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.
Highlights
Introduction the concept of the actuator disc is more than 100 years old, it is still the basis for rotor design codes using the blade element momentum theory developed over these 100 years, see [1]
In recent years the behaviour of actuator disc flows with a low rotational speed has been studied by several authors, providing several solutions depending on the type of load that is applied, see e.g. [2]
Research has focussed on rotors and discs having a constant circulation in the wake, known as the Joukowsky distribution [3], or the Betz distribution [4] yielding a helicoidal wake structure moving with a uniform axial velocity
Summary
The concept of the actuator disc is more than 100 years old, it is still the basis for rotor design codes using the blade element momentum theory developed over these 100 years, see [1]. The analysis shows that adding a disturbance parameter to the momentum balance removes the non-physical result of infinite efficiency for zero rotational speed, no matter how small this disturbance is. This is an indication that the momentum balance is very sensitive to small deviations in the flow parameters. A failed attempt to reproduce the results of [14] by the potential flow actuator disc code described in [15] initiated a re-analysis of the vortex core model and its impact on the momentum theory. The sign conventions are that the rotational speed Ω > 0 and Γ < 0 so ∆H < 0 implying that energy is extracted from the flow
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