Aerodynamics and Design for Ultra-Low Reynolds Number Flight

Chapter 4
modeled as a Gaussian distribution that varies with Reynolds number, distance aft from
the trailing edge, and distance above the trailing edge. This distribution is translated
downward based upon the local helix angle and blade spacing. The viscous swirl
velocity is then taken as the value of the Gaussian distribution at the intersection of the
translated profile and the next blade’s leading edge. The velocity distribution takes the
form:
There are several common problems with both of these models. Both treat each set of
62
----u
=
U ∞
umid ---------
U∞ (4.32)
leading and trailing blade sections as if isolated from the rest of the rotor and operating
in a uniform free-stream. Applying either model only once cannot account for the
coupled effect of each section on the total rotor system, but the models are unstable if
applied iteratively in an attempt to account for the other blades. The additive effects
continuously reduce the Reynolds number, increasing the viscous swirl component.
Neither model can account for the combined effects of viscous entrainment and rotor
downwash. The first model assumes no lift on the rotor, emulating the viscous
properties of a spinning solid disk. The Gaussian distribution model incorporates the
downwash, but since each pair of sections is treated in isolation there is no rotational
flow entrainment permitted ahead of the leading blade (above the rotor). The sharp roll-
off of the wake deficit velocity also makes the Gaussian model highly sensitive to the
prescribed induced velocities. Once again, this model has proven to be unsatisfactory for
reasons to be discussed in Chapter 6.
( y – µ ) 2
– ⎞
exp ------------------ ⎛
⎠ ⎝ 2σ
where σ is the standard deviation of the wake deficit and µ mean u ⎞ ⎛
⎝ ⎠
=
-------
U ∞