NACA Airfoil Plotter and Equations

Related Resources: calculators

NACA Airfoil Plotter and Equations

NACA Airfoil Plotter and Equations

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.

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

First digit describing maximum camber as percentage of the chord.
Second digit describing the distance of maximum camber from the airfoil leading edge in tenths of the chord.
Last two digits describing maximum thickness of the airfoil as percent of the chord.

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

y_{t} = 5 t [0.2969 \sqrt{x} - 0.1260 x - 0.3516 x^{2} + 0.2843 x^{3} - 0.1015 x^{4}]

where:

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

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

r = 1.1019t²

Now the coordinates (x_u, y_u) of the upper airfoil surface (x_L, y_L) of th elower airfoil surface are

x_U = x_L = x, y_U = +y_t, y_L = -y_t

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

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 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 (x_U, y_V) and (x_L, y_L) and ( , of respectively the upper and lower airfoil surface, become

where the chordwise location x and the ordinate y have been normalized by the chord. The constant r is chosen so that the maximum camber occurs at x=p; for example, for the 230 camber line, 0.3/2=0.15 and r = 0.2025. Finally, constant k₁ is determined to give the desired lift coefficient. For a 230 camber-line profile (the first 3 numbers in the 5-digit series), k₁ = 15.957 is used.

Reference:

National Advisory Committee for Aeronautics (NACA)