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Ansys CFX – NACA 4412 (Structured Mesh)

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Source: NASA

The NACA four-digit wing sections define the profile by:[1]

1. First digit describing maximum camber as percentage of the chord.
2. Second digit describing the distance of maximum camber from the airfoil leading edge in tenths of the chord.
3. Last two digits describing maximum thickness of the airfoil as percent of the chord.[2]

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

Plot of a NACA 0015 foil generated from formula

The formula for the shape of a NACA 00xx foil, with “x” being replaced by the percentage of thickness to chord, is

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

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/c = 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 radius of

${\displaystyle r=1.1019{\frac {t^{2}}{c}}.}$

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.

### Equation for a cambered 4-digit NACA airfoil

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

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

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 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

{\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

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

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

In this tutorial you will learn to simulate a NACA Airfoil (4412) using ANSYS CFX. First, we will import the points of the NACA profile and then we will generate the geometry using DesignModeler and SpaceClaim, the mesh using an unstructured mesh in Ansys Meshing. You can download the file in the following link.

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