Chapter 5, Problem 39P

The shape of a asymmetric four-digitseries NACA airfoil is described by the equations:

$\begin{array}{l}{x}_{\text{U}}=x-y_t\text{sin}\theta y_{\text{U}}=y_c+y_t\text{cos}\theta \\ x_{\text{L}}=x+y_t\text{sin}\theta y_L=y_c-y_t\text{cos}\theta \end{array}$

where the subscripts U and L corresponds to the upper and lower airfoil surface, respectively. y_tis half the thickness of the foil given by:

$y_t=5tc\left[0.2969\sqrt{\frac{x}{c}}-0.126\left(\frac{x}{c}\right)-0.3516{\left(\frac{x}{c}\right)}^2+0.2843{\left(\frac{x}{c}\right)}^3-0.1015{\left(\frac{x}{c}\right)}^4\right]$

where c is the cord length, t is the maximum thickness (as a fraction of the cord length), and x is the position along the cord, yc is the coordinate of the camber line given by:

$y_c=m\frac{x}{p^2}\left(2p-\frac{x}{c}\right)$ for $0\le x\le pc$ ,and $y_c=m\frac{c-x}{{(1-p)}^2}\left(1+\frac{x}{c}-2p\right)$for $pc\le x\le c$ where m and p are constants. The angle ? is given by:

$\theta =arc\tan \left[\frac{2m}{p^2}\left(p-\frac{x}{c}\right)\right]$ for $0\le x\le pc$ , and $\theta =arc\tan \left[\frac{2m}{{(1-p)}^2}\left(p-\frac{x}{c}\right)\right]$ for

$pc\le x\le c$. Plot the airfoil shown in the figure (NACA 4412) for which t=0.12, p=0.4, m=0.04,and c=1.5 m.