Ideal wind turbine performance at very low tip speed ratio

Abstract

At very low tip speed ratios, wind turbine rotors behave similarly to stationary wings for which the well-known lifting line analysis gives the optimal loading. Lifting line analysis is applied to a stationary rotor of N blades for N = 1, 2, and 4. Analytic (for N = 1 and 2) or semi-analytic solutions (for N = 4) agree with the classical results obtained by conformal mapping. The present solutions compare well with numerical solutions for the Goldstein function for optimally loaded propeller or wind turbine rotors at low tip speed ratio. In all cases, the induced velocity is linear with radius. Assuming this result applies for all N, lifting line analysis is recast as a singular integral equation whose solution agrees with Goldstein's obtained using the same conformal mappings as in his general analysis for any tip speed ratio. The implication is that the assumed induced velocity distribution is correct, and is, therefore, fundamentally different from that at high tip speed ratios when the induced velocity is inversely proportional to radius. For any N, the power and thrust coefficients become proportional to the square of the tip speed ratio and the tip loss factor alters significantly from the common form used at higher tip speed ratios. Extending the analysis to finite pitch was not achieved but several important results were obtained, including the behaviour of the tip loss as N varies at low tip speed ratio. The behaviour is complex and the resulting tip loss factor exceeds unity for a significant part of the blade.