Lift-to-drag ratio

The Wright brothers testing their gliders in 1901 (left) and 1902 (right). The angle of the tether reflects the dramatic improvement in the lift-to-drag ratio

In aerodynamics, the lift-to-drag ratio, or L/D ratio, is the amount of lift generated by a wing or vehicle, divided by the aerodynamic drag it creates by moving through the air. A higher or more favorable L/D ratio is typically one of the major goals in aircraft design; since a particular aircraft's required lift is set by its weight, delivering that lift with lower drag leads directly to better fuel economy, climb performance, and glide ratio.

The term is calculated for any particular airspeed by measuring the lift generated, then dividing by the drag at that speed. These vary with speed, so the results are typically plotted on a 2D graph. In almost all cases the graph forms a U-shape, due to the two main components of drag.

Lift-to-drag ratios can be determined by flight test, by calculation or by testing in a wind tunnel.

Drag

Lift-induced drag is a component of total drag that arises whenever a three-dimensional wing generates lift. At low speeds an aircraft has to generate lift with a higher angle of attack, thereby leading to greater induced drag. This term dominates the low-speed side of the L/D graph, the left side of the U.

Form drag is caused by movement of the aircraft through the air. This type of drag, also known as air resistance or profile drag varies with the square of speed (see drag equation). For this reason profile drag is more pronounced at higher speeds, forming the right side of the L/D graph's U shape. Profile drag is lowered primarily by streamlining and reducing cross section.

The drag curve

The peak L/D ratio doesn't necessarily occur at the point of least total drag, as the lift produced at that speed is not high, hence a bad L/D ratio. Similarly, the speed at which the highest lift occurs does not have a good L/D ratio, as the drag produced at that speed is too high. The best L/D ratio occurs at a speed somewhere in between (usually slightly above the point of lowest drag). Designers will typically select a wing design which produces an L/D peak at the chosen cruising speed for a powered fixed-wing aircraft, thereby maximizing economy. Like all things in aeronautical engineering, the lift-to-drag ratio is not the only consideration for wing design. Performance at high angle of attack and a gentle stall are also important.

Glide ratio

As the aircraft fuselage and control surfaces will also add drag and possibly some lift, it is fair to consider the L/D of the aircraft as a whole. As it turns out, the glide ratio, which is the ratio of an (unpowered) aircraft's forward motion to its descent, is (when flown at constant speed) numerically equal to the aircraft's L/D. This is especially of interest in the design and operation of high performance sailplanes, which can have glide ratios approaching 60 to 1 (60 units of distance forward for each unit of descent) in the best cases, but with 30:1 being considered good performance for general recreational use. Achieving a glider's best L/D in practice requires precise control of airspeed and smooth and restrained operation of the controls to reduce drag from deflected control surfaces. In zero wind conditions, L/D will equal distance traveled divided by altitude lost. Achieving the maximum distance for altitude lost in wind conditions requires further modification of the best airspeed, as does alternating cruising and thermaling. To achieve high speed across country, glider pilots anticipating strong thermals often load their gliders (sailplanes) with water ballast: the increased wing loading means optimum glide ratio at higher airspeed, but at the cost of climbing more slowly in thermals. As noted below, the maximum L/D is not dependent on weight or wing loading, but with higher wing loading the maximum L/D occurs at a faster airspeed. Also, the faster airspeed means the aircraft will fly at higher Reynolds number and this will usually bring about a lower zero-lift drag coefficient.

Theory

Mathematically, the maximum lift-to-drag ratio can be estimated as:

(L/D)_{max} = \frac{1}{2} \sqrt{\frac{\pi \epsilon AR}{C_{D,0}}}

,^[1]

where AR is the aspect ratio, $\epsilon$ the span efficiency factor, a number less than but close to unity for long, straight edged wings, and $C_{D,0}$ the zero-lift drag coefficient.

Most importantly, the maximum lift-to-drag ratio is independent of the weight of the aircraft, the area of the wing, or the wing loading.

It can be shown that two main drivers of maximum lift-to-drag ratio for a fixed wing aircraft are wingspan and total wetted area. One method for estimating the zero-lift drag coefficient of an aircraft is the equivalent skin-friction method. For a well designed aircraft, zero-lift drag (or parasite drag) is mostly made up of skin friction drag plus a small percentage of pressure drag caused by flow separation. The method uses the equation:

C_{D,0}=C_{fe}\frac{S_{wet}}{S_{ref}}

,^[2]

where $C_{fe}$ is the equivalent skin friction coefficient, $S_{wet}$ is the wetted area and $S_{ref}$ is the wing reference area. The equivalent skin friction coefficient accounts for both separation drag and skin friction drag and is a fairly consistent value for aircraft types of the same class. Substituting this into the equation for maximum lift-to-drag ratio, along with the equation for aspect ratio ( $b^2/S_{ref}$ ), yields the equation:

(L/D)_{max}=\frac{1}{2} \sqrt{\frac{\pi \epsilon}{C_{fe}}\frac{b^2}{S_{wet}}}

where b is wingspan. The term $b^2/S_{wet}$ is known as the wetted aspect ratio. The equation demonstrates the importance of wetted aspect ratio in achieving an aerodynamically efficient design.

Supersonic/hypersonic lift to drag ratios

At very high speeds, lift to drag ratios tend to be lower. Concorde had a lift/drag ratio of around 7 at Mach 2, whereas a 747 is around 17 at about mach 0.85.

Dietrich Küchemann developed an empirical relationship for predicting L/D ratio for high Mach:^[3]

L/D_{max}=\frac{4(M+3)}{M}

where M is the Mach number. Windtunnel tests have shown this to be roughly accurate.

Examples

The following table includes some representative L/D ratios.

Flight article	Scenario	L/D ratio
Virgin Atlantic GlobalFlyer	Cruise	37^[4]
Lockheed U-2	Cruise	~28
Rutan Voyager	Cruise^[4]	27
Albatross		20^[5]
Boeing 747	Cruise	17
Common tern		12^[5]
Herring gull		10^[5]
Concorde	M2 Cruise	7.14
Cessna 150	Cruise	7
Helicopter	100 kts speed	4.5^[6]
Concorde	Approach	4.35
House sparrow		4^[5]

In gliding flight, the L/D ratios are equal to the glide ratio (when flown at constant speed).

Flight article	Scenario	L/D ratio/ glide ratio
Modern sailplane	Gliding	40–60 (depending on span)
Hang glider		15
Gimli Glider	Boeing 767-200 with fuel exhaustion	~12
Paraglider	High performance model	11
Helicopter	Autorotation	4
Powered parachute	Rectangular/elliptical parachute	3.6/5.6
Space Shuttle	Approach	4.5^[7]
Wingsuit	Gliding	2.5
Northern flying squirrel	Gliding	1.98
Space Shuttle	Hypersonic	1^[7]
Apollo CM	Reentry	0.368^[8]

References

↑ Loftin, LK, Jr. "Quest for performance: The evolution of modern aircraft. NASA SP-468". Retrieved 2006-04-22.
↑ Raymer, Daniel (2012). Aircraft Design: A Conceptual Approach (5th ed.). New York: AIAA. |access-date= requires |url= (help)
↑ Aerospaceweb.org Hypersonic Vehicle Design
1 2 David Noland, "Steve Fossett and Burt Rutan's Ultimate Solo: Behind the Scenes", Popular Mechanics, Feb. 2005 (web version)
1 2 3 4 Fillipone
↑ Leishman, J. Gordon. Principles of helicopter aerodynamics p230, Cambridge University Press, 24 April 2006. Accessed: 27 February 2012. ISBN 0521858607. Quote: The maximum lift-to-drag ratio of the complete helicopter is about 4.5
1 2 Space Shuttle Technical Conference pg 258
↑ Hillje, Ernest R., "Entry Aerodynamics at Lunar Return Conditions Obtained from the Flight of Apollo 4 (AS-501)," NASA TN D-5399, (1969).

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