Momentum theory of Joukowsky actuator discs with swirl

Abstract

Actuator disc theory is the basis for most rotor design methods, be it 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 Cp → ∞ for Ω → 0. The momentum balance is very sensitive to the choice of the vortex core radius δ as the pressure and velocity gradients become infinite for δ → 0. Viscous vortex cores do not show this singular behaviour so an inviscid core model is sought which removes the momentum balance sensitivity to singular flow. A vortex core with a constant δ does so. Applying this results in Cp → 0 for Ω → 0, instead of Cp → ∞. The Joukowsky actuator disc theory is confirmed by a very good match with the numerically obtained results. It gives higher Cp values than corresponding solutions for discs with a Goldstein-based wake circulation published in literature.