Page 177 - Wind Energy Handbook

UNSTEADY FLOW – DYNAMIC INFLOW 151

1
c
4 Γ(t) Downwash
3
c w(t)
4
α(t) Continuous shed vorticity Starting vortex
W
W
–dΓ(t)
Wsinα(t)
dt
Figure 3.78 Wake Development after an Impulsive Change of Angle of Attack

After the impulsive change of angle of attack there is a sudden change in upwash
(W sin Æ) which must be matched by a sudden change of downwash. The change of
upwash implies an impulsive acceleration of the mass of the air that causes an
added mass force on the aerofoil. The sudden change of downwash must come
from a sudden increase in circulation that must be matched by an equal and
opposite starting vortex being shed into the wake and then convected downstream.
The influence of the starting vortex on the downwash gets gradually weaker as the
vortex moves away so the bound vortex must increase in strength to maintain that
the downwash matches the upwash. The increasing strength of the bound vortex
means that, to keep the overall angular momentum contained in the vorticity zero
(there was none before the impulse), continuous vorticity of the opposite sense
must be shed into the wake and this also contributes to the downwash.
The rate of increase of the bound vortex strength gradually reduces with a
corresponding reduction of the strength of the shed vorticity and eventually,
asymptotically, the steady-state bound circulation strength is developed.
The analytical solution to the problem was developed by Wagner (1925); it is
complex and expressed in terms of Bessel functions but several approximations to
the Wagner function exist, the most accurate of which is given by Jones (1945).

L c (ô) 0:0455ô 0:30ô
¼ Ö(ô) ¼ 1 0:165 e 0:335 e (3:214)
1 2 dC l
rW c sin(Æ)
2 dÆ
where ô ¼ tc=2W is the non-dimensional time based upon the half chord length c=2
of the aerofoil and dC l =dÆ is the slope of the static lift versus angle of attack
characteristic of the aerofoil. ô can also be regarded as the number of half-chord
lengths travelled downstream by the starting vortex after a time t has elapsed since
the impulsive change of angle of attack. Equation (3.214) is an example of an
indicial equation.
Figure 3.79 shows the progression of the growth of the lift as time proceeds from
the original impulsive change of angle of attack. The added mass lift gradually dies
away as the circulatory lift develops. Eventually, the steady state, full circulatory lift
is achieved. In the situation where the angle of attack is continuously changing,
which is the case for the wind turbine blade, the circulation never reaches an
equilibrium state and the added mass lift never dies away.