User:Eas4200c.f08/Airfoil

Airfoil geometry: Eastman Jacobs' family of NACA airfoil sections. NACA 4-digit series: $$\displaystyle (m, p, t_{a})$$


 * $$\displaystyle m$$ = maximum camber in percentage of chord length $$\displaystyle c$$
 * $$\displaystyle p$$ = position of maximum camber in tenths of the chord length $$\displaystyle c$$
 * $$\displaystyle t_a$$ = maximum airfoil thickness in percentage of the chord $$\displaystyle c$$

See a figure of the NACA 2415 = (2,4,15) airfoil.

Equation of NACA 4-digit airfoil: See the figure in this web page to understand the (modified) symbols used in the equations below.

Choose a value for the chord length $$\displaystyle c$$. The $$\displaystyle x$$ coordinate varies from $$\displaystyle 0$$ to $$\displaystyle c$$; we write $$\displaystyle x \in [0,c]$$, i.e., $$\displaystyle x$$ belongs to the interval $$\displaystyle [0,c]$$.

The camber mean line is the function $$\displaystyle y_c$$ of $$\displaystyle x$$ defined in two pieces as follows

\displaystyle y_c =  \frac {m} {p^2} \left(     2 p x - x^2   \right) {\ \rm for \ } x \in [0, p] $$



\displaystyle y_c =  \frac {m} {(1 - p)^2} \left[ (1 - 2 p) + 2 p x - x^2 \right] {\ \rm for \ } x \in [p, c] $$

The thickness distribution above (+) and below (-) the camber mean line, and in the direction perpendicular to the camber mean line (see figure), is given by

\displaystyle y_t =  \pm 5 t_a \left(     0.2969 \sqrt{x}      -      0.1260 x      -      0.3516 x^2       +       0.2843 x^3       -       0.1015 x^4   \right) $$ where $$\displaystyle t_a$$ is the maximum airfoil thickness in percentage of the chord $$\displaystyle c$$.

Let $$\displaystyle \theta$$ be the angle from the $$\displaystyle x$$ axis to the tangent to the camber mean line, defined as follows

\displaystyle \theta =  \tan^{-1} \left(     \frac      {d y_c}      {dx}   \right) $$

The coordinates of the airfoil upper surface $$\displaystyle (x_U, y_U)$$ are then given by

\displaystyle x_U =  x - y_t \sin \theta $$

\displaystyle y_U =  y_c + y_t \cos \theta $$

and the coordinates of the lower surface $$\displaystyle (x_L, y_L)$$ are given by

\displaystyle x_L =  x + y_t \sin \theta $$

\displaystyle y_L =  y_c - y_t \cos \theta $$

Since we want to use $$\displaystyle (y,z)$$, instead of $$\displaystyle (x,y)$$, as the coordinates in the plane of the airfoil, we can simply rewrite the above equations with the cyclic permutation of the symbols $$\displaystyle (x,y,z)$$, i.e., $$\displaystyle x \rightarrow y \rightarrow z \rightarrow x$$. Thus the above equations are now rewritten as follows.

Choose a value for the chord length $$\displaystyle c$$. The $$\displaystyle y$$ coordinate varies from $$\displaystyle 0$$ to $$\displaystyle c$$; we write $$\displaystyle y \in [0,c]$$.

The camber mean line is the function $$\displaystyle z_c$$ of $$\displaystyle y$$ defined in two pieces as follows


 * {| style="width:100%" border="0"

$$  \displaystyle z_c =  \frac {m} {p^2} \left(     2 p y - y^2   \right) {\ \rm for \ } y \in [0, p] $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (1)
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle z_c =  \frac {m} {(1 - p)^2} \left[ (1 - 2 p) + 2 p y - y^2 \right] {\ \rm for \ } y \in [p, c] $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (2)
 * }

The thickness distribution above (+) and below (-) the camber mean line, and in the direction perpendicular to the camber mean line (see figure), is given by


 * {| style="width:100%" border="0"

$$  \displaystyle z_t =  \pm 5 t_a \left(     0.2969 \sqrt{y}      -      0.1260 y      -      0.3516 y^2       +       0.2843 y^3       -       0.1015 y^4   \right) $$ where $$\displaystyle t_a$$ is the maximum airfoil thickness in percentage of the chord $$\displaystyle c$$.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (3)
 * }

Let $$\displaystyle \theta$$ be the angle from the $$\displaystyle y$$ axis to the tangent to the camber mean line, defined as follows


 * {| style="width:100%" border="0"

$$  \displaystyle \theta =  \tan^{-1} \left(     \frac      {d z_c}      {dy}   \right) $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (4)
 * }

The coordinates of the airfoil upper surface $$\displaystyle (y_U, z_U)$$ are then given by


 * {| style="width:100%" border="0"

$$  \displaystyle \begin{align} y_U & =     y - z_t \sin \theta \\     z_U & =     z_c + z_t \cos \theta \end{align} $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (5)
 * }

and the coordinates of the lower surface $$\displaystyle (y_L, z_L)$$ are given by


 * {| style="width:100%" border="0"

$$  \displaystyle \begin{align} y_L & =     y + z_t \sin \theta \\     z_L & =     z_c - z_t \cos \theta \end{align} $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (6)
 * }

See also NACA 4-Digit Airfoil Equations, Wing geometry definition.

UIUC Airfoil Data Site