There are many
factors
which influence the amount of aerodynamic
drag
which a body generates. Drag depends
on the shape,
size, and
inclination, of the object,
and on
flow conditions of the air passing the object.
For a three dimensional wing, there is an additional component
of drag, called induced drag, which will be discussed on this page.
For a lifting wing, the
air pressure
on the top of the wing is lower than the pressure below the wing.
Near the tips of the wing, the air is free to move from the region
of high pressure into the region of low pressure. The resulting
flow is shown on the figure by the two circular bluegrey arrows with
the arrowheads showing the flow direction. As the aircraft moves to
the lower left, a pair of counterrotating vortices are formed at the
wing tips. The line of the center of the vortices are
shown as blue vortex lines leading from the wing tips.
If the atmosphere has very high humidity,
you can sometimes see the vortex lines on an airliner
during landing as long thin "contrails" leaving
the wing tips.
The wing tip vortices produce a swirling flow
of air behind the wing which is very strong near the wing tips and
decreases toward the wing root. The effective
angle of attack
of the wing is decreased by the induced flow of the vortices
and varies from wing tip to wing root. The induced flow
produces an additional, downstreamfacing,
component
of aerodynamic force of the wing.
This additional force is called induced drag because it faces
downstream and has been "induced" by the action of the tip vortices.
It is also called "drag due to lift" because it only occurs
on finite, lifting wings and the magnitude of the drag depends on the lift of the wing.
The derivation of the equation for the induced drag is fairly tedious
and relies on some theoretical ideas which are beyond the scope
of the Beginner's Guide.
The induced drag coefficient Cdi is equal to
the square of the lift coefficient Cl divided by the quantity: pi(3.14159) times the
aspect ratio AR times an
efficiency factor e.
Cdi = (Cl^2) / (pi * AR * e)
The
aspect ratio
is the square of the span
s divided by the wing area A.
AR = s^2 / A
For a
rectangular wing this reduces to the ratio of the span to the chord
c.
AR (rectangle) = s / c
Considering the induced drag equation,
there are several ways to reduce the induced drag.
Wings with high aspect ratio have lower induced drag than wings with low aspect ratio
for the same wing area.
So wings with a long span and a short chord have lower induced drag than
wings with a short span and a long chord.
Lifting line theory shows that
the optimum (lowest) induced drag occurs for an elliptic distribution
of lift from tip to tip. The efficiency factor e is equal to 1.0
for an elliptic distribution and is some value less than 1.0 for any
other lift distribution. So an elliptical wing
planform has the lowest amount of induced drag
and all other
wing shapes
have higher induced drag than an elliptical wing.
For a rectangular wing, the efficiency factor is equal to .7.
For many years, wing designers have attempted to reduce the induced drag component
by special shaping of the wing tips. The Wright Brothers used curved trailing edges
on their rectangular wings based on
wind tunnel results.
The outstanding
aerodynamic performance of the British Spitfire of World War II is partially
attributable to its elliptic shaped wing which gave the aircraft a very low
amount of induced drag.
On modern airliners, the wing tips are often bent up to form
winglets.
Winglets were extensively studied by Richard Whitcomb of the NASA Langley Research Center in
an effort to reduce the induced drag on airliners.
For a wing, the total
drag coefficient, Cd
is equal to the base drag coefficient at zero lift Cdo
plus the induced drag coefficient Cdi.
Cd = Cdo + Cdi
The drag coefficient in this equation uses the wing
area for the reference area. Otherwise, we could not add it to the
square of the lift coefficient, which is also based on the wing
area.
You can further investigate the effect of induced drag and the other
factors affecting drag by using the
FoilSim III Java Applet.
You can also
download
your own copy of FoilSim to play with
for free.
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