Drag curve

The significant aerodynamic properties of aircraft wings are summarised by two dimensionless quantities, the lift and drag coefficients CL and CD.

L and D are at right angles, with D parallel to the free stream velocity (the relative velocity of the surrounding distant air), so the resultant force R lies at the same angle to D as the line from the origin of the graph to the corresponding CL, CD point does to the CD 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.

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.

In total, then The effect of CL0 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.

The forces involved are obtained from the coefficients by multiplication with (ρ/2).S V2, where ρ is the density of the atmosphere at the flight altitude, S is the wing area and V is the speed.

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 (which will also shed altitude), or drag is otherwise reduced (such as by retracting the landing gear).

Sustained flight behind the power curve requires alert piloting because inadequate thrust will cause a steady decrease in airspeed and altitude, which may go unnoticed, and can be difficult to correct close to the ground.

[8][9] 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.

Turboprop aircraft do produce some thrust through the turbine exhaust gases, however most of their output is as power through the propeller.

These are balanced by the force and lift components respectively, so Dividing one equation by the other shows that the glide angle is given by tan θ = CD/CL.

[12] One way of finding solutions to the two force equations is to square them both then add together; this shows the possible CL, CD values lie on a circle of radius 2.W / S.ρ.V2.

When this is plotted on the drag polar, the intersection of the two curves locates the solution and its θ value read off.

[12] A graph showing the sink rate of an aircraft (typically a glider) against its airspeed is known as a polar curve.

[13] The polar curve of a glider is derived from theoretical calculations, or by measuring the rate of sink at various airspeeds.

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.

[14] In subsiding air, the polar curve is shifted lower according the airmass sink rate, and a new tangent line drawn.

Correspondingly, the polar curve is displaced upwards according to the lift rate, and a new tangent line drawn.

Drag and lift coefficients for the NACA 63 3 618 airfoil. Full curves are lift, dashed drag; red curves have R e = 3·10 6 , blue 9·10 6 .
Coefficients of lift and drag against angle of attack.
Curve showing induced drag , parasitic drag and total drag as a function of airspeed.
Drag curve for the NACA 63 3 618 airfoil, colour-coded as opposite plot.
Drag curve for light aircraft. C D0 = 0.017, K = 0.075 and C L0 = 0.1. The tangent gives the maximum L/D point.
P R curve for the light aircraft with the drag curve above and weighing 2000 kg, with a wing area of 15 m² and a propeller efficiency of 0.8.
The same aircraft, without power. The tangent defines the minimum glide angle, for maximum range. The peak of the curve indicates the minimum sink rate, for maximum endurance (time in the air).
As airspeed increases, total drag decreases then increases.
Polar curve for a glider, showing glide angle for minimum sink rate. The origin of the graph is where the airspeed axis crosses the sink rate axis at zero airspeed and zero sink rate. The horizontal line is tangent to the top of the polar curve. That tangent point indicates the minimum sink airspeed (vertical line). The sink rate increases to the left or right of this point, corresponding to a lower or higher airspeed. This minimum sink airspeed has the lowest possible rate of sink, and allows the longest possible glide time before landing. [ 13 ] [ 14 ]
Polar curve for a glider, showing glide angle for the best glide speed (best L/D). It is the flattest possible glide angle through calm air, which will maximize the distance flown. This airspeed (vertical line) corresponds to the tangent point of a line starting from the origin of the graph. A glider flying faster or slower than this airspeed will cover less distance before landing. [ 14 ] [ 13 ]