Performance analysis of wind turbines at low tip-speed ratio using the Betz-Goldstein model

Analyzing wind turbine performance at low tip-speed ratio is challenging due to the relatively high level of swirl in the wake. This work presents a new approach to wind turbine analysis including swirl for any tip-speed ratio. The methodology uses the induced velocity field from vortex theory in the general momentum theory, in the form of the turbine thrust and torque equations. Using the constant bound circulation model of Joukowsky, the swirl velocity becomes infinite on the wake centreline even at high tip-speed ratio. Rankine, Vatistas and Delery vortices were used to regularize the Joukowsky model near the centreline. The new formulation prevents the power coefficient from exceeding the Betz-Joukowsky limit. An alternative calculation, based on the varying circulation for Betz-Goldstein optimized rotors is shown to have the best general behavior. Prandtl's approximation for the tip loss and a recent alternative were employed to account for the effects of a finite number of blades. The Betz-Goldstein model appears to be the only one resistant to vortex breakdown immediately behind the rotor for an infinite number of blades. Furthermore, the dependence of the induced velocity on radius in the Betz-Goldstein model allows the power coefficient to remain below Betz-Joukowsky limit which does not occur for the Joukowsky model at low tip-speed ratio.