Drag curve

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. 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, the lift and drag coefficients and. Like other such aerodynamic quantities, they are functions only of the angle of attack, the Reynolds number and the Mach number. 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, 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, and the magnitude and direction of the resulting force are measured, they can be plotted using polar coordinates. 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 and around 1910 by Gustav Eiffel, 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 and flaps are deployed.
The accompanying diagram shows against for a typical light aircraft. The minimum point is at the left-most point on the plot. One component of drag is induced drag. This is proportional to. The other drag mechanisms, parasitic and wave drag, 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.

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 equals drag, so
The extra factor of /η, with η the propeller efficiency, in the second equation enters because = ×/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 is about 86 kW ; 135 kW is required for a maximum speed of 300 km/h. 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
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. The corresponding maximum range condition is the maximum of, at =, and so the optimum speed is 244 km/h. 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" or "behind the power curve" where more thrust is required to sustain flight at lower speeds. It is an inefficient region of flight because a decrease in speed requires increased thrust and a resultant increase in fuel consumption. It is regarded as a "speed unstable" region of flight, because unlike normal circumstances, a decrease in airspeed due to a nose-up pitch control input will not correct itself if the controls are returned to their previous position. Instead, airspeed will remain low and drag will progressively accumulate as airspeed and altitude continue to decay, and this condition will persist until thrust is increased, angle of attack is reduced, or drag is otherwise reduced. Sustained flight behind the power curve requires alert piloting because inadequate thrust will cause a steady decrease in airspeed and a corresponding steady increase in descent rate, which may go unnoticed, and can be difficult to correct close to the ground. A not-infrequent result is the aircraft "mushing" and crashing short of the intended landing site because the pilot did not decrease angle of attack or increase thrust in time, or because adequate thrust is not available; the latter is a particular hazard during a forced landing after an engine failure.
Failure to control airspeed and descent rate while flying behind the power curve has been implicated in a number of prominent aviation accidents, such as Asiana Airlines Flight 214.

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
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, maximum range and therefore maximum fuel efficiency 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 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 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 is chosen based on the cost index, 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, 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. 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 against. 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 value read off. Alternatively, bearing in mind that glides are usually shallow, the approximation, good for 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.
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. This is not the lowest rate of sink but provides the greatest range, requiring a speed of 240 km/h ; the minimum sink rate of about 3.5 m/s is at 180 km/h, speeds seen in the previous, powered plots.

Sink rate

A graph showing the sink rate of an aircraft against its airspeed is known as a polar curve. Polar curves are used to compute the glider's minimum sink speed, best lift over drag, and speed to fly.
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 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, also known as the best "glide angle". These occur at different speeds. Knowing these speeds is important for efficient cross-country flying. 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.