Aerodynamics and Design for Ultra-Low Reynolds Number Flight

Chapter 4
This equation assumes that the helical wake is convected downstream at the idealized
constant downwash velocity based on actuator disk theory. The ring separation distance
is equivalent to the resulting pitch of the helix. This is consistent with the model
implemented for the Prandtl tip loss correction due to a cylindrical wake as described by
McCormick [26].
The Prandtl tip loss model depends upon the choice of a tip helix angle. This is based on
the radius, rotation rate, and induced velocities at the tip. This parameter could be
determined rigorously from iterative solution of the rotor and wake models, but for an
optimal hovering rotor it does not vary considerable from the actuator disk value. The
iteration required would also result in a significant increase in the computational cost.
Given the approximate nature of the governing wake model, the worth of this improved
fidelity is questionable. The additional computational cost would likely be better spent
on a more complex wake model such as a helical filament model.
The contracted wake is obtained iteratively by calculating the mass flux through each
ring due to the entire wake structure and resizing the ring radius in proportion to the flux
ratio of the rotor disk and the wake ring. A great advantage of the streamline
formulation is that the mass flux through any axisymmetric circle due to a vortex ring
can be directly calculated:
66
(4.42)
After calculating the flux through all rings including the rotor disk, the rings are resized
according to:
r new
S = – 2πψ(
x, r)
=
rold 1 +
Srotor – Sring ------------------------------
S rotor
⎝ ⎠
⎛ ⎞
(4.43)
The resized ring models the horizontal component of a helical filament with a modified
pitch. The helical filament must have a constant strength, so the strength of the
horizontal ring is modified by the ratio of the cosine of the local pitch angle to the cosine