Aerodynamics and Design for Ultra-Low Reynolds Number Flight

Chapter 4
Beyond the assumptions intrinsic in blade element / actuator ring theory, the only
additional assumptions are that lift is inviscid and plays no direct role in the viscous
swirl and any tip loss/wake corrections are neglected in the substitution of the thrust
equation. There are no small angle approximations and the simplicity of the final form is
due to exact cancellation. Viscous swirl as defined here also incorporates the pressure
drag of the section since both are included in the section drag coefficient. The versatility
of this model appears to extend reasonably well to the ultra-low Reynolds number
regime.
4.4 Development of Stream-Function-Based Vortex
Ring Wake Model
Blade-element theory and actuator ring theory alone provide a simple model for the
wake and its effects on the rotor. They do not account for any effects of discrete vorticity
in the wake due to a finite blade count, instead assuming the wake is composed of
continuously shed stream-wise vorticity along stream-tubes. The physical analog is the
presence of an infinite number of blades. This model typically overestimates the lift
generated near the blade tips.
The Prandtl tip loss correction described earlier is a significant improvement. It is based
upon helical vortices of constant strength and diameter emanating from each blade tip.
The vertical component of the shed vorticity is neglected and the wake model reduces to
a semi-infinite column of vortex rings. The spacing of the rings is determined from the
blade spacing and the wake tip helix angle, assuming uniform down-wash. From this
potential flow model, the vorticity distribution on the blade is determined and expressed
as a correction (κ) to the infinite blade solution. This model is well suited for lightly
loaded rotors and rotors with large advance ratios. In these two cases, the assumption of
a cylindrical wake is reasonable. The tendency of the helical wake to contract as it
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