Lift-induced drag

# Lift-induced drag

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In aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...

, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is a drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wing
Wing
A wing is an appendage with a surface that produces lift for flight or propulsion through the atmosphere, or through another gaseous or liquid fluid...

s or a lifting body
Lifting body
A lifting body is a fixed-wing aircraft configuration in which the body itself produces lift. In contrast to a flying wing, which is a wing with minimal or no conventional fuselage, a lifting body can be thought of as a fuselage with little or no conventional wing...

redirecting air to cause lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

and also in cars with airfoil wings that redirect air to cause a downforce
Downforce
Downforce is a downwards thrust created by the aerodynamic characteristics of a car. The purpose of downforce is to allow a car to travel faster through a corner by increasing the vertical force on the tires, thus creating more grip....

. With other parameters remaining the same, induced drag increases as the angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...

increases.

## Source of induced drag

Lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

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 atmospheric pressure, 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
Wingtip vortices
Wingtip vortices are tubes of circulating air that are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure...

. The resulting vortices change the speed and direction of the airflow behind the trailing edge, deflecting it downwards, and thus inducing downwash
Downwash
In aeronautics downwash is the air forced down by the aerodynamic action of a wing or helicopter rotor blade in motion, as part of the process of producing lift....

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.

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. At practical angles of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...

the lift greatly exceeds the drag. Unlike parasitic drag
Parasitic drag
Parasitic drag is drag caused by moving a solid object through a fluid medium . Parasitic drag is made up of many components, the most prominent being form drag...

on an object (which is proportional to the square of the airspeed), for a given lift, induced drag on an airfoil
Airfoil
An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....

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

Theoretically a wing of infinite span
Wingspan
The wingspan of an airplane or a bird, is the distance from one wingtip to the other wingtip. For example, the Boeing 777 has a wingspan of about ; and a Wandering Albatross caught in 1965 had a wingspan of , the official record for a living bird.The term wingspan, more technically extent, is...

and constant airfoil
Airfoil
An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....

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
Wind tunnel
A wind tunnel is a research tool used in aerodynamic research to study the effects of air moving past solid objects.-Theory of operation:Wind tunnels were first proposed as a means of studying vehicles in free flight...

, since the walls block spanwise flow and create what is effectively two-dimensional flow.

A rectangular wing produces much more severe wingtip vortices than a tapered or elliptical wing
Elliptical wing
An elliptical wing is a wing planform shape that minimizes induced drag. Elliptical taper shortens the chord near the wingtips in such a way that all parts of the wing experience equivalent downwash, and lift at the wing tips is essentially zero, improving aerodynamic efficiency due to a greater...

, 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
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...

Spitfire
Supermarine Spitfire
The Supermarine Spitfire is a British single-seat fighter aircraft that was used by the Royal Air Force and many other Allied countries throughout the Second World War. The Spitfire continued to be used as a front line fighter and in secondary roles into the 1950s...

and Thunderbolt
P-47 Thunderbolt
Republic Aviation's P-47 Thunderbolt, also known as the "Jug", was the largest, heaviest, and most expensive fighter aircraft in history to be powered by a single reciprocating engine. It was heavily armed with eight .50-caliber machine guns, four per wing. When fully loaded, the P-47 weighed up to...

. 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.

Similarly, a high aspect ratio wing will produce less induced drag than a wing of low aspect ratio because the size of the wing vortices will be much reduced on 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
Washout (aviation)
Washout refers to a feature of wing design to deliberately reduce the lift distribution across the span of the wing of an aircraft. The wing is designed so that angle of incidence is higher at the wing roots and decreases across the span, becoming lowest at the wing tip...

, a spanwise twist of the wing to reduce the incidence towards the wingtips, and by changing the airfoil
Airfoil
An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....

section near the wingtips. This allows more lift to be generated at 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 or wing fences to oppose the formation of 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 wing with an elliptical lift distribution, induced drag is calculated as follows:

where and

Thus

Hence

Where: is the aspect ratio
Aspect ratio (wing)
In aerodynamics, the aspect ratio of a wing is essentially the ratio of its length to its breadth . A high aspect ratio indicates long, narrow wings, whereas a low aspect ratio indicates short, stubby wings....

, is the induced drag coefficient
Drag coefficient
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...

(see Lifting-line theory
Lifting-line theory
Lifting-line theory or Lanchester-Prandtl wing theory was published by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk on the problem of a useful mathematical tool for examining lift from "real world" wings....

), is the lift coefficient
Lift coefficient
The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body, the dynamic pressure of the fluid flow around the body, and a reference area associated with the body...

, is the induced drag, is the wing span efficiency value by which the induced drag exceeds that of an elliptical lift distribution, typically 0.85 to 0.95, is the lift, is the gross wing area: the product of the wing span and the Mean Aerodynamic Chord.http://pilotsweb.com/principle/liftdrag.htm is the true airspeed, is the equivalent airspeed
Equivalent airspeed
Equivalent airspeed is the airspeed at sea level in the International Standard Atmosphere at which the dynamic pressure is the same as the dynamic pressure at the true airspeed and altitude at which the aircraft is flying. In low-speed flight, it is the speed which would be shown by an airspeed...

, is the air density and is 1.225 kg/m³, the air density at sea level, ISA
International Standard Atmosphere
The International Standard Atmosphere is an atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes. It has been established to provide a common reference for temperature and pressure and consists of tables of...

conditions.

### Combined effect with other drag sources

Induced drag must be added to the parasitic drag
Parasitic drag
Parasitic drag is drag caused by moving a solid object through a fluid medium . Parasitic drag is made up of many components, the most prominent being form 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.

• Aerodynamic force
• Drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

• Oswald efficiency number
Oswald efficiency number
The Oswald efficiency, similar to the span efficiency, is a correction factor that represents the change in drag with lift of a three dimensional wing or airplane, as compared with an ideal wing having the same aspect ratio and an elliptical lift distribution.-Definition:The Oswald efficiency is...

• Parasite drag
• Wave drag
Wave drag
In aeronautics, wave drag is a component of the drag on aircraft, blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects.- Overview :...