There are many
factors
which influence the amount of aerodynamic
drag
which a body generates. Drag depends
on the shape,
size,
inclination, and
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 blue lines with
the arrowheads showing the flow direction. As the aircraft moves to
the lower left, a pair of counter-rotating 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 atmospheric conditions
are right (high humidity), you can actually see the vortex lines on an airliner
during landing. The vortices produce a
down wash
of air behind the wing which is very strong near the wing tips and
decreases toward the wing root. The local
angle of attack
of the wing is increased by the **induced flow** of the down wash,
giving an additional, downstream-facing, component to the
aerodynamic force acting over the entire 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 varies with the square of the lift.

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**. For a
rectangular wing this reduces to the ratio of the span to the chord.
Long, slender, high aspect ratio wings have lower induced drag than
short, thick, low aspect ratio wings.
(Induced drag is a three dimensional effect related to the wing tips.
The longer the wing, the farther the tips are from the main portion of
the wing, and the lower the induced drag.)
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.
(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.)
A more typical value of e = .7 for a rectangular 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.

Navigation..

- Beginner's Guide to Aerodynamics
- Beginner's Guide to Propulsion
- Beginner's Guide to Model Rockets
- Beginner's Guide to Kites
- Beginner's Guide to Aeronautics

Go to...

- Beginner's Guide Home Page

*byTom
Benson
Please send suggestions/corrections to: benson@grc.nasa.gov *