Chapter 4, Problem 4.1P

Consider the data for the NACA 2412 airfoil given in Figure 4.10. Calculate the lift and moment about the quarter chord (per unit span) for this airfoil when the angle of attack is 4° and the frecstream is at standard sea level conditions with a velocity of 50 ft/s. The chord of the airfoil is 2 ft.

To determine

The lift per unit span and the moment about the quarter chord for NASA 2412 airfoil.

Answer to Problem 4.1P

The lift per unit span is 3.18 lb per unit span & the moment per unit span is -0.4277 lb.ft.per unit span.

Explanation of Solution

Given Information:

The velocity of free stream = 50 ft/s

The angle of attack = $4°$

The chord of airfoil = 2 ft

Calculation:

Assume the value of density of free stream as ${\rho }_{\infty }=0.002377slug/f{t}^3=1.225kg/m^3$

The velocity of air stream, $V_{\infty }=50ft/s$

The dynamic pressure acting on the foil:

  $q_{\infty }=\frac{1}{2}{\rho }_{\infty }{V}_{\infty }{}^2=\frac{1}{2}(0.002377){(50)}^2=2.971lb/f{t}^3$

The value of the lift coefficient and momentum coefficient is calculated from the standard values of NACA airfoil. (Reference 11 from textbook named as Fundamental of Aerodynamics, 6^th edition)

$\alpha$ (degrees)	Lift coefficient ( $c_l$)	Drag Coefficient ( $c_d$)	Momentum coefficient (   $c_{m,c/4}$)
-2	0.05	0.006	-0.042
0	0.25	0.006	-0.040
2	0.44	0.006	-0.038
4	0.64	0.007	-0.036
6	0.85	.0075	-0.034
8	1.08	0.0092	-0.034
10	1.26	.00115	-0.0.34

At $\alpha =4°$ (NASA Airfoil)

Lift coefficient, $c_l=0.64$

Momentum coefficient, $c_{m,c/4}=-0.036$

The lift per unit span is given by :

  $L_s=q_{\infty }S{c}_l$

Here S =surface area of the foil = 2 ft²

  $\begin{array}{l}{L}_s=q_{\infty }S{c}_l\\ =2.971\times 2\times 0.64\\ =3.803lb/\text{per}\text{unit}\text{span}\end{array}$

For moment about the quarter chord per unit span is given by:

  $\begin{array}{c}{M}_{c/4}=q_{\infty }.S.c.c_{m_{c/4}}\\ =2.971\times 2\times 2\times (-0.036)\\ =-0.42784lb.ft\text{per}\text{unit}\text{span}\end{array}$