Lift-induced drag
In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is an aerodynamic drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings or a lifting body redirecting air to cause lift and also in cars with airfoil wings that redirect air to cause a downforce. With other parameters remaining the same, induced drag increases as the angle of attack increases.[1]
1 Source of induced drag
2 Reducing induced drag
3 Calculation of induced drag
4 Combined effect with other drag sources
5 Minimum induced drag for generic non-planar systems
5.1 Optimal aerodynamic efficiency ratio
5.2 Augmented Munk's minimum induced drag theorem
5.3 Minimum induced drag curvature-invariance theorem
5.4 Quasi-closed C-wing zero gradient optimal circulation theorem
5.5 Optimal induced drag: minimization procedure and results
Source of induced drag[edit]
Induced drag is related to the amount of induced downwash in the vicinity of the wing. The grey vertical line labeled "L" is perpendicular to the free stream and indicates the orientation of the lift on the wing. The red vector labeled "Leff" is perpendicular to the actual airflow in the vicinity of the wing; it represents the lift on the airfoil section in two-dimensional flow at the same angle of attack. The lift generated by the wing has been tilted rearwards through an angle equal to the angle of the downwash in three-dimensional flow. The component of "Leff" parallel to the free stream is the induced drag on the wing.[2][3][4]
Lift is produced by the changing direction of the flow around a wing. The change of direction results in a change of velocity (even if there is no speed change, just as seen in uniform circular motion), which is an acceleration. To change the direction of the flow therefore requires that a force be applied to the fluid; lift is simply the reaction force of the fluid acting on the wing.
When producing lift, air below the wing is generally at a higher pressure than the air pressure above the wing, while air above the wing is generally at a lower than atmospheric pressure. On a wing of finite span, this pressure difference causes air to flow from the lower surface wing root, around the wingtip, towards the upper surface wing root. This spanwise flow of air combines with chordwise flowing air, causing a change in speed and direction, which twists the airflow and produces vortices along the wing trailing edge. The vortices created are unstable, and they quickly combine to produce wingtip vortices.[5] The resulting vortices change the speed and direction of the airflow behind the trailing edge, deflecting it downwards, and thus inducing downwash behind the wing.
Wingtip vortices modify the airflow around a wing. Compared to a wing of infinite span, vortices reduce the effectiveness of the wing to generate lift, thus requiring a higher angle of attack to compensate, which tilts the total aerodynamic force rearwards. The angular deflection is small and has little effect on the lift. However, there is an increase in the drag equal to the product of the lift force and the angle through which it is deflected. Since the deflection is itself a function of the lift, the additional drag is proportional to the square of the lift.[1]
The total aerodynamic force is usually thought of as two components, lift and drag. By definition, the component of force parallel to the oncoming flow is called drag; and the component perpendicular to the oncoming flow is called lift.[6] At practical angles of attack the lift greatly exceeds the drag.[7]
Reducing induced drag[edit]
According to the equations below, a wing of infinite aspect ratio (wingspan/chord length) and constant airfoil section would seemingly produce no induced drag per unit of surface area (although this depends on what happens to the factor e). The characteristics of such a wing can be measured on a section of wing spanning the width of a wind tunnel, since the walls block spanwise flow and create what is effectively two-dimensional flow. However, in reality, there will still be more drag (induced drag) when the angle of attack is high than when it is zero or slightly negative.
A rectangular wing produces much more severe wingtip vortices than a tapered or elliptical wing, therefore many modern wings are tapered. However, an elliptical planform is more efficient as the induced downwash (and therefore the effective angle of attack) is constant across the whole of the wingspan. Few aircraft have this planform because of manufacturing complications  -  the most famous examples being the World War II