Finite Wings - Theoretical Performance

The single most important difference between an airfoil and a wing is that the latter has a finite span (the airfoil therefore is an infinite wing)

This means that at the tips there is no barrier to the fluid from flowing from the botton surface (where it is at a relatively higher pressure) to the top surface (where it is at a relatively lower pressure). As the air flows around the tip it is also transported back due to the forward speed of the aircraft/wing. This rotating flow leaving the wing at the tips are called tip vortices. Very often these vortices are visible as vapor trails on aircraft flying high during dawn or dusk when the speeds and pressure are just right so that the surrounding water vapor condenses

The tip vortices spoil the lift at the regions of the tip. The flow over the middle section of the wing shown is representative of the 2D airfoil values. As we move outward to the tip the lift should decrease because of the cross flow effects. This is exaggerated in the figure below. The lift near the plane of symmetry is the maximum.

The actual physical tip vortex suggested a mathematical representation of the flow over a wing. The tip vortex can be replaced by a LINE VORTEX. The vortex line is a useful fluid element which was used by Prandtl to calculate the lift due to the wing. The flow was considered steady, ideal, and incompressible. The model was called the Prandtl Lifting Line Theory (PLL) and is illustrated below

In the PLL:

The wing was replaced by a single horse-shoe vortex line at about the c/4 line (going to infinity)
The strength of the vortex is unchanged along the line vortex (this has to be constant from Helmholtz Vortex Theorems)
The velocity induced by the vortex is a circular velocity (identified by the right hand rule). In the figure the vortex line direction is appropriate to explain the tip vortices
The Kutta-Jukowski theorem establishes the lift/span generated by the vortex line in the plane of the wing as
. This is shown as the Lift distribution in red on the figure
This lift distribution is therefore uniform along the span [Note: this is different from the real lift distribution as the lift is reduced at the tips - shown in black]

Nevertheless the PLL was a good start

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