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 Consider the integral
where
is a nonnegative integer. This integral is defined by its Cauchy principal value
As was demonstrated in Section 9.9,
Show that
and
and, hence, that
 Suppose that an airfoil of negligible thickness, and wingspan
, has a width whose
variation is expressed parametrically
as
for
, where
Show that the air circulation about the airfoil takes the form
where
. Here,
is the angle of attack (which is assumed to
be small).
Demonstrate that the downwash velocity at the trailing edge of the airfoil is
Hence, show that the lift and induced drag acting on the airfoil take the values
respectively.
Demonstrate that the drag to lift ratio can be written
where
is the aspect ratio.
Hence, deduce that the airfoil shape (in the

) plane that minimizes this ratio (at fixed aspect ratio) is an ellipse (i.e., such
that
for
).
 Consider a plane that flies with a constant angle of attack, and whose thrust is adjusted such that it cancels the induced
drag. The plane is effectively subject to two forces. First, its weight,
, and
second its lift
. Here,
and
are horizontal and
vertical coordinates, respectively,
is the plane's instantaneous velocity, and
is a positive constant. Note that the lift is directed at right angles to the
plane's instantaneous direction of motion, and has a magnitude proportional to the square of its airspeed. Demonstrate that the
plane's equations of motion can be written
where
is a positive constant with the dimensions of length. Show that
where
is a constant. Suppose that
and
, where
,
.
Demonstrate that, to first order in perturbed quantities,
Hence, deduce that if the plane is flying horizontally at some speed
, and is subject to a small perturbation, then its altitude oscillates
sinusoidally at the angular frequency
. This type of oscillation is known as a phugoid oscillation.
Next: Incompressible Viscous Flow
Up: Incompressible Aerodynamics
Previous: Simple Flight Problems
Richard Fitzpatrick
20160122