### The NACA 4 and 5 Digit aerofoils represent two families of aerofoil section that can be generated by the use of a set of simple polynomial equations. While these sections are slightly out of date in terms of current aircraft usage, they still represent useful sections and are easy to create. The aerofoils are created by summing a thickness distribution with a given mean line equation.

For both families of aerofoil section the thickness distribution is as follows,

$$y_t=t/{0.2}(0.2969√{x}-0.126x-0.3516x^2+0.2843x^3-0.1015x^4)$$

where (x) is a position along the chord line, given as a fraction of chord and t is the value of maximum thickness as given by the last two digits of the aerofoil designation number. (ie 0012 = symmetric section with t(max)=0.12c). For the 4-digit family, the mean line is given as,

$$\text"for "0<x<p\text" "y_c=m/p^2(2px-x^2)$$

$$\text"for "p<x<1\text" "y_c=m/{(1-p)^2}(1-2p+2px-x^2)$$

Values (p) and (m) are given from the first two digits of the designation number. (m) being the value of maximum camber height (1/100ths chord) and (p) being the position of maximum camber height (1/10ths chord).
(ie 2412 = maximum camber height =0.02c located at 0.4c).

For the 5-digit family, the mean line is given as,

$$\text"for "0<x<p\text" "y_c=k_1/6(x^3-3mx^2+m^2(3-m)x)$$

$$\text"for "p<x<1\text" "y_c=k_1/6m^3(1-x)$$

Values (p), (k1) and (m) are found from the following table based on the first three digits of the designation number.

 Mean Line No. p m k1 210 0.05 0.0580 361.4 220 0.10 0.1260 51.64 230 0.15 0.2025 15.957 240 0.20 0.2900 6.643 250 0.25 0.3910 3.230

The value of maximum camber height (m) and its position (p) will now be determined by the section construction process.
(ie 23012 = maximum camber height =0.02c located at 0.15c).

The construction of the section is then done numerically by identifying surface points which are the sum of camber and thickness effects. Points are normally generated using a cosine distribution of chord (x) coordinates. For each (x) coordinate an upper (xupper, yupper) and lower surface (xlower,ylower) data point is created by applying the above equations and construction method.

$$x_{upper}=x-y_t\sin(θ)\text" "x_{lower}=x+y_t\sin(θ)$$

$$y_{upper}=y_c+y_t\cos(θ)\text" "y_{lower}=y_c-y_t\cos(θ)$$

here (θ) is the angle of the mean line gradient ${dy_c}/{dx}$ at the coordinate (x) location. A leading edge radius (r) is applied to smooth the front data points.

$$r=1.1019t^2$$

### NACA 6 and 6A Series Aerofoil Sections

These aerofoil sections are designed to produce laminar flow and low drag over a reasonable range of angles of attack. The thickness distribution is thus based on a prescribed velocity distribution for the specific symmetric section required. The camber line is a polynomial function based on the desired ideal lift coefficient.

For 6 Series Sections the designation numbers represent the aerofoil aerodynamic properties as shown in the following example,

 64(1)-215 6 -- 6 series designation number. 4 -- location of Cp(min) as 1/10ths chord. (1) -- 1/2 width of drag bucket in CL counts (0.2) 2 -- Ideal (or Design) CL value. 15 -- Max thickness to chord ratio, 1/100ths chord

### References

• "Theory of Wing Sections" I.H.Abbott & A.E.Von Doenhoff, Dover, NY, 1959.

• "Computer Program to Obtain Ordinates for NACA Aerofoils." Ladson, Brooks, Hill & Sproles, NACA Langley, NASA TM-4741

### Software

The following applications can construct the various families of NACA 4,5 6 and 6A series sections using the techniques described by Ladson, Brooks, Hill and Sproles.

In these cases the data file contains 175 coordinate points with compact spacing closer to the leading edge of the section.

### Links to Other Aerofoil Section Data

Page with a large number of aerofoil sections. ( Martin Selig's pages at UIUC)