Analytical and numerical solutions to classical rotor designs

About a century ago, two different rotor models were proposed by Joukowsky (1912-1918) and Betz (1919). Both models were based on assumptions regarding the vortex structures of the wake and its induction on the rotor plane. However, due to complications of formulating the wake behavior into actual guidelines for the aerodynamic design of the rotor geometry, the models have until now not been used to design actual industrial wind turbine rotors. In this article, we propose a technique to solve analytically the induction problem of the two models, which enables the design of wind turbine rotors. We briefly present the theory behind the two rotor models, and show how this theory can be exploited to make actual designs of rotor planforms, i.e. chord-and twist-distributions. The designs are for three-bladed rotors optimized for different tip speed ratios and analyzed by comparing their performance to results using blade-element/momentum technique and lifting line theory.