NACA airfoil

Plot of a NACA 0015 foil generated from formula

The formula for the shape of a NACA 00xx foil, with "xx" being replaced by the percentage of thickness to chord, is[4]

${\displaystyle y_{t}=5t\left[0.2969{\sqrt {x}}-0.1260x-0.3516x^{2}+0.2843x^{3}-0.1015x^{4}\right],}$[5][6]

where:

x is the position along the chord from 0 to 1.00 (0 to 100%),

${\displaystyle y_{t}}$ is the half thickness at a given value of x (centerline to surface),

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 = 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

${\displaystyle r=1.1019t^{2}.}$[7]

Now the coordinates ${\displaystyle (x_{U},y_{U})}$ of the upper airfoil surface and ${\displaystyle (x_{L},y_{L})}$ of the lower airfoil surface are

${\displaystyle x_{U}=x_{L}=x,\quad y_{U}=+y_{t},\quad y_{L}=-y_{t}.}$

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

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[4]

${\displaystyle y_{c}={\begin{cases}{\dfrac {m}{p^{2}}}\left(2px-x^{2}\right),&0\leq x\leq p,\\{\dfrac {m}{(1-p)^{2}}}\left((1-2p)+2px-x^{2}\right),&p\leq x\leq 1,\end{cases}}}$

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 ${\displaystyle (x_{U},y_{U})}$ and ${\displaystyle (x_{L},y_{L})}$, of respectively the upper and lower airfoil surface, become[8]

${\displaystyle {\begin{aligned}x_{U}&=x-y_{t}\,\sin \theta ,&y_{U}&=y_{c}+y_{t}\,\cos \theta ,\\x_{L}&=x+y_{t}\,\sin \theta ,&y_{L}&=y_{c}-y_{t}\,\cos \theta ,\end{aligned}}}$

where

${\displaystyle \theta =\arctan {\frac {dy_{c}}{dx}},}$

${\displaystyle {\frac {dy_{c}}{dx}}={\begin{cases}{\dfrac {2m}{p^{2}}}\left(p-x\right),&0\leq x\leq p,\\{\dfrac {2m}{(1-p)^{2}}}\left(p-x\right),&p\leq x\leq 1.\end{cases}}}$

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

The camber line is defined as[10]

${\displaystyle y_{c}={\begin{cases}{\dfrac {k_{1}}{6}}{\big (}x^{3}-3rx^{2}+r^{2}(3-r)x{\big )},&0<x<r,\\{\dfrac {k_{1}r^{3}}{6}}(1-x),&r<x<1.\end{cases}}}$

with the camber line gradient

${\displaystyle {\frac {dy_{c}}{dx}}={\begin{cases}{\dfrac {k_{1}}{6}}{\big (}3x^{2}-6rx+r^{2}(3-r){\big )},&0<x<r,\\-{\dfrac {k_{1}r^{3}}{6}},&r<x<1.\end{cases}}}$

The following table presents the various camber-line profile coefficients for a theoretical design lift coefficient of 0.3 - the value of ${\displaystyle k_{1}}$ must be linearly scaled for a different desired design lift coefficient:[11]

Camber-line profile	${\displaystyle p}$	${\displaystyle r}$	${\displaystyle k_{1}}$
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

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 ${\displaystyle {\frac {x}{c}}\leq r}$

${\displaystyle {\frac {y}{c}}={\frac {k_{1}}{6}}\left[\left({\frac {x}{c}}-r\right)^{3}-{\frac {k_{2}}{k_{1}}}(1-r)^{3}{\frac {x}{c}}-r^{3}{\frac {x}{c}}+r^{3}\right].}$

From ${\displaystyle r<{\frac {x}{c}}\leq 1.0}$

${\displaystyle {\frac {y}{c}}={\frac {k_{1}}{6}}\left[{\frac {k_{2}}{k_{1}}}\left({\frac {x}{c}}-r\right)^{3}-{\frac {k_{2}}{k_{1}}}(1-r)^{3}{\frac {x}{c}}-r^{3}{\frac {x}{c}}+r^{3}\right].}$

The following table presents the various camber-line profile coefficients for a theoretical design lift coefficient of 0.3 - the value of ${\displaystyle r}$, ${\displaystyle k_{1}}$ and ${\displaystyle k_{2}/k_{1}}$ must be linearly scaled for a different desired design lift coefficient:[11]

Camber-line profile	${\displaystyle p}$	${\displaystyle r}$	${\displaystyle k_{1}}$	${\displaystyle k_{2}/k_{1}}$
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