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, induced drag increases as the angle of attack increases.[1]

Source of induced drag

Induced drag is directly related to the amount of induced downwash at the trailing edge of the wing. The lift generated by the wing is effectively tilted rearwards through an angle corresponding to the induced downwash angle. The total aerodynamic force created by the wing (Leff) is therefore reduced.[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

Theoretically a wing of infinite span 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 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 Spitfire and Thunderbolt. 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, 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 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:

$D_i = \frac{1}{2} \rho V^2 S C_{Di} = \frac{1}{2} \rho_0 V_e^2 S C_{Di}$

where

$C_{Di} = \frac{C_L^2}{ \pi e AR}$ and
$C_L = \frac{L}{ \frac{1}{2} \rho_0 V_e^2 S}$

Thus

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

Hence

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

Where:

$AR \,$ is the aspect ratio,
$C_{Di} \,$ 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 0.85 to 0.95,
$L \,$ is the lift,
$S \,$ is the gross wing area: the product of the wing span and the Mean Aerodynamic Chord.[1]
$V \,$ 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

Curves showing induced, parasitic, and combined drag vs airspeed

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 (VMD). 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.[8]

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. (1959), Theory of Wing Sections, Dover Publications Inc., New York, Standard Book Number 486-60586-8

Notes

1. ^ a b Clancy, L.J., Aerodynamics, Section 5.17
2. ^ Hurt, H. H. (1965) Aerodynamics for Naval Aviators, Figure 1.30, NAVWEPS 00-80T-80
3. ^ Clancy, L.J. (1975) Aerodynamics Fig 5.24. Pitman Publishing Limited, London. ISBN 0-273-01120-0
4. ^ Kermode, A.C. (1972). ‘’Mechanics of Flight’’, Figure 3.29, Ninth edition. Longman Scientific & Technical, England. ISBN 0-582-42254-X
5. ^ Clancy, L.J., Aerodynamics, Section 5.14
6. ^ Clancy, L.J., Aerodynamics, Section 5.3
7. ^ Abbott, Ira H., and Von Doenhoff, Albert E., Theory of Wing Sections, Section 1.2 and Appendix IV
8. ^ Clancy, L.J., Aerodynamics, Section 5.25