Definitions

# Lift-induced drag

In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is a 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, as the angle of attack increases, induced drag increases.

## Source of induced drag

It is not possible to have a wing of infinite span. However, 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 a two-dimensional flow. The aerodynamic force is resolved into two components. By definition, the component parallel to the vector representing the relative velocity between the wing and the air is the drag; and the component normal to that vector is the lift. At practical angles of attack the lift greatly exceeds the drag.

A wing produces lift by turning the airflow around the wing in a downwards direction. On a wing of finite span some air leaks around the wingtip from the lower surface to the upper surface producing a wingtip vortex. These vortices trail behind the wing for a great distance and can be powerful, producing hazardous conditions for following aircraft flying into them. The deflection of the airflow by the wing produces a down flow or 'downwash' behind the wing. The tip vortices also modify the airflow around the wing, relative to that on a wing of infinite span, reducing the effectiveness of the wing to generate lift, thus requiring a higher angle of attack to compensate, and tilting 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.

Unlike parasitic drag on an object (which is directly proportional to the square of the airspeed), for a given lift, induced drag on an airfoil is inversely proportional to the square of the airspeed. In straight and level flight of an aircraft, lift varies only slowly because it is approximately equal to the weight of the aircraft. Consequently in straight and level flight, the induced drag is inversely proportional to the square of the airspeed. At the speed for minimum drag, induced drag is equal to parasitic drag.

## Reducing Induced drag

Induced drag can be minimized by the following means:

• Increase the wing span. The effect of the wingtip vortices is greatest near the wing tips. With increased wingspan a lesser portion of the wing is in the most affected region. Increasing span with no other change would increase wing area. In practice, the wing area is kept constant by increasing the aspect ratio rather than the span.
• Optimise the spanwise load distribution. If the lift is diminished towards the wingtips there is less pressure differential near the wingtips to create wingtip vortices. For a given wingspan, minimum induced drag is achieved when the spanwise lift distribution is elliptical. The parameter with greatest effect on lift distribution is the wing planform. Thus, a wing with elliptical planform would have low induced drag. Few aircraft have this planform because of manufacturing complications — the most famous example is the World War II Spitfire. Tapered wings with straight leading and trailing edges can approximate to elliptical lift distribution. Typically, straight wings produce between 5–15% more induced drag than an elliptical wing. 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.
• Provide a physical barrier to vortex formation. Such a barrier might take several forms. Some early aircraft had fins mounted on the tips of the tailplane which served as endplates. More recent aircraft have wingtip mounted winglets to oppose the formation of vortices. Wingtip mounted fuel tanks may also provide some benefit.

## Calculation of Induced drag

For a wing with an elliptical lift distribution, induced drag is calculated as follows:

$D_i = frac\left\{1\right\}\left\{2\right\} rho V^2 S C_\left\{Di\right\} = frac\left\{1\right\}\left\{2\right\} rho_0 V_e^2 S C_\left\{Di\right\}$
where
$C_\left\{Di\right\} = frac\left\{k C_L^2\right\}\left\{ pi AR\right\}$ and

$C_L = frac\left\{L\right\}\left\{ frac\left\{1\right\}\left\{2\right\} rho_0 V_e^2 S\right\}$

Thus

$C_\left\{Di\right\} = frac\left\{k L^2\right\}\left\{frac\left\{1\right\}\left\{4\right\} rho_0^2 V_e^4 S^2 pi AR\right\}$

Hence

$D_i = frac\left\{k L^2\right\}\left\{frac\left\{1\right\}\left\{2\right\} rho_0 V_e^2 S pi AR\right\}$

Where:

$AR ,$ is the aspect ratio,
$C_\left\{Di\right\} ,$ is the induced drag coefficient (see Lifting-line theory),
$C_L ,$ is the lift coefficient,
$D_i ,$ is the induced drag,
$k ,$ is the factor by which the induced drag exceeds that of an elliptical lift distribution, typically 1.05 to 1.15,
$L ,$ 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,
$rho ,$ is the air density and
$rho_0 ,$ is 1.225 kg/m³, the air density at sea level, ISA conditions.

### Combined effect with other drag sources

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 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. An aircraft flying at this speed is at its optimal aerodynamic efficiency. The minimum drag speed occurs at the speed where the induced drag is equal to the parasitic drag. This is the speed at which the best gradient of climb, or for unpowered aircraft, minimum gradient of descent, is achieved.

The speed for best endurance, i.e. time in the air, is the speed for minimum fuel flow rate. The fuel flow rate is calculated as the product of the drag or power required and the engine specific fuel consumption. The engine specific fuel consumption will be expressed in units of fuel flow rate per unit of thrust or per unit of power depending on whether the engine output is measured in thrust, as for a jet engine, or power, as for a turbo-prop engine.

The speed for best range, i.e. distance travelled, occurs at the speed at which a tangent from the origin touches 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.

## References

• Clancy, L.J. (1975), Aerodynamics, Pitman Publishing Limited, London. ISBN 0 273 01120 0
• Abbott, Ira H., and Von Doenhoff, Albert E., Theory of Wing Sections, Dover Publications Inc., New York, Standard Book Number 486-60586-8