The lift coefficient and Reynolds number are usually the main constraints in the aerodynamic platform design during the design process. This work presents the airfoil shape equations, which achieve the best lift-drag ratio fulfilling specific lift coefficient and Reynolds number targets. The particle swarm optimization method is implemented and coupled with XFOIL and the open-source CFD OpenFOAM to optimize the airfoil shape parameterized by the NACA 4-digit equations. Several Optimizations with lift coefficients from 0.2 to 1.8 and Reynolds numbers from 80,000 to 500,000 are employed to be fitted by polynomial regressions, describing the best airfoil shape given the target lift coefficient and Reynolds number condition. The obtained analytical expressions are helpful for straightforwardly getting the optimal airfoil shape during a design process. The results provide insights into the thickness, maximum camber and its position and their relation to aerodynamic efficiency. The fits from XFOIL and CFD are compared and discussed. A linear relationship was found between the target lift coefficient and the maximum camber of the optimal airfoil, in addition to an impact of the target lift coefficient on the maximum achievable aerodynamic efficiency. Finally, the optimal airfoils are compared against the S1223 and E423 airfoils for low and medium Reynolds, showing better aerodynamic characteristics.
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