Aerodynamics and Design for Ultra-Low Reynolds Number Flight

Chapter 7
Similarly the thrust may be approximated by:
For an optimal rotor designed for hover, Gessow and Myers [47] indicate that the
constant C l approximation is reasonable, as the mean lift coefficient across the blade
should be close to the maximum lift coefficient in order to maximize the figure of merit.
The assumptions made here neglect variations in C lmax and C d that may occur with
Reynolds number variations across the blade, but still capture average Reynolds number
effects when comparing different rotors.
Finally, upon substitution of Eqn.7.2, Eqn.7.4, and Eqn.7.5 into Eqn.7.3, the approximate
expression for the rotor figure of merit becomes:
Equivalently, this may be expressed in terms of the thrust coefficient, C T :
The figure of merit can be related to the thrust specific power required for hover and the
disk loading by manipulating Eqn.7.1 and Eqn.7.2 to yield:
From the discussions of Chapter 3, the expected maximum steady state lift coefficients at
the Reynolds numbers seen by centimeter scale rotors are approximately 0.6 to 0.7 over
164
T ρσπCl ωR ( )2R 2
≈ --------------------------------------
6
1
M
3
1 --
2
C ≈ --------------------------------------d
----- ⎛ ⎞
+ --------
3
⎠
⎝ Cl σCl 1
M
3
1 --
2
C ≈ ---------------------------------------d
----- ⎛ ⎞ 1
+ ---------
⎠
⎝ Cl 2CT Preq ---------
T
=
1
----
M
-----
1
2ρ
T
-- ⎞ ⎛
⎠ ⎝A (7.5)
(7.6)
(7.7)
(7.8)