Modelling the NACA Four-Digit Series Airfoil

Let's start by choosing a couple of values in the two drop-down selection lists below.

camber and position airfoil thickness
Suffice it to say that the first two digits represent the amount of camber and its position respectively, the second two digits are the amount of thickness of the airfoil. If you're game, read on for an explanation of the values and functions applied to evaluate these profiles.


Early in the 20th Century, analytical methods were applied to the previous generation of empirical studies in the shape and characteristics of wing sections. In 1932, the National Advisory Committee for Aeronautics (NACA) hence derived the parameters for the four-digit wing section (ref. 3)

"When the NACA four digit wing sections were derived, it was found that the thickness distribution of efficient wing sections such as the G├Âttingen 398 and the Clarke Y were nearly the same when their camber was removed (mean line straightened) and they were reduced to the same maximum thickness. The thickness distribution for the NACA four digit sections was selected to correspond closely to that for these wing sections and given by the following equation:" (ref. 2, page 113)

thickness

where t = maximum thickness expressed as a fraction of the chord.

For example, the NACA 0020 section has a maximum thickness of 20% of chord, which happens to occur at 30% of chord on the abcissa, or x axis. The thickness distribution function defines the airfoil below, and all four digit airfoils with a symmetric profile. To reiterate, the last two digits of the NACA number, 20 in this case, represent the parameter t, which is the maximum thickness as a fraction of chord

NACA0020

Now that we have defined the thickness distribution of a symmetric 4-digit series airfoil we move forward to apply the Method of Combining Mean Lines and Thickness Distributions.

To understand the geometry of a cambered (non-symmetric) NACA four digit section we need to define the mean line and its parameters. The maximum camber, in % of chord, is represented by the variable (m). The position on the abcissa of the maximum camber, in fractions of chord, is represented by the variable (p).

Note that the terms Camber Line and Mean Line are synonymous. I'll use both from now on, interchangeably, for kicks.

Thus the NACA Four-Digit notation is as follows; "NACA (m)(p)(t)". For example, the NACA 2408 section has a maximum mean line ordinate (m) of 2% of chord, and a maximum mean line ordinate position (p) on the abcissa of 0.4 (fraction of chord), and a maximum thickness (t) of .08 (fraction of chord).

In the following diagram I have isolated the relationships between the thickness distribution and the mean line.

cambervariables

In order to apply a camber to the symmetric airfoil, ordinates of the cambered wing sections are obtained by laying off the thickness distributions perpendicular to the mean lines.

In other words, we begin by defining the mean line (or camber line) and then apply the thickness distribution to it. In the first example, a symmetric NACA 0020 profile, we simply applied the thickness function and were done. Get it ? OK, let's move on.

The shape of the camber line (the red Mean Line depicted in the previous graphic) is defined by two parabolic arc tangents at the position of maximum mean-line ordinate forward and aft of the maximum ordinate (p). The equations were taken to be: (ref. 3)

camberforward

Forward of maximum ordinate (p)

camberaft

Aft of maximum ordinate (p)

If xU and yU represent the abcissa and ordinate of a typical point of the upper surface of the wing section and yt is the ordinate of the symetric thickness distribution at chordwise position x or abcissa, the upper surface coordinates are given by the following relations: (ref. 2)

uppersurfacefunction

diagram

The corresponding expressions for the lower surface are:

lowersurfacefunction

In order to determine the angle Theta (¤┤) of the mean line at points along the abcissa we need to find the slope of the mean line function. This means solving the derivative of the mean line (camber) functions as follows:

forward camberForward of maximum ordinate (p)

aft camber

Aft of maximum ordinate (p)

We have now defined all the relationships necessary to iterate and render a numeric and/or visual presentation of the functions involved. Instead of using a pencil, paper and sliderule, we've done something fun with PHP and the JPGraph libraries.

I had originally coded this algorithm in MS QuickBasic, back in 1990, before the advent of more sophisticated code now available. The output was in DXF format for input to AutoCad, which enabled creation of 3d wireframe models of entire wing planforms. I present this example, along with code, for students and inquisitive types to use and learn from. I only ask that reference be given if you choose to copy any material herein.

//Blade Back
for($X=0; $X<=$pp; ++$X) {
$Ycfore[$X]=($m/pow($p,2))*((2*$p*$X*.001)-pow($X*.001,2));
$dYcfore=(2*$m/$p)-((2*$m*$X*.001)/pow($p,2));
$dYcslopefore=atan($dYcfore);
$Yt=($t/.2)*((.2969*sqrt($X*.001))-(.126*$X*.001)-(.3516*pow($X*.001,2))+(.2843*pow($X*.001,3))-(.1015*pow($X*.001,4)));
$camberforex[$X]=($X*.001);
$Xufore[$X]=($X*.001-($Yt*sin($dYcslopefore)));
$Yufore[$X]=($Ycfore[$X]+($Yt*cos($dYcslopefore)));
$Xlfore[$X]=($X*.001+($Yt*sin($dYcslopefore)));
$Ylfore[$X]=($Ycfore[$X]-($Yt*cos($dYcslopefore)));
}
// Blade Face
for($X=$pp; $X<=1000+$pp; ++$X) {
$Ycaft[$X]=($m/(pow((1-$p),2)))*((1-2*$p)+(2*$p*$X*.001)-(pow($X*.001,2)));
$dYcaft=((1-2*$p)*$m/pow((1-$p),2))+(2*$p*$m/pow((1-$p),2))-(2*$X*.001*$m/pow((1-$p),2));
$dYcslopeaft=atan($dYcaft);
$Yt=($t/.2)*((.2969*sqrt($X*.001))-(.126*$X*.001)-(.3516*pow($X*.001,2))+(.2843*pow($X*.001,3))-(.1015*pow($X*.001,4)));
$camberaftx[$X]=($X*.001);
$Xuaft[$X]=$X*.001-($Yt*sin($dYcslopeaft));
$Yuaft[$X]=$Ycaft[$X]+($Yt*cos($dYcslopeaft));
$Xlaft[$X]=$X*.001+($Yt*sin($dYcslopeaft));
$Ylaft[$X]=$Ycaft[$X]-($Yt*cos($dYcslopeaft));
}


References:

ref. 1 Basic Wing and Airfoil Theory; Pope, McGraw Hill 1951.

ref. 2 Theory of Wing Sections; Abbot & Doenhoff, Dover 1959.

ref. 3 Jacobs, Ward, and Pinkerton Characteristics of 78 Related Airfoil Sections from Tests in the Variable-density Wind Tunnel. NACA Rept. #617, 1941

James F. Horvath ~ all rights reserved.

Now that we have a data dump of our favorite NACA 4 Digit airfoil, we can pump it into Autocad or another CAD/CAM application using the DXF format.

NACAcad

For an explanation of the DXF file structure follow the link below. (Porting NACA data to Autocad with DXF)

teaserpic: 
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