Trefftz Plane Drag Derivation
Why does the contribution to drag from all but the front and back sides
vanish? The pressure terms are clearly 0 since the faces are parallel to
the x direction. But the momentum terms are not so easy. We argue that
the contribution goes to zero as the walls recede faster than the area goes
to infinity. Here is why: In the far field, the induced velocities of
a lifting system may be represented as the velocities induced by a single
transverse vortex filament and two trailing vortices. The two trailing
vortices cancel each other out in the far field, leaving only the piece
of bound vorticity. This piece induces a velocity that varies as 1/r2
while the area is increasing as r2. But because the flux (V·n) through
the top and bottom are opposite, no first order term exists on these surfaces.
Furthermore, this vorticity induces no V·n through the sides.
We start with the expression for the drag in terms of the perturbation velocities:
This result comes directly from the application of conservation of momentum
and the incompressible Bernoulli equation.
Actually, it also assumes that the wake extends infinitely far downstream
and trails back from the wing in the freestream direction. If we did not
go very far downstream or the wake were not assumed to be straight, the
more general expression would be:
But, if we assume that the streamwise perturbation velocities are small,
great simplifications are possible:
Now:
and outside the wake:
So:
Gauss' theorem states that:
So,
The contour integral is taken as shown below:
We thus obtain:
The jump in potential at the location y in the wake is just the integral
of V·ds from a point above the wake to a point below. Since the normal
velocity is continuous across the wake, the integral is just equal to the
circulation enclosed in the loop. This is just the circulation on the wing
at the point where this part of the wake left the trailing edge. Similarly,
the derivative of φ normal to the wake, is the induced normalwash, Vn.
So:
The last expression may be recognized as the result of lifting line theory,
but it has been derived in a much more general way.