Environmental Engineering Reference
In-Depth Information
lift approaches zero near 90 deg. Like a flat plate, the drag coefficients in Figure 6-2(b)
show a significant effect of
aspect ratio
(span length divided by chord width), with the low
aspect ratio case significantly lower in drag than the two-dimensional (
i.e.
,
infinite length)
situation.
It is further noted that the post-stall flow over the airfoils of an actual rotor experiences
two additional fluid-dynamic effects that are not represented in typical wind-tunnel tests.
One is the effect of
spanwise flow
caused by rotational effects. Generally, this will produce
a flow towards the blade tip, and the normal expectation is that this postpones stall nearer
the axis of rotation. Swept aircraft wings show a similar effect, with the spanwise
component of the flow causing the wing tips to stall prematurely. No rational quantitative
analysis is available to account for this. The second difference relates to the generally
non-uniform spanwise loading
on an actual rotor blade. A spanwise-constant lift coefficient
is seldom achieved, so stall will develop differently at different radial stations along the
blade. Again, no acceptable procedure is available to handle the influence of spanwise
loading variations on stall. The usual procedure is to assume each section behaves
independently, which effectively ignores spanwise interaction of sections.
Aspect Ratio Effects
The data in Figure 6-2 clearly indicate that lift and drag characteristics show a
significant aspect-ratio dependence at angles of attack larger than 30 deg. In the fully-
attached regime, airfoil section characteristics are not greatly affected by aspect ratio, so that
two-dimensional (
i.e.
infinite aspect ratio) data can be used in predicting performance at low
angles of attack. However, when two-dimensional data are used, a
tip-loss factor
must be
added, as described in Equations (5-35) and (5-36).
The size of the aspect-ratio effects on airfoil coefficients in the attached regime can be
estimated using the classical equations for correcting wind tunnel test data measured on a
finite-span airfoil, from the work of Munk, Glauert, and Prandtl [Jacobs and Abbot 1932].
In this case we are using these equations in reverse, starting from infinite-span data and
obtaining lift and drag curves for a finite aspect ratio. These formulas are as follows:
C
L
=
C
L
¢
(6-3a)
C
D
=
C
D
+
C
L
pm
¢
(6-3b)
a = a +
57.3
C
L
pm
¢
(6-3c)
where
C
L
¢,
C
D
¢= lift and drag coefficients for an infinite aspect ratio
C
L
,
C
D
= lift and drag coefficients for a finite aspect ratio
μ = aspect ratio
In Equations (6-3b) and (6-3c), minor corrections for the shape of the pressure distri-
bution on the airfoil (rectangular
vs.
elliptical) have been eliminated for convenience.
Inspection of these equations shows that finite length increases the angle of attack and the
drag coefficient for a given lift coefficient. Conversely, the lift coefficient is reduced for
the same angle of attack. From Figure 6-2, it is found that stall occurs at a lower lift
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