Trefftz Plane Lift Derivation

We have discussed the calculation of drag based on the velocities induced in the Trefftz plane, but can lift be calculated in a similar way?

The answer is not so easy. We start with the expression for force based on the momentum equation.

Let's assume (naively, for now) that the contribution of each of these integrals goes to zero on each side of the box, except for the back side, as the dimensions of the box are increased. This leaves the contribution in the Trefftz plane due to the wake.

In the Trefftz plane, if we assume that all of the induced velocities are normal to the plane, the lift becomes:

The evaluation of this integral seems straightforward.
But, it is not.

Consider the integral

when the wake is modeled simply by a pair of vortices.

The induced velocity, w, is given by:

Thus, the inner part of the above integral becomes:

So, the integral for lift is:

This looks exactly right; however, let's consider the same integral when the order of integration is reversed:

The inner part of the above integral now becomes:

This integrand is antisymmetric as shown in the plot below. So, the integral for lift is: L = 0.

Now we have a paradox. We get two values for the same integral.
Actually, this is not a paradox; it is rather a function that is not Lebesgue integrable.

In order to evaluate an integral unambiguously, the function must satisfy two conditions:
1. It must be continuous, except at a countable number of points.
2. The integral of its absolute value must be finite.

To avoid this problem, the integral for lift can be first evaluated over finite limits. Taking z from -A to A and y from -B to B we find:

L = (2 r U G / p) {(B+s) atan(A/(B+s)) - (B-s) atan(A/(B-s))
+ A/2 ln[ (A²+(B+s)²) / (A² + (B-s)²)] }

The limit as A and B get large depends on the ratio of A to B.
When A>>B the value goes to 2 r U G s, when B>>A the value is 0 and when
A = B the value is r U G s. Thus the integral remains ambiguous when evaluated over an infinite domain.

As noted by Larry Wigton of Boeing, this dilemma is resolved by using a different model of the flow field. When the vertical velocity associated with this vortex system is integrated over the Trefftz plane, no ambiguities arise. But, the results are surprising.

The result is that the contributions from the finite length trailing vortices goes to zero. The contribution from the bound vortex is found to be independent of the length of the trailing vortices and is:

The starting vortex contribution is similarly independent of the trailing vortex length and is equal to the bound vortex contribution. Thus, this lift is due to momentum flux, but not from the trailing vortices.

We finally need to look at how the pressure term on the upper and lower sides of the control volume is involved.

As might be expected, integrals once again are not unambiguous. They depend on the relative sizes of the box sides, even though everything is infinite.

A careful analysis leads to the following basic results:

1. If the wake length is small compared with the box width and height then the lift is associated with the momentum term of the starting and bound vortices.

2. If the wake length is large, and the box height is large compared with the width, then the lift is associated with the momentum term of the trailing vortices.

3. If the wake is long and the width is large compared with the height, then the lift is associated with the pressure terms on the top and bottom.

Nonplanar Wings

Even after the issues with infinite-domain integrals have been resolved, we must worry about the assumed wake position. Although we could argue that streamwise wakes can usually be used for far-field drag computataions, streamwise wakes can still support lift forces. When the wing or wake is substantially nonplanar, these effects can be significant. In fact, the vortex lift generated by highly swept wings can be estimated by far-field methods only when the roll-up of the wake sheet is accurately computed. The alternatives in such cases are to use near field methods or to compute the wake shape.