The drag curve or drag polar is the relationship between the
drag on an aircraft and other variables, such as lift, the coefficient of lift, angle-of-attack or speed. It may be described by an equation or displayed as a graph (sometimes called a "polar plot").
[ Drag may be expressed as actual drag or the coefficient of drag.
Drag curves are closely related to other curves which do not show drag, such as the power required/speed curve, or the sink rate/speed curve.
]
The drag curve
The significant aerodynamic properties of aircraft wings are summarised by two dimensionless quantities
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, the lift
Lift or LIFT may refer to:
Physical devices
* Elevator, or lift, a device used for raising and lowering people or goods
** Paternoster lift, a type of lift using a continuous chain of cars which do not stop
** Patient lift, or Hoyer lift, mobile ...
and drag coefficient
In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) 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 equ ...
s and . Like other such aerodynamic quantities, they are functions only of the angle of attack
In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is m ...
, the Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
and the Mach number
Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound.
It is named after the Moravian physicist and philosopher Ernst Mach.
: \mathrm = \frac ...
. and can be plotted against , or can be plotted against each other.[
The lift and the drag forces, and , are scaled by the same factor to get and , so = . and are at right angles, with parallel to the free stream velocity (the relative velocity of the surrounding distant air), so the resultant force lies at the same angle to as the line from the origin of the graph to the corresponding , point does to the axis.
If an aerodynamic surface is held at a fixed angle of attack in a ]wind tunnel
Wind tunnels are large tubes with air blowing through them which are used to replicate the interaction between air and an object flying through the air or moving along the ground. Researchers use wind tunnels to learn more about how an aircraft ...
, and the magnitude and direction of the resulting force are measured, they can be plotted using polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
. When this measurement is repeated at different angles of attack the drag curve is obtained. Lift and drag data was gathered in this way in the 1880s by Otto Lilienthal
Karl Wilhelm Otto Lilienthal (23 May 1848 – 10 August 1896) was a German pioneer of aviation who became known as the "flying man". He was the first person to make well-documented, repeated, successful flights with gliders, therefore making ...
and around 1910 by Gustav Eiffel
Alexandre Gustave Eiffel (born Bonickhausen dit Eiffel; ; ; 15 December 1832 – 27 December 1923) was a French civil engineer. A graduate of École Centrale des Arts et Manufactures, he made his name with various bridges for the French railway ...
, though not presented in terms of the more recent coefficients. Eiffel was the first to use the name "drag polar", however drag curves are rarely plotted today using polar coordinates.
Depending on the aircraft type, it may be necessary to plot drag curves at different Reynolds and Mach numbers. The design of a fighter will require drag curves for different Mach numbers, whereas gliders, which spend their time either flying slowly in thermals or rapidly between them, may require curves at different Reynolds numbers but are unaffected by compressibility effects. During the evolution of the design the drag curve will be refined. A particular aircraft may have different curves even at the same and values, depending for example on whether undercarriage
Undercarriage is the part of a moving vehicle that is underneath the main body of the vehicle. The term originally applied to this part of a horse-drawn carriage, and usage has since broadened to include:
*The landing gear of an aircraft.
*The ch ...
and flaps are deployed.[
]
The accompanying diagram shows against for a typical light aircraft
A light aircraft is an aircraft that has a maximum gross takeoff weight of or less.Crane, Dale: ''Dictionary of Aeronautical Terms, third edition'', page 308. Aviation Supplies & Academics, 1997.
Light aircraft are used as utility aircraft c ...
. The minimum point is at the left-most point on the plot. One component of drag is induced drag
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 ...
(an inevitable side-effect of producing lift, which can be reduced by increasing the indicated airspeed
Indicated airspeed (IAS) is the airspeed of an aircraft as measured by its pitot-static system and displayed by the airspeed indicator (ASI). This is the pilots' primary airspeed reference.
This value is not corrected for installation error, inst ...
). This is proportional to . The other drag mechanisms, parasitic
Parasitism is a close relationship between species, where one organism, the parasite, lives on or inside another organism, the host, causing it some harm, and is adapted structurally to this way of life. The entomologist E. O. Wilson has c ...
and wave drag
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
, have both constant components, totalling , and lift-dependent contributions that increase in proportion to . In total, then
::
The effect of is to shift the curve up the graph; physically this is caused by some vertical asymmetry, such as a cambered wing or a finite angle of incidence, which ensures the minimum drag attitude produces lift and increases the maximum lift-to-drag ratio
In aerodynamics, the lift-to-drag ratio (or L/D ratio) is the lift generated by an aerodynamic body such as an aerofoil or aircraft, divided by the aerodynamic drag caused by moving through air. It describes the aerodynamic efficiency under give ...
.[
]
Power required curves
One example of the way the curve is used in the design process is the calculation of the power required () curve, which plots the power needed for steady, level flight over the operating speed range. The forces involved are obtained from the coefficients by multiplication with , where ρ is the density of the atmosphere at the flight altitude, is the wing area and is the speed. In level flight, lift equals weight and thrust
Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that syst ...
equals drag, so
:::::::::: and
::::::::::.
The extra factor of /η, with η the propeller efficiency, in the second equation enters because = (required thrust)×/η. Power rather than thrust is appropriate for a propeller driven aircraft, since it is roughly independent of speed; jet engines produce constant thrust. Since the weight is constant, the first of these equations determines how falls with increasing speed. Putting these values into the second equation with from the drag curve produces the power curve. The low speed region shows a fall in lift induced drag, through a minimum followed by an increase in profile drag at higher speeds. The minimum power required, at a speed of 195 km/h (121 mph) is about 86 kW (115 hp); 135 kW (181 hp) is required for a maximum speed of 300 km/h (186 mph). Flight at the power minimum will provide maximum endurance; the speed for greatest range is where the tangent to the power curve passes through the origin, about 240 km/h (150 mph).[)
If an analytical expression for the curve is available, useful relationships can be developed by differentiation. For example the form above, simplified slightly by putting = 0, has a maximum at = . For a propeller aircraft this is the maximum endurance condition and gives a speed of 185 km/h (115 mph). The corresponding maximum range condition is the maximum of , at = , and so the optimum speed is 244 km/h (152 mph). The effects of the approximation = 0 are less than 5%; of course, with a finite = 0.1, the analytic and graphical methods give the same results.][
The low speed region of flight is known as the "back of the power curve"] (sometimes "back of the drag curve") where more power is required in order to fly slower. It is an inefficient region of flight because speed can be increased ''and'' power decreased; there is no trade-off between increased speed and increased power consumption. It is regarded as a "speed unstable" region of flight, because a decrease in speed will lead to a further increase in speed if power if not adjusted, unlike in normal circumstances.
Rate of climb
For an aircraft to climb at an angle θ and at speed its engine must be developing more power in excess of power required to balance the drag experienced at that speed in level flight and shown on the power required plot. In level flight = but in the climb there is the additional weight component to include, that is
:::::::::: = + .sin θ = + .
Hence the climb rate .sin θ = .[ Supposing the 135 kW engine required for a maximum speed at 300 km/h is fitted, the maximum excess power is 135 - 87 = 48 Kw at the minimum of and the rate of climb 2.4 m/s.
]
Fuel efficiency
For propeller aircraft (including turboprop
A turboprop is a turbine engine that drives an aircraft propeller.
A turboprop consists of an intake, reduction gearbox, compressor, combustor, turbine, and a propelling nozzle. Air enters the intake and is compressed by the compressor. Fuel ...
s), maximum range and therefore maximum fuel efficiency
Fuel efficiency is a form of thermal efficiency, meaning the ratio of effort to result of a process that converts chemical potential energy contained in a carrier (fuel) into kinetic energy or work. Overall fuel efficiency may vary per device, wh ...
is achieved by flying at the speed for maximum lift-to-drag ratio. This is the speed which covers the greatest distance for a given amount of fuel. Maximum endurance (time in the air) is achieved at a lower speed, when drag is minimised.
For jet aircraft, maximum endurance occurs when the lift-to-drag ratio is maximised. Maximum range occurs at a higher speed. This is because jet engines are thrust-producing, not power-producing. Turboprop aircraft do produce some thrust through the turbine exhaust gases, however most of their output is as power through the propeller.
"Long-range cruise" speed (LRC) is typically chosen to give 1% less fuel efficiency than maximum range speed, because this results in a 3-5% increase in speed. However, fuel is not the only marginal cost in airline operations, so the speed for most economical operation (ECON) is chosen based on the cost index (CI), which is the ratio of time cost to fuel cost.
Gliders
Without power, a gliding aircraft has only gravity to propel it. At a glide angle of θ, the weight has two components, at right angles to the flight line and parallel to it. These are balanced by the force and lift components respectively, so
:::::::::: and
::::::::::.
Dividing one equation by the other shows that the glide angle is given by tan θ = /. The performance characteristics of most interest in unpowered flight are the speed across the ground, say, and the sink speed ; these are displayed by plotting .sin θ = against .cos θ = . Such plots are generally termed polars, and to produce them the glide angle as a function of is required.[
One way of finding solutions to the two force equations is to square them both then add together; this shows the possible , values lie on a circle of radius / . When this is plotted on the drag polar, the intersection of the two curves locates the solution and its θ value read off. Alternatively, bearing in mind that glides are usually shallow, the approximation cos θ ≃ 1, good for θ less than 10°, can be used in the lift equation and the value of for a chosen calculated, finding from the drag polar and then calculating θ.][
The example polar here shows the gliding performance of the aircraft analysed above, assuming its drag polar is not much altered by the stationary propeller. A straight line from the origin to some point on the curve has a gradient equal to the glide angle at that speed, so the corresponding tangent shows the best glide angle ≃ 3.3°. This is not the lowest rate of sink but provides the greatest range, requiring a speed of 240 km/h (149 mph); the minimum sink rate of about 3.5 m/s is at 180 km/h (112 mph), speeds seen in the previous, powered plots.][
]
Sink rate
A graph showing the sink rate of an aircraft (typically a glider
Glider may refer to:
Aircraft and transport Aircraft
* Glider (aircraft), heavier-than-air aircraft primarily intended for unpowered flight
** Glider (sailplane), a rigid-winged glider aircraft with an undercarriage, used in the sport of glidin ...
) against its airspeed
In aviation, airspeed is the speed of an aircraft relative to the air. Among the common conventions for qualifying airspeed are:
* Indicated airspeed ("IAS"), what is read on an airspeed gauge connected to a Pitot-static system;
* Calibrated a ...
is known as a polar curve. Polar curves are used to compute the glider's minimum sink speed, best lift over drag (L/D), and speed to fly
Speed to fly is a principle used by soaring pilots when flying between sources of lift, usually thermals, ridge lift and wave. The aim is to maximize the average cross-country speed by optimizing the airspeed in both rising and sinking air. The ...
.
The polar curve of a glider is derived from theoretical calculations, or by measuring the rate of sink at various airspeeds. These data points are then connected by a line to form the curve. Each type of glider has a unique polar curve, and individual gliders vary somewhat depending on the smoothness of the wing, control surface drag, or the presence of bugs, dirt, and rain on the wing. Different glider configurations will have different polar curves, for example, solo versus dual flight, with and without water ballast, different flap settings, or with and without wing-tip extensions.[
Knowing the best ]speed to fly
Speed to fly is a principle used by soaring pilots when flying between sources of lift, usually thermals, ridge lift and wave. The aim is to maximize the average cross-country speed by optimizing the airspeed in both rising and sinking air. The ...
is important in exploiting the performance of a glider. Two of the key measures of a glider’s performance are its minimum sink rate and its best glide ratio
In aerodynamics, the lift-to-drag ratio (or L/D ratio) is the lift generated by an aerodynamic body such as an aerofoil or aircraft, divided by the aerodynamic drag caused by moving through air. It describes the aerodynamic efficiency under give ...
, also known as the best "glide angle". These occur at different speeds. Knowing these speeds is important for efficient cross-country flying
Cross-country flying ( XC flying) is a type of distance flying which is performed in a powered aircraft on legs over a given distance and in operations between two points using navigational techniques; and an unpowered aircraft (paraglider, hang g ...
. In still air the polar curve shows that flying at the minimum sink speed enables the pilot to stay airborne for as long as possible and to climb as quickly as possible, but at this speed the glider will not travel as far as if it flew at the speed for the best glide.
Effect of wind, lift/sink and weight on best glide speed
The best speed to fly in a head wind is determined from the graph by shifting the origin to the right along the horizontal axis by the speed of the headwind, and drawing a new tangent line. This new airspeed will be faster as the headwind increases, but will result in the greatest distance covered. A general rule of thumb is to add half the headwind component to the best L/D for the maximum distance. For a tailwind, the origin is shifted to the left by the speed of the tailwind, and drawing a new tangent line. The tailwind speed to fly will lie between minimum sink and best L/D.[
In subsiding air, the polar curve is shifted lower according the airmass sink rate, and a new tangent line drawn. This will show the need to fly faster in subsiding air, which gives the subsiding air less time to lower the glider's altitude. Correspondingly, the polar curve is displaced upwards according to the lift rate, and a new tangent line drawn.][
Increased weight does not affect the maximum range of a gliding aircraft. Glide angle is only determined by the lift/drag ratio. Increased weight will require an increased airspeed to maintain the optimum glide angle, so a heavier gliding aircraft will have reduced endurance, because it is descending along the optimum glide path at a faster rate.]
See also
*Drag coefficient
In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) 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 equ ...
*Lift coefficient
In fluid dynamics, the lift coefficient () is a dimensionless quantity that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. A lifting body is a foil or a com ...
*Angle of attack
In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is m ...
*Lift (force)
A fluid flowing around an object exerts a 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 force parallel to the flow direct ...
*Lifting-line theory
The Prandtl lifting-line theory is a mathematical model in aerodynamics that predicts lift distribution over a three-dimensional wing based on its geometry. It is also known as the Lanchester–Prandtl wing theory.
The theory was expressed indepen ...
External links
Glider Performance Airspeeds
An animated explanation of the basic polar curve, with modifications for sinking or rising air and for head- or tailwinds.
References
{{reflist, refs=
[{{cite book , title= Theory of wing sections , last1=Abbott, first1=Ira H., last2=von Doenhoff, first2=Albert E., year=1958, publisher=Dover Publications , location=New York , isbn=0-486-60586-8, pages=57–70;129–142]
[{{cite book , last=Shames , first=Irving H. , title=Mechanics of Fluids , year=1962 , publisher=McGraw-Hill , url=https://books.google.com/books?id=Ben6FKoTzSMC , access-date=8 November 2012 , lccn=61-18731 , page=364 , quote=Another useful curve that is commonly used in reporting wind-tunnel data is the CL vs CD curve, which is sometimes called the ''polar plot''.]
[{{cite book, title= Aircraft Performance and Design, pages=414–5]
[{{cite book, title= Aircraft Performance and Design, pages=199–252, 293–309]
[{{cite book, title= Aircraft Performance and Design, pages=282–7]
[{{cite book, title= Aircraft Performance and Design, pages=265–270]
Aerodynamics
Aircraft performance
Airspeed
Drag (physics)
Gliding technology
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