Lifting line theory of lift

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The "circulation theory of lift" develops into the "lifting line" theory of lift, by considering what happens to the circulation around a finite wing.

The circulation theory presented earlier deals with the simple case of circulation around a wing (sail, fin) of indefinite length.  In this simple case, the circulation, and the wing, stretch away into infinity.  In reality, a wing has a finite span, and the circulation around the wing doesn't stop at the wingtips.  The question is, what happens to it?

Circulation is conserved, like energy or momentum.  The circulation around a wing cannot simply appear by itself.  A circulation vortex must be a closed system, and the two ends of the circulation must meet up.  For some time, this seemed just a theoretical issue, but careful investigation demonstrated how this happens.  As the wing moves forward, it sheds a "starting" vortex, equal to but opposite in circulation to the wing vortex, called the "bound" vortex, bound as it is to the wing.  As the wing moves forward, tip vortices develop, with the result that a "closed" vortex loop develops.  If we think about an aircraft taking off, it leaves its starting vortex at the beginning of the runway, and the tip vortices then build up down the length of the runway and into the air as the aircraft climbs.  For a large aircraft, these vortices take several minutes to dissipate.

The bound vortex of the wing is considered bound to a "lifting line".  The lifting line theory considers the wing to be a single vortex filament.  The tip vortices are the "automatic" consequences of the lifting line and the circulation around a finite wing.

The tip vortices have profound consequences for the lift and drag of a wing.  Because of their action, they introduce additional downwash.  For most wing planforms, this additional downwash tends to be concentrated towards the wing tips.  This additional downwash means additional drag.  The diagram shows the increasing, additional downwash towards the wing tips due to the tip vortices.  This downwash is additional to the downwash due to the bound vortex, which is considered to be evenly distributed along the wing or lifting line.

The momentum theory of lift showed that the drag due to a lifting surface (called the induced drag) is given by the angle of downwash -- more downwash, more induced drag.  The extra downwash due to the tip vortices means a greater downwash angle, and hence a greater induced drag coefficient.  The diagram shows that, if there is more downwash at the wing tip than at the centre of the wing, there will be more induced drag there.

According to the momentum theory, the increased drag varies as the square of the lift coefficient.  As the wing develops more lift, the induced drag increases proportionately more at the wing tips, if this is where the tip vortices are adding extra downwash.

Notice also that the induced drag depends upon the wing aspect ratio.  Long thin wings, with high aspect ratio, have less induced drag.  This is the same as saying that their tip vortices are weaker, and their tip vortex downwash is less.  In a way, this is obvious, since the tip vortices on a long wing are further apart, and their downwash doesn't affect as much of the airflow behind the wing.

An important question concerns the wing planform which generates the least amount of additional induced drag.  This is the same question as the wing planform which generates a constant downwash from root to tip, such a constant downwash yielding the least induced drag.  The answer turns out to be an elliptical planform.

An elliptical wing planform has the interesting property of yielding tip vortices which are the least "concentrated", that is, the downwash they yield is spread most evenly along the wingspan.


2022 Lester Gilbert