On the simple actuator disk

Abstract

The standard textbook model of a helicopter rotor in vertical translation, a disk loaded with a uniform pressure jump in inviscid fluid, is revisited in search of correct descriptions of the far-field velocity and of the vortex sheet, allowing a rigorous control-volume analysis. The translation rate is not required to be large compared with the induced velocity. The classical results for induced power are unchanged, and now have a strong foundation: they are exact within the steady inviscid problem statement, instead of depending on a quasi-one-dimensional approximation as in the literature. Conversely, even with a uniform pressure jump the induced velocity is far from uniform over the disk, again in conflict with common beliefs and with any quasi-one-dimensional argument: the flow is upwards near the rim, both inside and outside it. The cross-section of the vortex sheet probably begins with a 45° spiral, as opposed to the smooth funnel shape that has been sketched, in the literature and below. A viscous numerical solution supports this conjecture. Plausible boundaries between the translation rates that produce the two ‘clean’ streamtube flow types, namely climb/hover and rapid descent, and those in-between that produce the vortex-ring state are also discussed.