Induced Drag and the Trefftz Plane
Fundamentals
The 2-D paradox that surfaces in inviscid flow produce no drag no longer
applies in 3-D. The downwash created by the trailing wake changes the direction
of the force generated by each section:
In three dimensions the force per unit length acting on a vortex filament
is
Here, the local velocity V includes the component from the freestream and
a component from the induced downwash. This latter component produces a
component of force in the direction of the freestream: the induced drag.
The induced drag is related to the lift by: 
From the results of lifting line theory for lift and downwash in terms of
the Fourier coefficients of the lift distribution:
so we have:
The induced drag is often written as:
This may be written in coefficient form as:
with the same definition of e. Note that e simply depends on the shape
of the lift distribution. It is called the span efficiency factor or Oswald's
efficiency factor. Note also that the induced drag force depends principally
on the lift per unit span, L/b.
We can determine quickly, from the expression for induced drag above that
drag is a minimum for a given lift and span when all of the Fourier coefficients
except the A1 term (which produces lift) are zero. This corresponds to
the elliptic loading case mention previously. In this case the downwash
is constant and e = 1.
Far Field Analysis and the Trefftz Plane
The analysis above works quite well for analyzing the drag of wings given
the distribution of lift. It was invented by Prandtl and Betz around 1920.
It does involve a bit of hand-waving, though, and requires some very approximate
idealizations of the flow. For example: when we talk about the downwash
induced by the wake on the wing, just where on the wing do we mean? We
assume a single bound vortex line and compute velocities there, but a real
wing does not have a single bound vortex and the velocity induced by the
wake varies along the chord. Fortunately, the answers from this model are
more general than the model itself appears. The induced drag formulas can
also be derived from very fundamental momentum considerations.
If the box is made large, contributions from certain sides vanish. In
the limit as the box sides go to infinity we obtain the following expressions
for lift and drag:
Here, u, v, and w are the perturbation velocities induced by the wing and
its wake. Note that the drag only depends on the velocities induced in
the "Trefftz Plane" -- a plane far behind the wing.
The drag can be expressed as the integral over the infinite plane of the
perturbation velocities squared. But, using Gauss' theorem (derivation)
it can be expressed as a line integral over the wake itself:
This simplifies the calculation of the drag.
The normalwash, Vn, is just the downwash if the wake is flat, but the downwash
far behind the wing, not at the wing itself.
Thus we would obtain the same expression as from lifting line theory if
the downwash due to the wake at the wing is half the downwash at infinity.
This is indeed the case for unswept wings modeled with a lifting line.
Nonplanar Wakes
All of the comments above apply to nonplanar wings as well as to simple planar ones. But we must be
careful about the assumed wake shape when evaluating forces in the far field. The integrals above
actually give the force on the wing and wake combination. Of course, in reality, there is no force
on the wake sheet, but if we assume a shape a priori, it is not likely to be a force-free wake. However,
since forces act in a direction perpendicular to the vortex, extending wakes streamwise always
yields a drag-free wake and nearly correct answers for drag using far field methods.
Far-field velocities can also be used to compute
the lift. The results are subtle, but rather interesting.
Munk's Stagger Theorem
The result that the drag of a lifting system depends only on the distribution
of circulation shed into the wake leads to some very useful results in classical
aerodynamics.
Perhaps the most useful of these is called Munk's stagger theorem. It states
that:
The total induced drag of a system of lifting surfaces is not changed
when the elements are moved in the streamwise direction.
The theorem applies when the distribution of circulation on the surfaces
is held constant by adjusting the surface incidences as the longitudinal
position is varied.
This implies that the drag of an elliptically-loaded swept wing is the same
as that of an unswept wing. It also is very useful in the study of canard
airplanes for which the canard downwash on the wing is quite complicated.
Moving the canard very far behind the wing does not change the drag, but
makes its computation much easier. One may use the stagger theorem to prove
several other useful results. One of these is the mutual induced drag theorem
which states that: The interference drag caused by the downwash of one
wing on another is equal to that produced by the second wing on the first,
when the surfaces are unstaggered (at the same streamwise location).
These results are especially useful in analyzing multiple lifting surfaces.