Aerofoils: Lift, Drag, and Circulation - Small Wind Turbines: Analysis, Design, and Application - page 67

Fig. 4.7 Relationship
between C l and C d at high
angles and Reynolds number
below 2 9 10 5 . Data sources
listed in Table 4.3

1.5

1

0.5

0

20 30 40 50 60 70 80 90

angle of attack ( ° )

Fig. 4.8 Circulation around
a cascade of blade elements
at radius r. The four legs of
the rectangular circuit used to
determine the circulation are
numbered

U 0 or U T

2 r/Nc

π

2

2 π

r/Nc

1 3

4

aerofoil produces only lift, which must have a component up the page to produce
torque to rotate the blades. To determine the blade circulation, C, the clockwise
circuit around the rectangle in Fig. 4.8 is divided into four legs. The analysis is
made easier if the radius r is also the radius of the streamtube intersecting that
element. As a result, legs 1 and 3 must be close to the blade.
The circulation is defined for any closed contour by the line integral of the
velocity:

C ¼ I

U dl ð 4 : 8 Þ

where U is the velocity vector, and d l is the increment along the curve, and has the
units of velocity 9 length or m 2 /s in the SI system. Thus the circulation around a
circular contour downstream of the blades and centred on the turbine's axis (in a
plane parallel to the rotor disk) is 2prW where W is the circumferential velocity.
This shows the close connection between C and angular momentum. One of the
other very useful properties of C is that by Gauss' theorem it is equal to the area
integral of the vorticity within the contour, which is normally confined to the
boundary layers on the upper (suction) and lower (pressure) surfaces of the blade.