Lift induced drag

Source of induced drag

The total aerodynamic force acting on a body is usually thought of as having 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. At practical angles of attack the lift greatly exceeds the drag.

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.

To produce lift, air below the wing is at a higher pressure than the air pressure above the wing. 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. 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, reducing wing's ability to generate lift, so that it requires a higher angle of attack for the same lift, which tilts the total aerodynamic force rearwards and increases the drag component of that force. 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.

Reducing induced drag

According to the equations below, a wing of infinite aspect ratio (wingspan/chord length) and constant airfoil section would produce no induced drag. 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.

A rectangular planform wing produces stronger wingtip vortices than does 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 Spitfire and Thunderbolt. Tapered wings with straight leading and trailing edges can approximate to elliptical lift distribution. Typically, straight edged non-tapered wings produce 5%, and tapered wings produce 1-2% more induced drag than an elliptical wing.

Similarly, for a given wing area, a high aspect ratio wing will produce less induced drag than a wing of low aspect ratio because there is less air disturbance at the tip of a longer, thinner wing. Induced drag can therefore be said to be inversely proportional to aspect ratio. The lift distribution may also be modified by the use of washout, a spanwise twist of the wing to reduce the incidence towards the wingtips, and by changing the airfoil section near the wingtips. This allows more lift to be generated nearer the wing root and less towards the wingtip, which causes a reduction in the strength of the wingtip vortices.

Some early aircraft had fins mounted on the tips of the tailplane which served as endplates. More recent aircraft have wingtip mounted winglets to reduce the intensity of wingtip vortices. Wingtip mounted fuel tanks may also provide some benefit, by preventing the spanwise flow of air around the wingtip.

Calculation of induced drag

For a planar wing with an elliptical lift distribution, induced drag is often calculated as follows. These equations make the induced drag depend on the square of the lift, for a given aspect ratio and surface area (while varying the angle of attack), but as the accompanying graph shows, this is only an approximation and is not valid at high angles of attack (and probably not for very high values of aspect ratio either).

D i = 1 2 ρ V 2 S C D i = 1 2 ρ 0 V e 2 S C D i

where

C D i = C L 2 π e A R and C L = L 1 2 ρ 0 V e 2 S

Thus

C D i = L 2 1 4 ρ 0 2 V e 4 S 2 π e A R

Hence

D i = L 2 1 2 ρ 0 V e 2 S π e A R

where:

A R is the aspect ratio, C D i is the induced drag coefficient (see Lifting-line theory), C L is the lift coefficient, D i is the induced drag, e is the wing span efficiency value by which the induced drag exceeds that of an elliptical lift distribution, typically 0.95-0.99, L is the lift, S is the gross wing area: the product of the wing span and the Mean Aerodynamic Chord. V is the true airspeed, V e is the equivalent airspeed, ρ is the air density and ρ 0 is 1.225 kg/m³, the air density at sea level, ISA conditions.