For the NACA 2412 airfoil, the lift coefficient and moment coefficient about the quarter-chord at −6◦ angle of attack are −0.39 and −0.045, respectively. At 4◦ angle of attack, these coefficients are 0.65 and −0.037, respectively.
Calculate the location of the aerodynamic center.

Location of aerodynamic centre is; X_ac = 0.242c

Explanation:

We are given;

First Lift coefficient; C_L1 = -0.39

First Moment Coefficient about the quarter-chord; C_mc/4 = -0.045

First Angle of attack; α_1 = -6°

2nd Lift coefficient; C_L2 = 0.65

2nd Moment Coefficient about the quarter-chord; C'_mc/4 = -0.037

2nd Angle of attack; α_2 = 4°

Now, the formula for location of chord at aerodynamic centre is given as;

X_ac = -(m/α) + 0.25

Used 0.25 because moment is about quarter chord which is 1/4

Where;

m is slope of moment coefficient curve

α is lift curve slope

Now, formula for lift curve slope is given as;

m = (C_L2 - C_L1)/(α_2 - α_1)

Now, plugging in the relevant values to get;

α = (0.65 - (-0.39))/(4 - (-6))

α = (0.65 + 0.39)/(4+6)

α = (1.04)/10

α = 0.104 per°

Now formula to calculate slope of moment coefficient curve is given as;

m = [(C'_mc/4) - (C_mc/4)]/(α_2 - α_1)

Thus, plugging in relevant values, we have;

m = [-0.037 - (-0.045)]/(4 - (-6))

m = [-0.037 + 0.045)]/(4 + 6)

m = 0.008/10 = 0.0008 per°

Now, plugging the relevant values into X_ac = -(m/α) + 0.25 ;we have;

X_ac = -(0.0008/0.104) + 0.25

X_ac = -0.00769 + 0.25

X_ac = 0.2423c ≈ 0.242c

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