Aerofoils: Lift, Drag, and Circulation - Small Wind Turbines: Analysis, Design, and Application - page 62

Figure 4.3 shows that the large change in lift:drag is due to a more gradual
increase in C l and decrease in C d with increasing Re. This trend continues above
Re = 500,000. A review of the large number of wind tunnel measurements of
the NACA 0012 section, McCroskey [ 1 ] suggested the following empirical for-
mulae for the linear part of the lift curve:

dC l = da ¼ 0 : 1025 þ 0 : 00485 log 10 Re = 10 6

ð 4 : 3 Þ

and for the minimum drag coefficient, C d0 :

C d0 ¼ 0 : 0044 þ 0 : 018 Re 0 : 15

ð 4 : 4 Þ

for Re C 500,000 approximately. Unfortunately he did not provide a data corre-
lation for C d (a). From Eq. 4.3 the lift increases and, from ( 4.4 ), drag decreases as
Re increases. It is generally the case, however, that the changes are not great in
regions where relations like ( 4.3 ) and ( 4.4 ) apply. Equations 4.3 and 4.4 were used
in the program LandD_0012.m described in Chap. 3 . Please note that the final
equation in the last part of that program where C d (a) is determined from C d0 ,is
only a rough approximation developed by the author. It is used to complete a very
simple determination of lift and drag so that attention can focus on the other issues
of BET implementation. The next chapter shows that incorporating measured
aerofoil characteristics can seriously increase the size of a blade element program.
From the answers to the Exercises in Chap. 1 , it can be inferred that most large
wind turbine blades operate at Re [ 500,000, at least near the tip, the region where
Chap. 5 shows that most power is produced. However, this Re is the maximum
reached on the 5 kW turbine described in that chapter. From Exercise 1.10, the
minimum operating Re of a typical 500 W turbine is less than 8,400. Thus many
small wind turbines operate at Reynolds numbers below 10 5 , the minimum of the
data in Fig. 4.3 . To understand the effects of low Re, it is important to be clear about
the relationship between the lift and the pressure distribution around an aerofoil.
Recall that there are two forces acting at any point on a two-dimensional body
immersed in a fluid flow: that due to the pressure is normal to the surface and that
due to the shear stress is tangential. The magnitude of the pressure is usually by far
the larger. For an aerofoil, and for most streamlined bodies with no separation, the
pressure does not contribute significantly to the drag. This is why the lift can be much
higher than drag as seen in Fig. 4.3 c. As the Re decreases, the boundary layers
attached to the aerofoil surface increase in thickness and so change the effective
shape of the aerofoil in a way that decreases lift and increases drag.
Figure 4.4 shows the computed surface pressures on a NACA0012 aerofoil at
a = 0,4 , and 8 obtained using the Matlab program Pablo. 1 C p , the pressure
coefficient, is defined as the gauge pressure, P - P 0 (where the latter is the
free-stream static pressure) at the position x along the chord defined in Figs. 4.1
and 4.2 , divided by the free-stream dynamic pressure:

1

Can be downloaded from: http://www.nada.kth.se/*chris/pablo/ (accessed 4 Mar 2010).