Wing Models

One may apply the results of 3-D potential theory in several ways. We first consider the theory of finite wings.

We might start out by saying that each section of a finite wing behaves as described by our 2-D analysis. If this were true then we would still find that the lift curve slope was 2p per radian, that the drag was 0, and the distribution of lift would vary as the distribution of chord. Unfortunately, things do not work this way. There are several reasons for this:

One explanation is that the high pressure on the lower surface of the wing and the low pressure on the upper surface causes the air to leak around the tips, causing a reduction in the pressure difference in the tip regions. In fact, the lift must go to zero at the tips because of this effect. We will next see how and why we must model the 3-D wing differently from 2-D.

If we were to take the naive view that the 2-D model would work in 3-D, we might have the picture shown on the right. If each section had the distribution of vorticity along its chord that it had in 2-D, the lift would be proportional to the chord, and would not drop off at the tips as we know it must.

This sort of model does not conform to our physical picture of what happens at the wing tips. And indeed, it does not satisfy the equations of 3-D fluid flow. The reason that this does not work is that in this case the streamlines are not confined to a plane. They move in 3-D and the flow pattern is quite different.

We could go back to the governing equations and start simply with the linear Laplace equation. By superimposing known solutions we could obtain a simple model of a 3-D wing. We might start by superimposing vortices on the wing itself:

But this is no more than the strip theory model that did not work. The reason that this model (which seems just to be a superposition of known solutions) is not adequate is that it violates the governing equations in certain regions. The model does not satisfy the Helmholtz laws since vorticity ends in the flow near the tips. Some additional requirements must be imposed on the model. The requirements for such a model are just the Helmholtz vortex theorems, discussed previously.

Our simple 3-D model above may be modified as shown below to satisfy the first of the Helmholtz theorems.

In fact, as can be seen from the picture here, this vortex model is not too far from reality.

The downwash field and the existence of trailing vortices are not just some strange mathematical result. They are necessary for the conservation of mass in a 3-D flow.

Air is pushed downward behind the wing, but this downward velocity does not persist far from the wing. Instead it must move outward. The outward-moving air is then squeezed upward outboard of the wing and the flow pattern shown above develops.

The trailing vortex is visualized by NASA engineers by flying an agricultural airplane through a sheet of smoke.

The main effect of this vortex wake is to produce a downwash field on the wing.

This downwash field has several very significant effects:

It changes the effective angle of attack of the airfoil section. This changes the lift curve slope and has many implications.
Induced drag: Lift acts normal to flow in 2D. This accounts for about 40% of the fuel used in a commercial airplane, and as much as 80% of the drag in the critical climb segments.
It produces interference effects that are important in the analysis of stability and control.

The magnitude of the downwash can be estimated using the Biot-Savart law, discussed previously.

When applied to our simple model with two discrete trailing vortices, the equation predicts infinite downwash at the wing tips, a result that is clearly wrong. In fact, the induced downwash is not even very large.

The failure of this simple model led Prandtl to develop a slightly more sophisticated one in 1918. Rather than representing the wing with just one horseshoe-shaped vortex, the wing is represented by several of them:

In this way the circulation on the wing can vary from the root to the tip. The strength of the trailing vortex filaments is related to the circulation on the wing then by:
G_wake = DG_wing
A vortex is shed from the wing whenever the circulation changes.

In the limit as the number of horseshoe vortices goes to infinity, the trailing wake is a sheet of vorticity.

The trailing vortex strength per unit length in the y direction (vorticity) is the derivative of the total circulation on the wing at that station. From this model, we can derive the basic relations for finite wings.

The vorticity strength in the trailing vortex sheet is given by:
g = d G/dy
and since the wing circulation changes most quickly near the tips, the trailing vorticity is strongest in this region. This is why we see tip vortices, and not a complete vortex sheet, as in this NASA photo of an F-111 in a 4-g turn. The vortices are visible in this picture because the low pressure in this region lowers the temperature and we see the condensed water vapor.