# 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 in aircraft, 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.[citation needed]

## Drag

Lift-induced drag is a component of total drag that arises whenever a finite span 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 lift versus velocity graph.

The drag curve
Drag polar for light aircraft. The tangent gives the maximum L/D point.

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 lift/velocity graph's U shape. Profile drag is lowered primarily by streamlining and reducing cross section.

Lift, like drag, increases as the square of the velocity and the ratio of lift to drag is often plotted in terms of the lift and drag coefficients CL and CD. Such graphs are referred to as drag polars. Speed increases from left to right. The lift/drag ratio is given by the slope from the origin to some point on this curve and so the peak L/D ratio does not occur at the point of least drag, the leftmost point. Instead it occurs at a slightly higher speed. 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:

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

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

${\displaystyle C_{D,0}=C_{fe}{\frac {S_{wet}}{S_{ref}}}}$,[2]

where ${\displaystyle C_{fe}}$ is the equivalent skin friction coefficient, ${\displaystyle S_{wet}}$ is the wetted area and ${\displaystyle S_{ref}}$ 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 (${\displaystyle b^{2}/S_{ref}}$), yields the equation:

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

where b is wingspan. The term ${\displaystyle b^{2}/S_{wet}}$ 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]

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

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

## Examples

A House sparrow has a 4:1 L/D ratio, a Herring gull a 10:1 one, a Common tern 12:1 and an Albatross 20:1, to be compared to 8.3:1 for the Wright Flyer to 17.7:1 for a Boeing 747 in cruise.[4] A cruising Airbus A380 reaches 20:1.[5] The Concorde at takeoff and landing had a 4:1 L/D ratio, increasing to 12:1 at Mach 0.95 and 7.5:1 at Mach 2.[6] A Helicopter at 100 kn (190 km/h) has a 4.5:1 L/D ratio.[7] A Cessna 172 glides at a 10.9:1 ratio.[8] A cruising Lockheed U-2 has a 25.6 L/D ratio.[9] The Rutan Voyager had a 27:1 ratio and the Virgin Atlantic GlobalFlyer 37:1.[10]

Computed aerodynamic characteristics[11]
Jetliner cruise L/D First flight
L1011-100 14.5 Nov 16, 1970
DC-10-40 13.8 Aug 29, 1970
A300-600 15.2 Oct 28, 1972
MD-11 16.1 Jan 10, 1990
B767-200ER 16.1 Sep 26, 1981
A310-300 15.3 Apr 3, 1982
B747-200 15.3 Feb 9, 1969
B747-400 15.5 Apr 29, 1988
B757-200 15.0 Feb 19, 1982
A320-200 16.3 Feb 22, 1987
A330-300 18.1 Nov 2, 1992
A340-200 19.2 Apr 1, 1992
A340-300 19.1 Oct 25, 1991
B777-200 19.3 Jun 12, 1994

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
Eta (glider) Gliding 70[12]
Great frigatebird Soaring over the ocean 15-22 at typical speeds[13]
Hang glider Gliding 15
Air Canada Flight 143 (Gimli Glider) a Boeing 767-200 with all engines failed caused by fuel exhaustion ~12
British Airways Flight 9 a Boeing 747-200B with all engines failed caused by volcanic ash ~15
US Airways Flight 1549 a Airbus A320-214 with all engines failed caused by bird strike ~17
Paraglider High performance model 11
Helicopter Autorotation 4
Powered parachute Rectangular/elliptical parachute 3.6/5.6
Space Shuttle Approach 4.5[14]
Hypersonic Technology Vehicle 2 Equilibrium hypersonic gliding estimate[15] 2.6
Wingsuit Gliding 2.5
Northern flying squirrel Gliding 1.98
Space Shuttle Hypersonic 1[14]
Apollo CM Reentry 0.368[16]

## References

1. ^ Loftin, LK, Jr. "Quest for performance: The evolution of modern aircraft. NASA SP-468". Retrieved 2006-04-22.
2. ^ Raymer, Daniel (2012). Aircraft Design: A Conceptual Approach (5th ed.). New York: AIAA.
3. ^ Aerospaceweb.org Hypersonic Vehicle Design
4. ^ Antonio Filippone. "Lift-to-Drag Ratios". Advanced topics in aerodynamics. Archived from the original on March 28, 2008.
5. ^ Cumpsty, Nicholas (2003). Jet Propulsion. Cambridge University Press. p. 4.
6. ^ Christopher Orlebar (1997). The Concorde Story. Osprey Publishing. p. 116. ISBN 9781855326675.
7. ^ Leishman, J. Gordon (24 April 2006). Principles of helicopter aerodynamics. Cambridge University Press. p. 230. ISBN 0521858607. The maximum lift-to-drag ratio of the complete helicopter is about 4.5
8. ^ a b Cessna Skyhawk II Performance Assessment [temporal.com.au]
9. ^ "U2 Developments transcript". Central Intelligence Agency. 1960. Lay summarytranscript.
10. ^ David Noland (February 2005). "The Ultimate Solo". Popular Mechanics.
11. ^ Rodrigo Martínez-Val; et al. (January 2005). "Historical evolution of air transport productivity and efficiency". 43rd AIAA Aerospace Sciences Meeting and Exhibit. doi:10.2514/6.2005-121.
12. ^ Eta aircraft Eta aircraft performances plots - accessed 2004-04-11
13. ^ Flight performance of the largest volant bird
14. ^ a b Space Shuttle Technical Conference pg 258
15. ^ [scienceandglobalsecurity.org]
16. ^ Hillje, Ernest R., "Entry Aerodynamics at Lunar Return Conditions Obtained from the Flight of Apollo 4 (AS-501)," NASA TN D-5399, (1969).