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. It is symbolized as „${\textstyle D_{\text{i}}}$”, and the lift-induced drag coefficient as „${\textstyle C_{D,i}}$”.

Samuel Langley observed higher aspect ratio flat plates had higher lift and lower drag and stated in 1902 “A plane of fixed size and weight would need less propulsive power the faster it flew”, the counter-intuitive effect of induced drag.

Source of induced drag

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.

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 for wings generating the same lift the induced drag is inversely proportional to the square of the wingspan. A wing of infinite span and uniform airfoil section would experience no induced drag. The drag characteristics of such an airfoil section can be measured on a scale-model wing spanning the width of a wind tunnel.

An increase in wingspan or a solution with a similar effect is only way. Some early aircraft had fins mounted on the tips. More recent aircraft have wingtip mounted winglets to reduce the induced drag. Wingtip mounted fuel tanks and wing washout may also provide some benefit.

Typically, the elliptical spanload (spanwise distribution of lift) produces the minimum of the induced drag for planar wings. A small number of aircraft have a planform approaching the elliptical — the most famous examples being the World War II Spitfire and Thunderbolt. Tapered wings with straight leading edges can also approximate an elliptical lift distribution. For modern wings with winglets, the ideal spanload is not elliptical.

Similarly, for a given wing area, a high aspect ratio wing will produce less induced drag than a wing of low aspect ratio. Therefore for wings of a given area, induced drag can be said to be inversely proportional to aspect ratio.

Calculation of induced drag

For a planar wing with an elliptical lift distribution, induced drag Di can be calculated as follows:

${\displaystyle D_{\text{i}}={\frac {L^{2}}{{\frac {1}{2}}\rho _{0}V_{E}^{2}\pi b^{2}}}}$,

where

${\displaystyle L\,}$ is the lift,
${\displaystyle \rho _{0}\,}$ is the standard density of air at sea level,
${\displaystyle V_{E}\,}$ is the equivalent airspeed,
${\displaystyle \pi \,}$ is the ratio of circumference to diameter of a circle, and
${\displaystyle b\,}$ is the wingspan.

From this equation it is clear that the induced drag decreases with flight speed and with wingspan. Deviation from the non-planar wing with elliptical lift distribution are taken into account by dividing the induced drag by the span efficiency factor ${\displaystyle e}$.

To compare with other sources of drag, it can be convenient to express this equation in terms of lift and drag coefficients:

${\displaystyle C_{D,i}={\frac {D_{\text{i}}}{{\frac {1}{2}}\rho _{0}V_{E}^{2}S}}={\frac {C_{L}^{2}}{\pi A\!\!{\text{R}}e}}}$, where
${\displaystyle C_{L}={\frac {L}{{\frac {1}{2}}\rho _{0}V_{E}^{2}S}}}$

and

${\displaystyle A\!\!{\text{R}}={\frac {b^{2}}{S}}\,}$ is the aspect ratio,
${\displaystyle S\,}$ is a reference wing area.

This indicates how high aspect ratio wings are beneficial to flight efficiency. With ${\displaystyle C_{L}}$ being a function of angle of attack, induced drag increases as the angle of attack increases.

The above equation can be derived using Prandtl's lifting-line theory. Similar methods can also be used to compute the minimum induced drag for non-planar wings or for arbitrary lift distributions.

Combined effect with other drag sources

Total drag is parasitic drag plus induced drag

Induced drag must be added to the parasitic drag to find the total drag. Since induced drag is inversely proportional to the square of the airspeed (at a given lift) whereas parasitic drag is proportional to the square of the airspeed, the combined overall drag curve shows a minimum at some airspeed - the minimum drag speed (VMD). An aircraft flying at this speed is operating at its optimal aerodynamic efficiency. According to the above equations, the speed for minimum drag occurs at the speed where the induced drag is equal to the parasitic drag. This is the speed at which for unpowered aircraft, optimum glide angle is achieved. This is also the speed for greatest range (although VMD will decrease as the plane consumes fuel and becomes lighter). The speed for greatest range (i.e., distance travelled) is the speed at which a straight line from the origin is tangent to the fuel flow rate curve. The curve of range versus airspeed is normally very flat and it is customary to operate at the speed for 99% best range since this gives about 5% greater speed for only 1% less range. (Of course, flying higher where the air is thinner will raise the speed at which minimum drag occurs, and so permits a faster voyage for the same amount of fuel. If the plane is flying at the maximum permissible speed, then there is an altitude at which the air density will be what is needed to keep it aloft while flying at the angle of attack that minimizes the drag. The optimum altitude at maximum speed, and the optimum speed at maximum altitude, may change during the flight as the plane becomes lighter.)

The speed for maximum endurance (i.e., time in the air) is the speed for minimum fuel flow rate, and is less than the speed for greatest range. The fuel flow rate is calculated as the product of the power required and the engine specific fuel consumption (fuel flow rate per unit of power). The power required is equal to the drag times the speed.