Joukowsky actuator disc momentum theory

Abstract. Actuator disc theory is the basis for most rotor design methods, albeit with many extensions and engineering rules added to make it a well-established method. However, the off-design condition of a very low rotational speed Ω of the disc is still a topic for scientific discussions. Several authors have presented solutions of the associated momentum theory for actuator discs with a constant circulation, the so-called Joukowsky discs, showing the efficiency C_p → ∞ for the tip speed ratio λ → 0. The momentum theory is very sensitive to the choice of the radius δ of the core of the centreline vortex as the pressure and velocity gradients become infinite for δ → 0. Usually the vortex core area is not included in the momentum balance, as it vanishes for δ → 0. However, the pressure in the vortex core behaves as a Delta function and so contributes to the balance, thereby cancelling the singular behaviour. Applying this in the momentum balance results in C_p → 0 for λ → 0, instead of C_p → ∞. The Joukowsky actuator disc theory is confirmed by a very good match with numerically obtained results. At the disc the velocity in the meridian plane is shown to be constant. The Joukowsky calculations give higher C_p values than corresponding solutions for discs with a Goldstein-based wake circulation published in literature.