Lift-induced drag

Image:Induce drag downwash.png in the vicinity of the wing.

The grey vertical line labeled "L" is the force required to counteract the weight of the aircraft. The red vector labeled "L_eff" is the actual lift on the wing; it is perpendicular to the effective relative airflow in the vicinity of the wing. The lift generated by the wing has been tilted rearwards through an angle equal to the downwash angle in three-dimensional flow. The component of "L_eff" parallel to the free stream is the induced drag on the wing.Hurt, H. H. (1965) Aerodynamics for Naval Aviators, Figure 1.30, NAVWEPS 00-80T-80Clancy, L.J. (1975) Aerodynamics. Pitman Publishing Limited, London. {{ISBN|0-273-01120-0}}{{rp|Fig 5.24.}}Kermode, A.C. (1972). Mechanics of Flight, Figure 3.29, Ninth edition. Longman Scientific & Technical, England. {{ISBN|0-582-42254-X}}{{cite conference |last=McLean |first=Doug |date=2005 |title=Wingtip Devices: What They Do and How They Do It |conference=2005 Boeing Performance and Flight Operations Engineering Conference |url=http://www.smartcockpit.com/docs/Wingtip_Devices.pdf }}{{rp|4.4|quote=While the air more than about one wingspan ahead of the wing is essentially undisturbed, the general flow pattern of Figure 3.1 reaches practically full strength at a distance of about one wingspan behind the wing and generally persists over long distances downstream. At the location of the wing itself, the flow pattern has reached roughly half of its maximum strength, and the wing is flying through air that is already moving generally downward between the wingtips. Thus the wing can be thought of as flying in a downdraft of its own making. Because of the apparent downdraft, or "downwash," the total apparent lift vector is tilted backward slightly. It is the backward component of the apparent lift that is felt as induced drag.}}]]

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.{{cite book |last1=Anderson |first1=John D. Jr. |title=Fundamentals of aerodynamics |date=2017 |publisher=McGraw-Hill Education |location=New York, NY |isbn=978-1-259-12991-9 |page=20 |edition=Sixth}}{{rp|Section 5.3}} At practical angles of attack the lift greatly exceeds the drag.Abbott, Ira H., and Von Doenhoff, Albert E., Theory of Wing Sections, Section 1.2 and Appendix IV

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), which is an acceleration. To change the direction of the flow therefore requires that a force be applied to the fluid; the total aerodynamic force is simply the reaction force of the fluid acting on the wing.

An aircraft in slow flight at a high angle of attack will generate an aerodynamic reaction force with a high drag component. By increasing the speed and reducing the angle of attack, the lift generated can be held constant while the drag component is reduced. At the optimum angle of attack, total drag is minimised. If speed is increased beyond this, total drag will increase again due to increased profile drag.

When producing 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, around the wingtip, towards the upper surface.{{cite book |last=McLean |first=Doug |date=2012 |title=Understanding Aerodynamics: Arguing from the Real Physics |isbn=978-1119967514}}{{rp|8.1.1}} This spanwise flow of air combines with chordwise flowing air, which twists the airflow and produces vortices along the wing trailing edge.{{rp|4.6|quote=The induction myth is more complicated and involves a serious misunderstanding of cause and effect. The trailing vortex sheet and the rolled-up vortex cores are often talked about as if they were the direct cause of the velocities everywhere else in the flowfield, and of induced drag, but this is misleading. It is true that in order for the large-scale flow pattern of Figure 3.1 to exist, there must be a vortex sheet shed from the trailing edge, but the vortex sheet is not a direct physical cause of the large-scale flow; it is more of a manifestation.}}{{rp|4.7|quote=The induction myth leads us to think of induced drag as being "caused" by the vortex wake, and thus to think that by doing something very local to change the flow in the core of the “tip vortex” we can have a large effect on the induced drag.}}{{rp|8.1.4, 8.3, 8.4.1}}

The vortices reduce the 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.{{rp|Section 5.17}}

The vortices created are unstable,{{what|date=March 2022}} and they quickly combine to produce wingtip vortices which trail behind the wingtip.{{rp|Section 5.14}}

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

:$D\_\backslash text\{i\}\; =\; \backslash frac\{L^2\}\{\backslash frac\{1\}\{2\}\backslash rho\_0\; V\_E^2\; \backslash pi\; b^2\}$,

where

:$L\; \backslash ,$ is the lift,

:$\backslash rho\_0\; \backslash ,$ is the standard density of air at sea level,

:$V\_E\; \backslash ,$ is the equivalent airspeed,

:$\backslash pi\; \backslash ,$ is the ratio of circumference to diameter of a circle, and

:$b\; \backslash ,$ is the wingspan.

From this equation it is clear that the induced drag varies with the square of the lift; and inversely with the square of the equivalent airspeed; and inversely with the square of the 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 $e$.

To compare with other sources of drag, it can be convenient to express this equation in terms of lift and drag coefficients:Anderson, John D. (2005), Introduction to Flight, McGraw-Hill. {{ISBN|0-07-123818-2}}. p318

:$C\_\{D,i\}\; =\; \backslash frac\{D\_\backslash text\{i\}\}\{\backslash frac\{1\}\{2\}\backslash rho\_0\; V\_E^2\; S\}\; =\; \backslash frac\{C\_L^2\}\{\backslash pi\; A\backslash !\backslash !\backslash text\{R\}\; e\}$, where

:$C\_L\; =\; \backslash frac\{L\}\{\; \backslash frac\{1\}\{2\}\; \backslash rho\_0\; V\_E^2\; S\}$

and

:$A\backslash !\backslash !\backslash text\{R\}=\backslash frac\{b^2\}\{S\}\; \backslash ,$ is the aspect ratio,

:$S\; \backslash ,$ is a reference wing area,

:$e\; \backslash ,$ is the span efficiency factor.

This indicates how, for a given wing area, high aspect ratio wings are beneficial to flight efficiency. With $C\_L$ being a function of angle of attack, induced drag increases as the angle of attack increases.{{rp|Section 5.17}}

The above equation can be derived using Prandtl's lifting-line theory.{{cn|date=April 2022}} Similar methods can also be used to compute the minimum induced drag for non-planar wings or for arbitrary lift distributions.{{cn|date=April 2022}}

According to the equations above, 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 (or a 2D wing) would experience no induced drag.{{cite book |last1=Houghton |first1=E. L. |title=Aerodynamics for engineering students |date=2012 |publisher=Elsevier Science |location=Waltham, MA |isbn=978-0-08-096632-8 |page=61 |edition=Sixth |chapter=1.6 |quote=For a two-dimensional wing at low Mach numbers, the drag contains no induced or wave drag}} The drag characteristics of a wing with infinite span can be simulated using an airfoil section the width of a wind tunnel.{{cite book |last1=Molland |first1=Anthony F. |title=Marine rudders and control surfaces : principles, data, design and applications |date=2007 |publisher=Elsevier/Butterworth-Heinemann |location=Amsterdam |isbn=9780750669443 |page=41 |edition=1st |chapter=Physics of control surface operation |quote=With infinite span, fluid motion is 2-D and in the direction of flow perpendicular to the span. Infinite span can, for example, be simulated using a foil completely spanning a wind tunnel.}}

An increase in wingspan or a solution with a similar effect is one way to reduce induced drag.{{rp|4.10|quote=Based on our general appreciation of the physics, we can anticipate that drag-reduction devices need to be fairly large as viewed in the Trefftz plane, since any significant reduction in induced drag requires changing the global flowfield associated with the lift, so as to reduce its total kinetic energy. We know that we can’t do this just by tinkering with the "tip vortex" and thus that having a significant effect on the drag requires a significant change in the way the lift is distributed spatially. If our starting point is a wing on which the lift is already advantageously distributed, the only way to improve will be to provide a significant increase in the horizontal span or to introduce a nonplanar element that has a similar effect.}} The Wright brothers used curved trailing edges on their rectangular wings. Some early aircraft had fins mounted on the tips. More recent aircraft have wingtip-mounted winglets to reduce the induced drag.{{cite tech report |author=Richard T. Whitcomb |title=A design approach and selected wind-tunnel results at high subsonic speeds for wing-tip mounted winglets |publisher=NASA |date=July 1976 |id=19760019075 |url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760019075.pdf |quote-page=1 |quote=Winglets, which are small, nearly vertical, winglike surfaces mounted at the tips of a wing, are intended to provide, for lifting conditions and subsonic Mach numbers, reductions in drag coefficient greater than those achieved by a simple wing-tip extension with the same structural weight penalty.}} Winglets also provide some benefit by increasing the vertical height of the wing system.{{rp|4.10|quote=Trefftz-plane theory tells us that we can reduce the ideal induced drag by increasing the vertical height of the lifting system, as well as by increasing the horizontal span. A vertical fin or winglet that adds vertical height to the system will reduce the ideal induced drag if it is placed anywhere along the span of the wing off of the airplane center plane, but it is most effective by far when it is placed at the station of maximum span; that is, at the tip.}} Wingtip mounted fuel tanks and wing washout may also provide some benefit.{{cn|date=January 2022}}

Typically, the elliptical spanwise distribution of lift produces the minimum induced dragGlauert, H. The Elements of Aerofoil and Airscrew Theory (1926); referenced in Fig. 5.4 of Airplane Aerodynamics by Daniel O. Dommasch, Sydney S. Sherby, Thomas F. Connolly, 3rd ed. (1961) for a planar wing of a given span. A small number of aircraft have a planform approaching the elliptical — the most famous examples being the World War II Spitfire{{cite web |title=Induced Drag Coefficient |url=https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/induced-drag-coefficient/ |website=www.grc.nasa.gov |access-date=9 February 2023}} and Thunderbolt. For modern wings with winglets, the ideal lift distribution is not elliptical.{{rp|4.9|quote=The well-known elliptic spanload is "ideal" for a planar (flat) wing. For nonplanar configurations, the ideal spanload is not generally elliptic, but it is easily calculated for a given geometry. With a vertical winglet added, for example, the ideal spanload shows less lift inboard and more lift outboard, relative to elliptic, with a certain, optimum distribution on the winglet itself, as shown in Figure 3.5... Relative to these "ideal" spanloads, the spanloads used on real wings are usually modified somewhat to reduce bending loads and allow a lighter wing structure, at the expense of a slight increase in drag. The presence a fuselage and wing-mounted engines also tends to alter the spanload on real wings.}}

For a given wing area, a high aspect ratio wing will produce less induced drag than a wing of low aspect ratio.{{cite web |url=http://www.skybrary.aero/index.php/Induced_Drag |title=Skybrary: Induced Drag|access-date=5 May 2015}} While induced drag is inversely proportional to the square of the wingspan, not necessarily inversely proportional to aspect ratio, if the wing area is held constant, then induced drag will be inversely proportional to aspect ratio. However, since wingspan can be increased while decreasing aspect ratio, or vice versa, the apparent relationship between aspect ratio and induced drag does not always hold.{{cite web |last1=Illsley |first1=Michael |title=Why Aspect Ratio doesn't Matter – Understanding Aerospace |url=https://bigsynthesis.com/understandingaerospace/index.php/21-why-aspect-ratio-doesn-t-matter/ |website=Understanding Aerospace |date=4 July 2017 |access-date=25 March 2022}}{{rp|489}}

For a typical twin-engine wide-body aircraft at cruise speed, induced drag is the second-largest component of total drag, accounting for approximately 37% of total drag. Skin friction drag is the largest component of total drag, at almost 48%.{{cite journal |last1=Robert |first1=JP |editor1-last=Cousteix |editor1-first=J |title=Drag reduction: an industrial challenge |journal=Special Course on Skin Friction Drag Reduction |volume=AGARD Report 786 |page=2-13 |date=March 1992 |publisher=AGARD |url=https://www.sto.nato.int/publications/_layouts/mobile/view.aspx?List=03e8ea21%2D64e6%2D4d37%2D8235%2D04fb61e122e9&View=7e9c814c%2D056a%2D4d31%2D8392%2D7c6752b2af2b&RootFolder=%2Fpublications%2FAGARD%2FAGARD%2DR%2D786&ViewMode=Detail }}{{cite journal |last1=Coustols |first1=Eric |editor1-last=Meier |editor1-first=GEA |editor2-last=Schnerr |editor2-first=GH |title=Control of Turbulent Flows for Skin Friction Drag Reduction |journal=Control of Flow Instabilities and Unsteady Flows |date=1996 |page=156 |isbn=9783709126882 |url=https://books.google.com/books?id=w1ruCAAAQBAJ&pg=PA156 |access-date=24 March 2022}}{{cite conference |last1=Marec |first1=J.-P. |title=Aerodynamic Drag Reduction Technologies. Proceedings of the Ceas/Dragnet European Drag Reduction |chapter=Drag Reduction: A Major Task for Research |date=2001 |pages=17–27 |doi=10.1007/978-3-540-45359-8_3 |chapter-url=https://link.springer.com/chapter/10.1007/978-3-540-45359-8_3 |access-date=22 March 2022 |editor=Peter Thiede |isbn=978-3-642-07541-4 |issn=0179-9614 |publisher=Springer |bibcode=2001adrt.conf...17M |language=en}}{{rp|20}} Reducing induced drag can therefore significantly reduce cost and environmental impact.{{rp|18}}

File:Drag curves for aircraft in flight.svg plus induced drag]]

In 1891, Samuel Langley published the results of his experiments on various flat plates. At the same airspeed and the same angle of attack, plates with higher aspect ratio produced greater lift and experienced lower drag than those with lower aspect ratio.

His experiments were carried out at relatively low airspeeds, slower than the speed for minimum drag.{{cite book |last1=Hallion |first1=Richard |title=Taking Flight: Inventing the Aerial Age, from Antiquity Through the First World War |date=8 May 2003 |publisher=Oxford University Press, USA |isbn=978-0-19-516035-2 |page=147 |url=https://books.google.com/books?id=YRqV_PayIKIC |access-date=13 April 2022 |language=en}} He observed that, at these low airspeeds, increasing speed required reducing power.{{cite book |last1=Hansen |first1=James R. |title=The Bird Is on the Wing: Aerodynamics and the Progress of the American Airplane |date=2004 |publisher=Texas A&M University Press |location=College Station |isbn=978-1-58544-243-0 |page=23 |url=https://books.google.com/books?id=GDDQ8jQmPTEC |access-date=13 April 2022 |language=en}} (At higher airspeeds, parasitic drag came to dominate, causing the power required to increase with increasing 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 (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 (V_MD). 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.{{rp|Section 5.25}} This is the speed at which for unpowered aircraft, optimum glide angle is achieved. This is also the speed for greatest range (although V_MD 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 shallow and it is customary to operate at the speed for 99% best range since this gives 3-5% greater speed for only 1% less range. 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 sufficient to keep it aloft while flying at the angle of attack that minimizes the drag. The optimum altitude will increase 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 always 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{{efn|The engine specific fuel consumption is normally 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 shaft horsepower, as for a propeller engine. To convert fuel rate per unit thrust to fuel rate per unit power one must divide by the speed.}}). The power required is equal to the drag times the speed.

• Aerodynamic force
• Drag
• Oswald efficiency number
• Parasitic drag
• Wave drag
• Wingtip vortices

{{notelist}}

{{reflist}}

• L. J. Clancy (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, Standard Book Number 486-60586-8
• Luciano Demasi, Antonio Dipace, Giovanni Monegato, and Rauno Cavallaro. Invariant Formulation for the Minimum Induced Drag Conditions of Nonplanar Wing Systems, AIAA Journal, Vol. 52, No. 10 (2014), pp. 2223–2240. [https://doi.org/10.2514/1.J052837 doi: 10.2514/1.J052837]

• {{YouTube|id=QKCK4lJLQHU|t=30m46s|title=Doug McLean, Common Misconceptions in Aerodynamics}}

Category:Aircraft aerodynamics

Category:Drag (physics)

Category:Gliding technology