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. 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.
Reducing induced drag
According to the equations below, a wing of infinite aspect ratio 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 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 or in extreme cases, propellers, 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 can be calculated as follows: where 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. To compare with other sources of drag, it can be convenient to express this equation in terms of lift and drag coefficients: and This indicates how high-aspect ratio wings are beneficial to flight efficiency. With being a function of angle of attack, induced drag increases as the angle of attack increases, up to the stall angle. 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
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 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 would also be the speed for greatest range if the efficiency of the engine were constant with speed. But in fact, the efficiency of jet engines increases with speed, so greatest range is achieved at a speed greater than the speed of minimum drag. The speed for greatest range 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 optimal angle of attack. The optimum altitude at maximum speed, and the optimum speed at maximum altitude, change during the flight as the plane consumes fuel and becomes lighter. For this reason airliners normally climb during long flights. The speed for maximum endurance 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. The power required is equal to the drag times the speed.
