# Design and Testing of a LUT Airfoil for Straight-Baded Vertical Axis Wind Turbines
## 2 Basis and Method of Design for the LUT Airfoil
### LUT Airfoil Optimization and Design Method
\begin{array}{cc}B^{k}&-\
abla c\Big{(}\mathrm{X}^{k}\Big{)}\\ -\
abla c\Big{(}\mathrm{X}^{k}\Big{)}^{T}&0\end{array} \right]\left[ \begin{array} []{c}\mathrm{d}\\ \lambda\end{array} \right]=\left[ \begin{array}{c}-\
abla f\Big{(}\mathrm{X}^ {k}\Big{)}\\ c\Big{(}\mathrm{X}^{k}\Big{)}\end{array} \right] \tag{9}\]  
This equation is the exact K-T condition of the quadratic programming problem:  
\[\left\{ \begin{array}{cc}Min&\frac{1}{2}\mathrm{d}^{T}B^{k}\mathrm{d}+\
abla f \Big{(}\mathrm{X}^{k}\Big{)}^{T}\mathrm{d}\\ s.t.&\
abla c\Big{(}\mathrm{X}^{k}\Big{)}^{T}\mathrm{d}+c\Big{(}\mathrm{X}^{k} \Big{)}=0\end{array} \right. \tag{10}\]  
where the vector \(\mathrm{d}\) represents the search direction, and \(B^{k}\) is the approximation of the Hessen matrix. In this way, a common constraint problem is transformed into a quadratic programming problem. Moreover, on this basis, it can be naturally extended to inequality constraints:\[\left\{ \begin{array}{ll}Min&\frac{1}{2}\mathrm{d}^{T}B^{k}\mathrm{d}+\
abla f \left(\mathrm{X}^{k}\right)^{T}\mathrm{d}\\ &\
abla c\left(\mathrm{X}^{k}\right)^{T}\mathrm{d}+c\left(\mathrm{X}^{k}\right) =0\\ s.t.&\
abla c\left(\mathrm{X}^{k}\right)^{T}\mathrm{d}+c\left(\mathrm{X}^{k} \right)\geq 0\end{array} \right. \tag{11}\]  
The above SQP method is only locally convergent in theory. To converge to the optimal solution, a one-dimensional search is required to ensure its overall convergence. The search direction \(\mathrm{d}\) is obtained by Equation (11), which is the downward direction of many penalty functions. For example:  
\[F_{\sigma}(\mathrm{X})=f(\mathrm{X})+\sigma\left[\sum_{i=1}^{m_{e}}|c_{i}( \mathrm{X})|+\sum_{i=m_{e}+1}^{m}|min\left\{c_{i}(\mathrm{X}),0\right\}|\right] \tag{12}\]  
where \(\sigma\) is the penalty factor that satisfies \(\sigma>\max\{|\lambda_{i}|\ i=1,2,\cdots,m\}\). A one-dimensional search based on \(F_{\sigma}\left(\mathrm{X}^{k},\mathrm{d}\right)<0\) is carried out along with the \(\mathrm{d}\) direction, and then the step length \(t^{k}\) that satisfies \(F_{\sigma}\left(\mathrm{X}^{k}+t^{k}\mathrm{d}\right)<F_{\sigma}\left(\mathrm{ X}^{k}\right)\) is obtained. Meanwhile, the next iteration point can be obtained:  
\[\mathrm{X}^{k+1}=\mathrm{X}^{k}+t^{k}\mathrm{d}^{k} \tag{13}\]  
The flowchart describing the optimization of the airfoil's aerodynamic shape based on the SQP method is shown in Appendix A.  
Additionally, the aerodynamic performance of the LUT airfoil is obtained under free transition by employing ANSYS Fluent code during the process of optimization; the numerical solution of the Reynolds-averaged Navier-Stokes (RANS) equation is used, and the Shear-Stress-Transport (\(SST\)) \(k\)-\(\omega\) turbulence model is employed [27]. Pressure-velocity coupling is achieved by the SIMPLEC scheme, where the second-order upwind discretization scheme is utilized to calculate the pressure, and the second-order implicit formula is adopted to test temporal discretization. Figure 2a demonstrates the geometric scheme and the boundary conditions of the numerical simulation. The walls of the airfoil are set as non-slip walls, while the other walls are set as free-slip walls. The O-grids are positioned near the airfoil, and the first grid's spacing is less than \(1\times 10^{-5}\) to ensure that the \(\mathrm{Y}^{+}\) is less than \(1\). The design results are introduced in the results and discussion section. This study required a LUT airfoil with a good aerodynamic performance; meanwhile, to ensure the structural strength of the blade for the LUT airfoil, the corresponding airfoil thickness and bending  
Figure 2: Scheme of the geometry and boundary conditions for the numerical simulation: (**a**) The semicircular far field is set as a velocity inlet located at 12.5 c (c is chord length) from the trailing edge of the airfoil, while the outlet condition is the pressure outlet located at 20 c from the trailing edge of the airfoil.