Lifting Line Theory

Basic Theory

We could try using 2-D flow results for each section, but correct them for the influence of the trailing vortex wake and its downwash. This is the idea of lifting line theory.

We use the 2-D result that:

together with the relation:

to obtain:

But the angle of attack used here is reduced through the effects of downwash so that the effective angle of attack is the true angle* minus the downwash angle:

Where the induced downwash, Wind, is given by the Biot-Savart Law:

Combining the expression for gamma:

with the expression for the downwash angle:

provides an integral equation for the circulation distribution along the wing.

Just as in thin airfoil theory, the integral equation can be solved by assuming a Fourier series representation for the distribution.

Substitution of the expression for circulation into the integral equation leads to:

After integrating we have:

The solution of this equation for all values of y is not quite so easy as in the case of thin airfoil theory where we could get closed form expressions for the A_n's. This is generally done numerically. However, several interesting and simple results appear from this analysis without ever actually computing the A_n's from the distribution of local angle of attack. Some of these are discussed in the next section.

Elliptic Wing Results

If, for example, we represent the lift distribution with only a single term in the Fourier series, then:

This represents an elliptic distribution of lift.

The downwash angle is, in this case:

The integral is constant when |y| < b/2.

In this domain:

Since the downwash distribution is constant the C_l distribution is just:

If the angle of attack is also constant along the wing (no twist) then the C_l is constant and since:

Then in this case the section C_l is equal to the wing C_L and:

or:

Recall that this holds for unswept elliptical wings.

General Lift Distributions

If we are given the lift distribution we can compute the A_n's as we would with any Fourier expansion. And once we know the Fourier coefficients, we may compute the downwash distribution and the induced drag:

Substitution and evaluation of the definite integral** leads to:

This formula gives the downwash in the plane of the wing for arbitrary load distributions. For the simple elliptical case, closed form solutions for the downwash and sidewash at the start of the wake sheet exist. The simple relation for the velocity induced by an elliptic wing tailing vortex sheet is:

Here, the variable Z is the complex coordinate y + iz and wo is the downwash at the wing root: y = z = 0.

This formula permits computation of induced velocities behind a wing as they effect downstream surfaces such as horizontal tails.

Note that the downwash is only constant in the plane of the wing and behind the wing. As we move outboard of the wing or out of the plane of the wake, the downwash varies considerably and there is a rather large upwash beyond the wing tips.

This downwash field produces several important effects. It changes the lift of surfaces in other surfaces wakes. This is important in the analysis of airplane stability and the effectiveness of horizontal tails. As can be seen from the downwash plot, the interference of a canard wake with a wing is extreme: the wing lift is reduced behind the canard and the part of the wing outboard of the canard has increased lift.

The downwash also produces induced drag as discussed in the next section.

*Note that what we have called the geometric angle of attack is just that for flat plates but it is, in general, the angle of attack from zero lift.

**The integral is not in all tables of integrals. It is sometimes called the Glauert integral and is given, for example, in Kuethe and Chow page 146.