Editing Lift (force) (section)

==Three-dimensional flow==
[[Image:Wing isobars.jpg|thumb|right|300px|Cross-section of an airplane wing-body combination showing the isobars of the three-dimensional lifting flow]]
 
[[Image:Wing velocity vectors.jpg|thumb|right|300px|Cross-section of an airplane wing-body combination showing velocity vectors of the three-dimensional lifting flow]]

The flow around a three-dimensional wing involves significant additional issues, especially relating to the wing tips. For a wing of low [[Aspect ratio (aeronautics)|aspect ratio]], such as a typical [[delta wing]], two-dimensional theories may provide a poor model and three-dimensional flow effects can dominate.<ref>Milne-Thomson (1966), Section 12.3</ref> Even for wings of high aspect ratio, the three-dimensional effects associated with finite span can affect the whole span, not just close to the tips.

===Wing tips and spanwise distribution===
The vertical pressure gradient at the wing tips causes air to flow sideways, out from under the wing then up and back over the upper surface. This reduces the pressure gradient at the wing tip, therefore also reducing lift. The lift tends to decrease in the spanwise direction from root to tip, and the pressure distributions around the airfoil sections change accordingly in the spanwise direction. Pressure distributions in planes perpendicular to the flight direction tend to look like the illustration at right.<ref>McLean (2012), Section 8.1.3</ref> This spanwise-varying pressure distribution is sustained by a mutual interaction with the velocity field. Flow below the wing is accelerated outboard, flow outboard of the tips is accelerated upward, and flow above the wing is accelerated inboard, which results in the flow pattern illustrated at right.<ref>McLean (2012), Section 8.1.1</ref>

There is more downward turning of the flow than there would be in a two-dimensional flow with the same airfoil shape and sectional lift, and a higher sectional angle of attack is required to achieve the same lift compared to a two-dimensional flow.<ref>Hurt, H. H. (1965) ''Aerodynamics for Naval Aviators'', Figure 1.30, NAVWEPS 00-80T-80</ref> The wing is effectively flying in a downdraft of its own making, as if the freestream flow were tilted downward, with the result that the total aerodynamic force vector is tilted backward slightly compared to what it would be in two dimensions. The additional backward component of the force vector is called [[lift-induced drag]].

[[Image:Tip vortex rollup.png|thumb|300px|Euler computation of a tip vortex rolling up from the trailed vorticity sheet]]

The difference in the spanwise component of velocity above and below the wing (between being in the inboard direction above and in the outboard direction below) persists at the trailing edge and into the wake downstream. After the flow leaves the trailing edge, this difference in velocity takes place across a relatively thin shear layer called a vortex sheet.

===Horseshoe vortex system===
[[Image:Wing horseshoe vortex.jpg|thumb|right|300px|Planview of a wing showing the horseshoe vortex system]]
The wingtip flow leaving the wing creates a tip vortex. As the main vortex sheet passes downstream from the trailing edge, it rolls up at its outer edges, merging with the tip vortices. The combination of the [[wingtip vortices]] and the vortex sheets feeding them is called the vortex wake.

In addition to the vorticity in the trailing vortex wake there is vorticity in the wing's boundary layer, called 'bound vorticity', which connects the trailing sheets from the two sides of the wing into a vortex system in the general form of a horseshoe. The horseshoe form of the vortex system was recognized by the British aeronautical pioneer Lanchester in 1907.<ref>Lanchester (1907)</ref>

Given the distribution of bound vorticity and the vorticity in the wake, the [[Biot–Savart law]] (a vector-calculus relation) can be used to calculate the velocity perturbation anywhere in the field, caused by the lift on the wing. Approximate theories for the lift distribution and lift-induced drag of three-dimensional wings are based on such analysis applied to the wing's horseshoe vortex system.<ref>Milne-Thomson (1966), Section 10.1</ref><ref>Clancy (1975), Section 8.9</ref> In these theories, the bound vorticity is usually idealized and assumed to reside at the camber surface inside the wing.

Because the velocity is deduced from the vorticity in such theories, some authors describe the situation to imply that the vorticity is the cause of the velocity perturbations, using terms such as "the velocity induced by the vortex", for example.<ref>Anderson (1991), Section 5.2</ref> But attributing mechanical cause-and-effect between the vorticity and the velocity in this way is not consistent with the physics.<ref>Batchelor (1967), Section 2.4</ref><ref>Milne-Thomson (1966), Section 9.3</ref><ref>Durand (1932), Section III.2</ref> The velocity perturbations in the flow around a wing are in fact produced by the pressure field.<ref>McLean (2012), Section 8.1</ref>