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, or drag due to lift.
Induced drag is a three dimensional effect related to the distribution of lift
across the wing. Flow near the wing tips have a strong influence on the amount of induced drag
because of the tip vortices that are generated there.
The aspect ratio
is the ratio of the square of the span to the wing area. Induced drag is inversely related
to the aspect ratio of the wing. Long thin wings have low induced drag.
Wings with an elliptical
planform also have lower induced drag than
rectangular wings, as expressed in the efficiency factor in the induced drag equation.
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.

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.
On modern airliners, the wing tips are often bent up to form
winglets.
Winglets were wind tunnel tested and computer analyzed by Richard Whitcomb
of the NASA Langley Research Center in the mid 1970's.
The idea behind the winglet is to reduce the strength of the tip vortex and therefore cause
the flow across the wing to be more two-dimensional. Flight tests at the NASA Dryden Flight Research
Center have found a 6.5% reduction in the fuel use of a Boeing 707 type airliner when using winglets.
Winglets must be carefully integrated into the total wing design, which explains why many different
winglet designs appear on various airliners.

There is a mathematical equation that quantifies the effects of induced drag.
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.

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. The value of the efficiency factor is 1.0 for an elliptical wing and
some smaller number for any other planform. The value is about .7 for a rectangular wing.

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

To help you understand the effects of winglets on the drag of a wing, we have performed some simple
wind tunnel tests
with a variety of winglet models.
The winglet design, construction, and testing was performed by three high school "shadows".
The wind tunnel that
was used is a
low speed tunnel
that has been used for several student projects.
The winglets were attached to a 1/72 scale model of a Cessna 172.
Five different winglet configurations were wind tunnel tested in May, 2014, by
Austin Vorisek, a graduate of Solon High School, who will be attending Ohio State University in the fall of 2014, and
Alex Mann, a graduate of Orange High School, who will be attending Akron Univerity in the fall of 2014.
An additional nine configurations were studied in June, 2014, by
Dominic Roberts II, a home-schooled senior from Lansing, Michigan, who is already workng at the aero lab at Michigan State University.

The students prepared this small Java computer program to display the experimental results.
Simply select the wing design from the drop menu. The design of the winglet as viewed from the side
will be shown at the far right. Enter the drag coefficient for the wing without winglets at the left
and the program will calculate a change to the drag coefficient with winglets.
Some winglet designs have little or no effect,
some actually make the drag worse because they increase the overall surface area of the aircraft model.
But some of the models do produce an effect
in the correct range as indicated by the flight tests described above.
Entering the speed, altitude, and wing area provides the program with enough data to calculate the
drag
of the wing.
As a naming convention, "SW" denotes single winglet pointing up from the wing, "DW" is a double winglet; one pointing up and one pointing down.

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.