New Empirical Relationship between Thrust Coefficient and Induction Factor for the Turbulent Windmill State

Wind turbines sometimes experience the turbulent windmill state during startup or shutdown. This rarely happens during normal operation, so it has little effect on power curves or energy production. However, for completeness we need to be able to model situations where the axial induction factor exceeds 0.5. Classical momentum theory, which shows a relationship between the thrust coefficient andthe axial induction factor, is not valid in this region. Glauert plotted some experimental data taken by Lock in the 1920s against this parabolic relationship and found very poor agreement for operation in this high-induction state. He proposed a new empirical relationship to fit the experimental data. Unfortunately, the new empirical curve does not account for tip or hub losses. Others haveproposed multiplying the axial induction factor by the loss factor to correct the curve, but this still leaves a mathematical no-man's-land between the classical curve and the modified version of Glauert's empirical curve. The purpose of this paper is to document the derivation of a new curve that accounts for tip and hub losses and eliminates the numerical problems of the previous approaches.