NACA airfoil

The NACA airfoils are airfoil shapes for aircraft wings developed by the National Advisory Committee for Aeronautics (NACA). The shape of the NACA airfoils is described using a series of digits following the word "NACA". The parameters in the numerical code can be entered into equations to precisely generate the cross-section of the airfoil and calculate its properties.

Airfoil geometry
Profile geometry – 1: Zero lift line; 2: Leading edge; 3: Nose circle; 4: Max. thickness; 5: Camber; 6: Upper surface; 7: Trailing edge; 8: Camber mean-line; 9: Lower surface
Airfoil lines
Profile lines – 1: Chord, 2: Camber, 3: Length, 4: Midline
NACA Profil 0
A: blue line = chord, green line = camber mean-line, B: leading edge radius, C: x-y-coordinates for the profile geometry (Chord = x-Axis; y-Axis line on that leading edge)

Four-digit series

The NACA four-digit wing sections define the profile by:[1]

  1. First digit describing maximum camber as percentage of the chord.
  2. Second digit describing the distance of maximum camber from the airfoil leading edge in tens of percents of the chord.
  3. Last two digits describing maximum thickness of the airfoil as percent of the chord.[2]

For example, the NACA 2412 airfoil has a maximum camber of 2% located 40% (0.4 chords) from the leading edge with a maximum thickness of 12% of the chord.

The NACA 0015 airfoil is symmetrical, the 00 indicating that it has no camber. The 15 indicates that the airfoil has a 15% thickness to chord length ratio: it is 15% as thick as it is long.

Equation for a symmetrical 4-digit NACA airfoil

NACA0015 a
Plot of a NACA 0015 foil, generated from formula

The formula for the shape of a NACA 00xx foil, with "x" being replaced by the percentage of thickness to chord, is:[3]



  • x is the position along the chord from 0 to 1.00, (0 to 100%)
  • is the half thickness at a given value of x (centerline to surface), and
  • t is the maximum thickness as a fraction of the chord (so t gives the last two digits in the NACA 4-digit denomination divided by 100).

Note that in this equation, at (x/c) = 1 (the trailing edge of the airfoil), the thickness is not quite zero. If a zero-thickness trailing edge is required, for example for computational work, one of the coefficients should be modified such that they sum to zero. Modifying the last coefficient (i.e. to −0.1036) will result in the smallest change to the overall shape of the airfoil. The leading edge approximates a cylinder with a radius of:


Now the coordinates of the upper airfoil surface, and of the lower airfoil surface are:

Symmetrical 4-digit series airfoils by default have maximum thickness at 30% of the chord from the leading edge.

Equation for a cambered 4-digit NACA airfoil

NACA 2412
Plot of a NACA 2412 foil. The camber line is shown in red, and the thickness – or the symmetrical airfoil 0012 – is shown in purple.

The simplest asymmetric foils are the NACA 4-digit series foils, which use the same formula as that used to generate the 00xx symmetric foils, but with the line of mean camber bent. The formula used to calculate the mean camber line is:[3]


  • m is the maximum camber (100 m is the first of the four digits),
  • p is the location of maximum camber (10 p is the second digit in the NACA xxxx description).

For this cambered airfoil, because the thickness needs to be applied perpendicular to the camber line, the coordinates and , of respectively the upper and lower airfoil surface, become:[7]


Five-digit series

The NACA five-digit series describes more complex airfoil shapes.[8] Its format is: LPSTT, where:

  • L: a single digit representing the theoretical optimum lift coefficient at ideal angle-of-attack CLI = 0.15*L (this is not the same as the lift coefficient, CL)
  • P: a single digit for the x-coordinate of the point of maximum camber (max camber at x = 0.05*P)
  • S: a single digit indicating whether the camber is simple (S=0) or reflex (S=1)
  • TT: the maximum thickness in percent of chord, as in a four-digit NACA airfoil code

For example, the NACA 23112 profile describes an airfoil with design lift coefficient of 0.3 (0.15*2), the point of maximum camber located at 15% chord (5*3), reflex camber (1), and maximum thickness of 12% of chord length (12).

The camber-line is defined in two sections:[9]

where the chordwise location and the ordinate have been normalized by the chord. The constant is chosen so that the maximum camber occurs at ; for example, for the 230 camber-line, and . Finally, constant is determined to give the desired lift coefficient. For a 230 camber-line profile (the first 3 numbers in the 5 digit series), is used.

Non-reflexed 3 digit camber lines

3 digit camber lines provide a very far forward location for the maximum camber.

The camber line is defined as:[9]

The following table presents the various camber line profile coefficients:

Camber Line Profile
210 0.05 0.0580 361.40
220 0.10 0.126 51.640
230 0.15 0.2025 15.957
240 0.20 0.290 6.643
250 0.25 0.391 3.230

Reflexed 3-digit camber lines

Camber lines such as 231 makes the negative trailing edge camber of the 230 series profile to be positively cambered. This results in a theoretical pitching moment of 0.



The following table presents the various camber line profile coefficients:

Camber Line Profile
221 0.10 0.130 51.990 0.000764
231 0.15 0.217 15.793 0.00677
241 0.20 0.318 6.520 0.0303
251 0.25 0.441 3.191 0.1355


Four- and five-digit series airfoils can be modified with a two-digit code preceded by a hyphen in the following sequence:

  1. One digit describing the roundness of the leading edge with 0 being sharp, 6 being the same as the original airfoil, and larger values indicating a more rounded leading edge.
  2. One digit describing the distance of maximum thickness from the leading edge in tens of percent of the chord.

For example, the NACA 1234-05 is a NACA 1234 airfoil with a sharp leading edge and maximum thickness 50% of the chord (0.5 chords) from the leading edge.

In addition, for a more precise description of the airfoil all numbers can be presented as decimals.


A new approach to airfoil design pioneered in the 1930s in which the airfoil shape was mathematically derived from the desired lift characteristics. Prior to this, airfoil shapes were first created and then had their characteristics measured in a wind tunnel. The 1-series airfoils are described by five digits in the following sequence:

  1. The number "1" indicating the series
  2. One digit describing the distance of the minimum pressure area in tens of percent of chord.
  3. A hyphen.
  4. One digit describing the lift coefficient in tenths.
  5. Two digits describing the maximum thickness in percent of chord.

For example, the NACA 16-123 airfoil has minimum pressure 60% of the chord back with a lift coefficient of 0.1 and maximum thickness of 23% of the chord.


An improvement over 1-series airfoils with emphasis on maximizing laminar flow. The airfoil is described using six digits in the following sequence:

  1. The number "6" indicating the series.
  2. One digit describing the distance of the minimum pressure area in tens of percent of chord.
  3. The subscript digit gives the range of lift coefficient in tenths above and below the design lift coefficient in which favorable pressure gradients exist on both surfaces
  4. A hyphen.
  5. One digit describing the design lift coefficient in tenths.
  6. Two digits describing the maximum thickness as percent of chord.

For example, the NACA 612-315 a=0.5 has the area of minimum pressure 10% of the chord back, maintains low drag 0.2 above and below the lift coefficient of 0.3, has a maximum thickness of 15% of the chord, and maintains laminar flow over 50% of the chord.


Further advancement in maximizing laminar flow achieved by separately identifying the low pressure zones on upper and lower surfaces of the airfoil. The airfoil is described by seven digits in the following sequence:

  1. The number "7" indicating the series.
  2. One digit describing the distance of the minimum pressure area on the upper surface in tens of percent of chord.
  3. One digit describing the distance of the minimum pressure area on the lower surface in tens of percent of chord.
  4. One letter referring to a standard profile from the earlier NACA series.
  5. One digit describing the lift coefficient in tenths.
  6. Two digits describing the maximum thickness as percent of chord.
  7. "a=" followed by a decimal number describing the fraction of chord over which laminar flow is maintained. a=1 is the default if no value is given.

For example, the NACA 712A315 has the area of minimum pressure 10% of the chord back on the upper surface and 20% of the chord back on the lower surface, uses the standard "A" profile, has a lift coefficient of 0.3, and has a maximum thickness of 15% of the chord.


Supercritical airfoils designed to independently maximize airflow above and below the wing. The numbering is identical to the 7-series airfoils except that the sequence begins with an "8" to identify the series.

See also


  1. ^ E.N. Jacobs, K.E. Ward, & R.M. Pinkerton. NACA Report No. 460, "The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel". NACA, 1933.
  2. ^ "Fundamentals of aerodynamics", John D. Anderson,Jr, third ed, chap 4
  3. ^ a b Moran, Jack (2003). An introduction to theoretical and computational aerodynamics. Dover. p. 7. ISBN 0-486-42879-6.
  4. ^ | Ask Us - NACA Airfoil Series
  5. ^ Payne, Greg (8 Jul 1994), NACA 6, 7, and 8 series, archived from the original on April 27, 2009
  6. ^ Gordon J. Leishman. Principles of Helicopter Aerodynamics. p. 361.
  7. ^ Marzocca, Pier. "The NACA airfoil series" (PDF). Clarkson University. Retrieved July 5, 2016.
  8. ^ E. N. Jacobs & R. M. Pinkerton 1936 Test in the variable-density wind tunnel of related airfoils having the maximum camber unusually far forward, NACA Report No. 537.
  9. ^ a b Abbott, Ira (1959). Theory of Wing Sections: Including a Summary of Airfoil Data. New York: Dover Publications. p. 115. ISBN 978-0486605869.

External links

2D Flow Aerofoil Sections Source for NACA Java Java Applet Source Code for NACA 4 & 5-digit aerofoil generator

Broburn Wanderlust

The Broburn Wanderlust was a small, wooden, single-seat glider designed in the United Kingdom just after World War II. Only one was built, though it was well used.

CVT4 Strale

The CVT4 Strale (English: Arrow) was a one-off, experimental, high performance glider designed and built in Italy. Completed in 1961, it was a refined version of the earlier CVT2 Veltro.

Camber (aerodynamics)

In aeronautics and aeronautical engineering, camber is the asymmetry between the two acting surfaces of an aerofoil, with the top surface of a wing (or correspondingly the front surface of a propeller blade) commonly being more convex (positive camber). An aerofoil that is not cambered is called a symmetric aerofoil. The benefits of cambering were discovered and first utilized by George Cayley in the early 19th century.

Eastman Jacobs

Eastman Jacobs (1902–1987) was a leading aerodynamicist who worked for NACA's Langley Memorial Aeronautical Laboratory (renamed NASA Langley Research Center in 1958) from the 1920s to the 1940s. He was responsible for advancing many fields in aerodynamics, dealing particularly with wind tunnels, airfoils, turbulence, boundary layers, and Schlieren photography.

Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time.

Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.

Foil (fluid mechanics)

A foil is a solid object with a shape such that when placed in a moving fluid at a suitable angle of attack the lift (force generated perpendicular to the fluid flow) is substantially larger than the drag (force generated parallel to the fluid flow). If the fluid is a gas, the foil is called an airfoil or aerofoil, and if the fluid is water the foil is called a hydrofoil.


The IIL IS-10 was a high-performance, single-seat glider, designed and built in Romania in the early 1960s. It was the first Romanian aircraft to use laminar flow airfoils.

Index of physics articles (N)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

MMPL Kanpur

The MMPL Kanpur was an Indian light four-seat aircraft, designed for service and agricultural work in the early 1960s. It is a rare example of an aircraft designed and built by a national air force for its own use.

Merville SM.31

The Merville SM.31 is a French high performance glider with a laminar flow wing, first flown in 1960. Only one was built.

Miles M.76

In 1947 a British Gliding Association design competition, for a two-seat sailplane, was won by Hugh Kendall, Miles' assistant test pilot. It was a side-by-side two seater called the Kendall Crabpot I, with a 60 ft. span and an aspect ratio of 18. A version with a novel asbestos fibre-polymer wing and a wooden fuselage with a butterfly tail was proposed by Miles, but the wing failed under low loads. Elliotts of Newbury built a conventional wooden wing to use with Miles' fuselage. The resulting glider flew, but not well and development was abandoned with just one example built.

NACA Report No. 964

NACA Report No. 964 - The effects of variations in Reynolds number between 3.0 x 106 and 25.0 x 106 upon the aerodynamic characteristics of a number of NACA 6-series airfoil sections was published by the United States National Advisory Committee for Aeronautics in 1950. It contained a series of graphs showing the resulting lift and drag of several NACA 6-series airfoil sections from tests performed in a variable-density wind tunnel, in which the Reynolds number (RN) was set at three different values.

National Advisory Committee for Aeronautics

The National Advisory Committee for Aeronautics (NACA) was a U.S. federal agency founded on March 3, 1915, to undertake, promote, and institutionalize aeronautical research. On October 1, 1958, the agency was dissolved, and its assets and personnel transferred to the newly created National Aeronautics and Space Administration (NASA). NACA was an initialism, i.e. it was pronounced as individual letters, rather than as a whole word (as was NASA during the early years after being established).Among other advancements, NACA research and development produced the NACA duct, a type of air intake used in modern automotive applications, the NACA cowling, and several series of NACA airfoils which are still used in aircraft manufacturing.

During World War II, NACA was described as "The Force Behind Our Air Supremacy" due to its key role in producing working superchargers for high altitude bombers, and for producing the laminar wing profiles for the North American P-51 Mustang. NACA was also key in developing the area rule that is used on all modern supersonic aircraft, and conducted the key compressibility research that enabled the Bell X-1 to break the sound barrier.

SCAMP (boat)

The SCAMP (acronym of Small Craft Advisor Magazine Project) is a wooden or fiberglass hulled Balanced Lug rigged sailing dinghy. The boat is 11 ft 11 in (3.63 m) long, and capable of accommodating four persons on a daysail or one to two for overnighting or extended cruising. Craig Wagner and Josh Colvin, editors of Small Craft Advisor Magazine, teamed up with noted New Zealand boat designer, John Welsford, to create what they call a "Mini Microcruiser" sailboat. Welsford considers it possibly the best boat he's designed, based on "suitability for purpose". Some early plans details received subsequent revisions by Kees Prins and the Northwest Maritime Center. While no particular feature of the boat is unprecedented, the combination of design elements has produced a "new genre of sailboat".

Schweizer SGS 1-34

The Schweizer SGS 1-34 is a United States Standard Class, single-seat, high-wing glider built by Schweizer Aircraft of Elmira, New York.The 1-34 was designed over a number of years in the mid-1960s and first flew in 1969.By the time the 1-34 was introduced it was no longer competitive in the Standard Class, but has proven very successful as a club aircraft.

Streamlines, streaklines, and pathlines

Streamlines, streaklines and pathlines are field lines in a fluid flow.

They differ only when the flow changes with time, that is, when the flow is not steady.

Considering a velocity vector field in three-dimensional space in the framework of continuum mechanics, we have that:

Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction in which a massless fluid element will travel at any point in time.

Streaklines are the loci of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline.

Pathlines are the trajectories that individual fluid particles follow. These can be thought of as "recording" the path of a fluid element in the flow over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time.

Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time, creating a line or a curve that is displaced in time as the particles move.By definition, different streamlines at the same instant in a flow do not intersect, because a fluid particle cannot have two different velocities at the same point. Similarly, streaklines cannot intersect themselves or other streaklines, because two particles cannot be present at the same location at the same instant of time; unless the origin point of one of the streaklines also belongs to the streakline of the other origin point. However, pathlines are allowed to intersect themselves or other pathlines (except the starting and end points of the different pathlines, which need to be distinct).

Streamlines and timelines provide a snapshot of some flowfield characteristics, whereas streaklines and pathlines depend on the full time-history of the flow. However, often sequences of timelines (and streaklines) at different instants—being presented either in a single image or with a video stream—may be used to provide insight in the flow and its history.

If a line, curve or closed curve is used as start point for a continuous set of streamlines, the result is a stream surface. In the case of a closed curve in a steady flow, fluid that is inside a stream surface must remain forever within that same stream surface, because the streamlines are tangent to the flow velocity. A scalar function whose contour lines define the streamlines is known as the stream function.

Dye line may refer either to a streakline: dye released gradually from a fixed location during time; or it may refer to a timeline: a line of dye applied instantaneously at a certain moment in time, and observed at a later instant.

Supercritical airfoil

A supercritical airfoil is an airfoil designed primarily to delay the onset of wave drag in the transonic speed range. Supercritical airfoils are characterized by their flattened upper surface, highly cambered ("downward-curved") aft section, and larger leading-edge radius compared with NACA 6-series laminar airfoil shapes. Standard wing shapes are designed to create lower pressure over the top of the wing. The camber of the wing determines how much the air accelerates around the wing. As the speed of the aircraft approaches the speed of sound, the air accelerating around the wing reaches Mach 1 and shockwaves begin to form. The formation of these shockwaves causes wave drag. Supercritical airfoils are designed to minimize this effect by flattening the upper surface of the wing.

The supercritical airfoils were suggested first in Germany in 1940, when K. A. Kawalki at Deutsche Versuchsanstalt für Luftfahrt Berlin-Adlershof designed airfoils characterised by elliptical leading edges, maximal thickness located downstream up to 50% chord and a flat upper surface. Testing of these airfoils was reported by B. Göthert and K. A. Kawalki in 1944. Kawalki's airfoil shapes were identical to Richard Whitcomb's. Hawker-Siddeley in Hatfield, England, designed in 1959–1965 improved airfoil profiles known as rooftop rear-loaded airfoils, which were the basis of the Airbus A300 supercritical wing, which first flew in 1972.In the U.S., supercritical airfoils were studied in the 1960s, by then NASA engineer Richard Whitcomb, and were first tested on a modified North American T-2C Buckeye. After this first test, the airfoils were tested at higher speeds on the TF-8A Crusader. While the design was initially developed as part of the supersonic transport (SST) project at NASA, it has since been mainly applied to increase the fuel efficiency of many high-subsonic aircraft. The supercritical airfoil shape is incorporated into the design of a supercritical wing.

Kawalki's research was the basis for the objection in 1984 against the US-patent specification for the supercritical airfoil.

Trojan 750

The Trojan 750 is a trailer yacht made in New Zealand. It was designed by Ferris de Joux and Alan Warwick and built by Trojan Yachts Ltd from the late 1970s to the early 1980s.It is on the large side for a trailer yacht, with room for 4–6 people.

In the forward cabin it has a double "V" berth; aft, two large single beds that go under the cockpit floor, normally equipped with a head in its own room just behind the "V" berth, and a small integrated galley with a sink, stove and storage behind that, joined to the centre board case.


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