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.
Origins.
NACA initially developed the numbered airfoil system which was further refined by the United States Air Force at Langley Research Center. According to the NASA website:
Four-digit series.
The NACA four-digit wing sections define the profile by:
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.
The formula for the shape of a NACA 00xx foil, with "xx" being replaced by the percentage of thickness to chord, is
where:
Note that in this equation, at "x" = 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 chord-normalized radius of
Now the coordinates formula_4 of the upper airfoil surface and formula_5 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.
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
where
For example, a NACA 2412 airfoil uses a 2% camber (first digit) 40% (second digit) along the chord of a 0012 symmetrical airfoil having a thickness 12% (digits 3 and 4) of the chord.
For this cambered airfoil, because the thickness needs to be applied perpendicular to the camber line, the coordinates formula_4 and formula_5, of respectively the upper and lower airfoil surface, become
where
Five-digit series.
The NACA five-digit series describes more complex airfoil shapes. Its format is LPSTT, where:
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:
where the chordwise location formula_14 and the ordinate formula_15 have been normalized by the chord. The constant formula_16 is chosen so that the maximum camber occurs at formula_17; for example, for the 230 camber line, formula_18 and formula_19. Finally, constant formula_20 is determined to give the desired lift coefficient. For a 230 camber-line profile (the first 3 numbers in the 5-digit series), formula_21 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
The following table presents the various camber-line profile coefficients:
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.
From formula_23
From formula_25
formula_26
The following table presents the various camber-line profile coefficients:
Modifications.
Four- and five-digit series airfoils can be modified with a two-digit code preceded by a hyphen in the following sequence:
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.
1-series.
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:
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.
6-series.
An improvement over 1-series airfoils with emphasis on maximizing laminar flow. The airfoil is described using six digits in the following sequence:
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.
7-series.
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:
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.
8-series.
Supercritical airfoils designed to independently maximize laminar flow 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.