[\overrightarrow{\mathbf{u}}_{\mathbf{n}}^{*}=\frac{\text{d}}{\text{ds}}( \mathbf{u}_{\mathbf{e}},\pi\mathbf{\xi}^{*})\text{ and }[[\overrightarrow{\mathbf{u}}_{\mathbf{n}}^{*}]]_{\text{w}}=\frac{\text{d}}{ \text{ds}}(\mathbf{u}_{\mathbf{e}},\pi\mathbf{\xi}^{*})^{\text{upper}}+\frac{ \text{d}}{\text{ds}}(\mathbf{u}_{\mathbf{e}},\pi\mathbf{\xi}^{*})^{\text{lower}} \tag{3}\]  
As seen in Equation (3), in the wake, the transpiration velocity jump is calculated by summing up the mass deficit of the upper and lower side of the wake shear layer. The boundary layer solution is supplemented by a transition prediction model based on the \(\mathbf{e}^{\text{N}}\) spatial amplification theory [18]. It is noted that Foil2w has been thoroughly validated against measurements on an airfoil undergoing harmonic pitching motion [9]. The flow chart of the optimization process is illustrated in Figure 1. Random combinations (ensembles) of control points are generated and create Bezier curves that approximate the LE and TE regions.