Page 176 - Wind Energy Handbook

150 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

(1) A lift force with the centre of pressure at the mid-chord point of an amount
equal to the apparent mass times the acceleration normal to the chord line of the
mid-chord point:

2
1 2 @v e 1 @ â e
L m1 (ô) ¼ ðc r c h (3:211)
4 @t 2 @t 2

(2) A lift force with the centre of pressure at the 3=4 chord point, of the nature of a
centrifugal force, of an amount equal to the apparent mass times W(t)(@â e =@t)
1 2 @â e
L m2 (ô) ¼ ðc rW(t) (3:212)
4 @t
There is also a nose-down pitching moment equal to the apparent moment of inertia
4
(ð=128)c r (which, actually, is only a quarter of the moment of inertia per unit
2
length of the cylinder of air of diameter c) times the pitching acceleration @ â e =@t 2
2
1 4 @ â e
M m ¼ ðrc (3:213)
128 @t 2
The added masses are determined by a process similar to that of Section 3.13.2.

3.13.7 The effect of the wake on aerofoil aerodynamics in unsteady
flow

If the angle of attack of the flow relative to an aerofoil changes, the strength of the
circulation also changes, but the process is not instantaneous because the circulation
can only develop gradually. To determine how the lift on an aerofoil actually
develops with time after an impulsive change of angle of attack occurs it is
necessary to include the wake in the analysis. The sudden change of a causes a
build up of circulation around the aerofoil that is matched by an equal and opposite
vorticity being shed into the wake.
The bound circulation on an aerofoil is actually distributed along the chord but,
for simplicity, can be assumed to be a concentrated vortex ˆ at the aerodynamic
centre 1=4 chord point). In steady flow conditions, the velocity induced by the
vortex, normal to the chord-line, at the 3=4 chord point is exactly equal and opposite
to the component of the flow velocity normal to the chord-line. The two opposed
velocities ensure that no flow passes through the aerofoil at the 3=4 chord point, a
condition which, of course, must be true everywhere along the chord-line but the
3=4 chord point is used as a control point. The simplified situation assumes that the
aerofoil can be represented geometrically by its chord-line, this known as thin
aerofoil representation.
In unsteady flow conditions the velocity induced at the 3=4 chord point (often
referred to as downwash) is caused jointly by the bound vortex and the wake
vorticity, see Figure 3.78, but must still be equal and opposite to the component of
the flow velocity normal to the chord-line.