Optimal performance of actuator disc models for horizontal-axis turbines

This study analyzes actuator disc (AD) models of horizontal-axis turbines to determine optimal performance, defined as the maximum power extracted at any tip speed ratio. We use the calculus of variations to maximize rotor torque relative to the thrust without making any assumptions about the rotor loading. The torque was obtained from the angular momentum equation and the thrust from the Kutta-Joukowsky equation which depends on the circumferential velocity and tip speed ratio. The optimality requirement is that the pitch of the vorticity exiting the rotor must be constant across the wake and equal to the ratio of torque to thrust. This result generalizes the classical finding of Betz and Goldstein that optimal lightly-loaded ADs have constant pitch. Optimizing the torque in the far-wake, well downstream of the rotor, leads to the same requirement of constant pitch. This implies that the pitch of an optimal rotor is constant everywhere in the wake at all tip speed ratios. We show that it is not possible for the pitch to reach its optimal value because of the vorticity distribution in the wake, and propose modifications to the pitch at the rotor and in the far-wake. The axial and circumferential velocities in the far-wake, which are easily determined, were used to find those at the rotor from the “disc loading equation” for the angular momentum which is also the normalized bound circulation at the rotor. For the simplest case of a lightly-loaded rotor at zero tip speed ratio, the induced circumferential velocity is linear in radius and the axial component is quadratic, As the tip speed ratio increases, the optimal power and thrust asymptote to the familiar Betz-Joukowsky values, and the induced axial velocity and rotor bound circulation become constant. At low tip speed ratios, the optimal wakes are constrained by the need to avoid breakdown of the flow at high swirl, and the conventional thrust equation, involving the axial velocity only, is inaccurate. As found in previous studies, the power coefficient increases monotonically with tip speed ratio, but the thrust coefficient reaches a maximum value slightly above the Betz-Joukowsky limit at a tip speed ratio of two, before decreasing towards the limit.