Aerodynamics and Design for Ultra-Low Reynolds Number Flight

Chapter 4
4.2.4 Prandtl Tip Loss Correction
Up to this point, the actuator ring equations have assumed constant induced velocities
across any particular annulus. In reality, the presence of a finite number of blades can
result in a circulation distribution that is markedly different from the infinite blade limit.
A simple correction to this assumption is obtained by applying a form of the Prandtl tip
loss factor. This correction is based on a cylindrical vortex helices in the wake.
Contraction of the wake is not considered in this model, but a modification for
contraction effects will be introduced later in this chapter. For details of the
development, the reader is referred to the text by McCormick [26]. Defining the Prandtl
tip loss factor as κ:
This correction is applied to the local bound circulation as modeled by the inviscid
portions of the actuator ring equations (Eqns. 4.9 and 4.10):
4.2.5 Swirl Velocity Considerations
54
BΓ actual
=
κΓ ∞blades
κ ( 2 ⁄ π)
e f
= acos(
)
⎛ ⎞⎛ ⎞
⎝ ⎠⎝
⎠
f
r
= ( B ⁄ 2)
1 – --
R
---------------
1
sin φtip dT = ( 2κρu( u+ U∞) ( 2πr)dr)
– ( BC ( d ⁄ Cl) ( u+ U∞)ρΓdr) dQ =
( 2κρv( u+ U∞) ( 2πr )rdr) + ( BC ( d ⁄ Cl) ( Ωr – v)ρΓrdr)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
The formulation, at this stage, incorporates only the inviscid induced tangential velocity,
also referred to as inviscid swirl. In most conventional large scale, high Reynolds
number applications, this is sufficient. For the small scale, very low Reynolds number