Glauert empirical formula
A wind turbine is expected to slow/retard down a blowing wind and in
this process extracting a portion of its kinetic energy while a propeller/fan
accelerate a wind and transfer a portion of energy to the wind. A Darrieus
wind turbine exhibits both behaviour of retarding and accelerating the
wind speed. At the wind assisted side where the airfoil moves nearly parallel
and in direction of wind speed (i.e. 180degree), the faster moving symmetry
airfoil actually ‘drags’ the wind to move faster. However,
for the rest of the angular position, the Darrieus act normally like a
wind turbine retarding the incoming wind. This retardation is especially
strong around the front (90degree) and around the back (270 degree) position.
When solidity and or tip speed ratio is high, this retardation became
even stronger.
Thrust coefficient versus axial interference factor
The momentum equation predicts that thrust force increase with increasing
retardation (or induction factor) of wind speed. However, after induction
factor of 0.5, the thrust is predicted to decrease, as shown in figure
above, which is not logical. The real situation is actually the thrust
keeps on increasing with more retardation of the wind speed. Data collected
from wind turbines, autogiros, and helicopters (as shown in the figure
above) proves that the thrust coefficient increases with induction factor
in this turbulent wake state instead of decreasing as predicted by momentum
theory. An empirical formula published by Glauert also predicts increasing
thrust coefficient between the induction factor range of 0.4 to 1.0 with
thrust increasing from 0.96 to 2.0. Therefore, the momentum equation is
not applicable after 0.5 or more precisely 0.4.
The streamtube models must be modified to address the failure of momentum
equation at that region. A crude approximation is made by modeling the
Glauert empirical formula with a straight line between the 0.4 to 1.0
induction factor range. It has to be noted that the thrust coefficient
data was very probably collected for the whole rotor swept area so the
thrust coefficient and induction factor was probably an average value
References
Spera, D. A. (Editor) (1994), Wind turbine technology: Fundamental concepts
of wind turbine engineering. New York, ASME Press.
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