Real airplane wings are not infinite in extent, but they have a finite *wingspan* , denoted by *b*. It is useful to define the
*aspect ratio* AR of the wing as

with the surface area of the wing. For a wing with total lift *L*
the lift coefficient is

For a wing of finite span, the mere existence of lift leads to an additional
source of drag, known as *induced drag* . This
drag is due to the high pressure fluid on the lower wing surface moving
outward and around the wingtip to meet the low pressure fluid on the upper
surface of the wing. When viewed from the front or rear of the aircraft, we
see that this flow produces vortices on the wingtips, which are often called
*trailing vortices*.
See figures 7.9 and 7.10 in *Melissinos*.

These trailing vortices connect with the starting vortex, or downwash, to
form a U-shaped vortex connected to the wingtips. Now a vortex can contain a great deal of kinetic energy, and it
requires energy to produce a vortex. Since the wings are constantly
supplying energy to the fluid to create the vortices, the fluid is doing
work on the wings, resulting in a drag on the wings. There is no way to
eliminate this drag--it is the price to be paid for the generation of lift.
Quite generally, the coefficient of induced drag is

where *e* is the *span efficiency factor*. We see that the coefficient
of induced drag varies as the *square* of the lift coefficient. As a
result, the induced drag can be a substantial fraction of the total drag on
an airplane. We also see that the induced drag is minimized for high aspect
ratio wings--for instance, the long, narrow wings of a glider are designed
to produce a small induced drag. The total coefficient of drag on the
airplane is

where contains contributions from both the skin friction and the pressure drag (due to boundary layer separation).

Although it isn't possible to eliminate the induced drag, it is possible to
minimize it by suitably choosing the shape of the wings. In fact, it was
shown by Prandtl that for a fixed coefficient of drag and aspect ratio
AR the induced drag is minimized if the lift per unit span along the wing is
elliptical. This can be achieved in a number of ways--for instance, making
the chord vary elliptically along the wing (known as an *elliptical
planform* ). This leads to *e*=1; any other
geometry has *e*<1, leading to a higher drag. Typical subsonic aircraft have
0.85 to 0.95.

In addition to producing drag the trailing vortices have another important effect--they create considerable turbulence which can be of great danger to nearby aircraft. Apparently the trailing vortices of a large jet can actually flip a small aircraft which approaches too closely. These considerations are important in air traffic control.

Finally, what about the flaps and slats on an airplane wing? These are
extended on take-off in order to increase the wing camber, thereby producing
a greater lift. The flaps and slats are retracted for cruising, since this
reduces the lift and therefore the induced drag on the plane. On landing the
flaps on the trailing edge are extended even farther, inducing separation
and producing a large drag to slow the plane. Air brakes on the top surface
of the wing are also projected at right angles to the wing to produce
additional drag. Once on the runway reversal of the thrust from the jets
produces an additional braking effect, slowing the plane to a point where
the brakes on the wheels can be safely applied.

FootnoteIn fact in a nonviscous fluid vortices can begin and end only at a free surface of the fluid or on a solid body, so we see that the existence of the starting vortex requires the existence of trailing vortices.

Tue Oct 21 21:23:27 EDT 1997