Aerofoils: Lift, Drag, and Circulation - Small Wind Turbines: Analysis, Design, and Application - page 71

solely from the change in velocity; it was assumed that the starting velocity (for a
stationary blade) is 3 m/s, and the rated speed is 10 m/s. At the latter speed, the tip
speed ratio, k, is around 7, so the tip velocity is over 70 m/s, giving a 20:1 range in
Re. The figure also shows that most commercial aviation occurs at Re [ 10 6 —the
Cessna 180 is a single engine, four-seater aircraft—and that the lower limit of
Re = 10 5 for the measured aerofoil data for the SG6040, SG6041, and SG6043, is
too high for many small wind turbine applications.
Mueller and DeLaurier [ 16 ] explain that aerofoil performance below about
Re = 500,000 is governed primarily by effects of the laminar separation bubble
that forms on the upper surface as mentioned in Sect. 4.3 . (In this general
description of Reynolds number effects, it must be appreciated that the actual
values are not precise and will be shape dependent.) As Re increases from 200,000
to about 500,000, the bubble gets shorter and the drag it causes reduces, leading to
higher l/d. Between about 70,000 and 200,000 it is possible to achieve laminar
flow without a bubble, which can lead to impressive performance, as shown in
Fig. 4.3 for the SG6043 at Re = 200,000. Note that each of the SG6040, SG6041,
and SG6043, suffer from obvious effects of laminar separation bubbles, at their
lowest measured Re. Between 30,000 and 70,000, aerofoil thickness has a direct
influence on bubble formation, which is why thin sections behave better in this
region. Below about 50,000, transition in the separated flow may not occur before
the trailing edge and there is no reattachment.
It is again emphasised that aerofoils are two-dimensional and that the con-
version of aerofoil lift and drag (in N/m per unit width) into actual forces (in N) on
an aerodynamic body involves considering the three-dimensional effects, which
usually scale with the aspect ratio, AR, defined as the planform area of a blade or
wing divided by its span. For a rectangular wing of span b and chord c,
AR = b/c. Thus AR ? ? is the aerofoil ''limit''. Most basic aeronautics texts,
such as Bertin and Cummings [ 19 ], give the following formula for the total drag on
a rectangular (or similarly shaped) wing:

C L
p A ð e

C D ¼ C D0 þ

ð 4 : 13 Þ

where the use of capital letters for the subscripts emphasise that the lift and drag
are of three-dimensional bodies. C D0 is the aerofoil drag converted into a three-
dimensional value by multiplying both the drag and the denominator of Eq. 4.1 by
the span, and the second term on the right is often called the induced drag. e is the
''span efficiency factor'' and has a value of 0.8-0.9 for most wings. For typical C L
values of around unity, the induced drag dominates at sufficiently low AR. How-
ever, AR for most wind turbine blades is in the range 10-30, and the calculations of
blade circulation to be discussed in Chap. 5 , suggest a much more uniform dis-
tribution than the elliptical loading on wings that underlies Eq. 4.13 . This equation
probably over-estimates AR effects for wind turbine blades, and is not used.
One area where AR effects can be important is at high incidence. As demon-
strated in Sect. 4.4 , aerofoils should behave as a thin flat plate when a ? 90