


Aerodynamics
of Wind Turbines: Drag
Both aircraft engineers who build aircraft wings and propellers,
and wind turbine engineers who design rotor blades are concerned
with aerodynamic drag. Drag is always
important when an object moves rapidly through the air. Aircraft
should have good fuel economy, and wind turbine rotor blades
must have high tip speeds to work efficiently. Therefore it is
critical, that both aircraft wings and rotor blades have low
aerodynamic drag.
Drag
The black
arrow above the picture shows the direction of the drag force,
when the cross section of the airplane wing or rotor blade is
moving towards the right.


Drag
Increases with the Area Facing the Wind
Drag increases in proportion to the frontal area of an
object facing the wind. The frontal area is the area of the hole
in a wall if the object were thrown through a wall in a cartoon.
It is a good idea to design objects that have to move quickly
through fluids with low energy use to have small cross sections
facing the current. That is the reason why submarines, missiles,
cars, or bicyclists' helmets often are built using elongated
drop shapes.
Drag
Depends on the Shape of the Object

A
Parachute shape has a very high drag coefficient of 1.42 
Different shapes
have very different drag. The size of the drag for a given shape
is usually measured by the drag coefficient, C_{D}, which is defined as the drag force
per square metre frontal area of the object. *)
The picture shows a hollow hemisphere open
towards the wind, just like the cup of a cup anemometer
or like a parachute. Such a shape has a very high C_{D} of 1.42, whereas its C_{D}
is only 0.38 if you turn it 180° around a vertical axis.
A modern automobile typically has a C_{D}
in the range 0.270.35. A runner has a C_{D}
about 0.5, a racing bicyclist about 0.4. An airfoil shape used
on aircraft wings or rotor blades, typically has an extremely
small C_{D} about 0.04. **)
*) For aircaft airfoils the drag coefficient
is usually defined relative to the wing area.
**) The drag coefficients are valid for Reynolds
numbers above 10^{}4, a subject which is further discussed at the
bottom of this page.


Drag
Increases with the Square of the Wind Speed
Both lift and drag increase with the square
of the wind speed.
The reason why drag is much more of a problem
for a racing bicyclist than for, say, a runner, is that a fast
runner will be moving at speed of some 6 m/s (21 km/h, 13 mph),
whereas a racing bicyclist is moving at a speed of some 12 m/s
(42 km/h, 26 mph).
A modern car (drag coefficient 0.34) with
a 110 kW (150 HP) engine will be using about 4.6 kW (6 HP) of
power to overcome air drag and 11 kW (14 HP) for mechanical propulsion
(rolling resistance etc.) when it is being driven at a constant
80 km/h (22.2 m/s, 49 mph). When driving at its top speed of
210 km/h (58 m/s, 128 mph) it will be using 82 kW (112 HP) to
overcome drag and the remaining 28 kW (38 HP) to overcome rolling
resistance etc.
High
Lift to Drag Ratio Needed for airfoils for Rotor Blades
The rotors on modern wind turbines have very high tip speeds
for the rotor blades, usually around 75 m/s (270 km/h, 164 mph).
In order to obtain high efficiency, it is therefore essential
to use airfoil shaped rotor blades with a very high lift to
drag ratio, i.e. rotor blades which provide a lot of lift
with as little drag as possible. This is particularly necessary
in the section of the blade near the tip, where the speed relative
to the air is much higher than close to the centre of the rotor.
For wind turbines with a low tip speed it
is not necessary to use top quality airfoils. The "Western"
type windmill rotors can easily be manufactured from flat metal
plate.
Drag
Increases with the density of air
Both lift and drag increase in proportion
to with the density of air. Cold air will thus give more drag
than hot air. You may find the density of air at different temperatures
in the Reference
Manual to study how important this effect is.
Drag
Coefficient Varies with the roughness of the object surface
Just as it is case for lift, drag may
vary quite dramatically with the surface roughness of the object.
Normally it is desirable to have smooth, clean surfaces in order
to minimise drag. There are some curious exceptions to this rule,
as you can see on the page about Research
and Development in Wind Energy.


Drag
Force Formula
The drag force for a given object can be found using the formula
below:
F_{D} = C_{D} 0.5
A v^{2}
F_{D} = The drag force measured
in N (Newton).
C_{D} = The drag coefficient,
measured in N/ m^{2}, i.e. the
drag force per square metre frontal area of the object shape.
You will have to find this figure in a table. This value is usually
found through (very expensive) wind tunnel measurements.
Warning: Airfoil tables frequently give the
drag coefficient per square metre wing or rotor blade area, rather
than per square metre frontal area. You can recalculate that
figure, since you know the relative thickness of the airfoil.
= The density of the fluid measured
in kg/m^{3}. In the case of dry
air at sea surface level at 15° C the figure is 1.225 kg/m^{3}. You can find air densities for other
temperatures in the Reference
Manual.
A = Frontal area of the object in m^{2}.
v = Relative wind speed in m/s, i.e. the speed of the fluid moving
past the object. If the object itself is moving while facing
a headwind, you have to add the headwind speed to the speed of
the object to obtain v.
The formula for the drag force is fairly
logical when you compare it with the formula for the power
of the wind. The reason why it is v^{2}
which is used in the formula is that we are dealing with a force.
If we had looked at the power loss from drag instead,
we should have multiplied by v^{3}.


Drag
Depends on the Reynolds Number
In reality, there is not just one, but two kinds of drag: Pressure
drag, and friction drag. At very low speeds, and for
small objects, say dust particles, the friction drag dominates.
At high speeds and/or large object sizes pressure differences
dominate. The drag coefficient for an object will therefore depend
on which type of flow is dominating. A microscopic parachute
will not work like a large parachute.
Fortunately we are able to predict which
type of flow we are dealing with if we know the socalled Reynolds
number for the experiment. (Named after the British engineer
Sir Osborne Reynolds 18421912).
The Reynolds number is defined as
Re = v L / (μ /
)
Re = The Reynolds number, which is dimensionless, i.e. it
is a ratio of two quantities with the same unit.
v = The relative velocity of the fluid in m/s.
L = The characteristic length, in this case the largest cross
section of the frontal area in m.
μ = The viscosity of the fluid in Ns/m^{2}.
The viscosity of air, also called the dynamic viscosity of
air is 1.8 x 10^{ 5} at 15°
C and atmospheric pressure at sea level. The value for the viscosity of air at other
temperatures may be found in the Reference Manual.
= The density of the fluid in
kg/m^{3}.
The value in the denominator of the fraction
(μ / ) is called the
kinematic viscosity of air. When the kinematic viscosity
is high, the laminar flows dominate.
Thus, if the Reynolds number is very small,
below 1, one can ignore pressure drag and concentrate on friction
drag. If the number is large, above 100, one can ignore the friction
drag and look at pressure drag only. Close to the surface of
the object friction drag and viscosity are always important.
If you are interested in more detail on this subject, you
should consult a textbook on fluid mechanics such as the ones
mentioned in the bibliography.



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© Copyright 2002 Danish Wind Industry Association
Updated 11 September 2002
http://www.windpower.org/tour/wtrb/drag.htm
