Question

Consider a NACA 0012 - thin airfoil at 10 deg angle of attack. From the results of thin airfoil theory, calculate the lift coefficient, lift force, net circulation, the moment coefficient about the leading edge and the moment about the leading edge. Where is the center of pressure and aerodynamic center of this airfoil? What is the moment across the center of pressure and aerodynamic center of this airfoil?

Answer 1

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Answer 2

To calculate the lift coefficient, lift force, net circulation, moment coefficient, and moment of a NACA 0012 airfoil at 10 deg angle of attack using thin airfoil theory, we need to use the following equations:

1. Lift coefficient:

CL = 2 * pi * alpha

where alpha is the angle of attack.

2. Lift force:

L = 0.5 * rho * V^2 * c * CL

where rho is the air density, V is the air velocity, and c is the chord length of the airfoil.

3. Net circulation:

Gamma = 2 * pi * CL

4. Moment coefficient about the leading edge:

CmLE = -CL * (0.25 - c/4 * (xcp/c - 0.5))

where xcp/c is the distance of the center of pressure from the leading edge, and the coefficient -0.25 is the moment coefficient of the airfoil when the angle of attack is zero.

5. Moment about the leading edge:

MLE = 0.5 * rho * V^2 * c^2 * CmLE

where c^2 is the moment reference length.

Using the NACA 0012 airfoil, which has a maximum thickness-to-chord ratio of 0.12, we can assume that the chord length c is equal to 1. We can also assume a typical air density of 1.2 kg/m^3 and an air velocity of 30 m/s.

1. Lift coefficient:

CL = 2 * pi * alpha = 2 * pi * 10 deg = 0.349

2. Lift force:

L = 0.5 * rho * V^2 * c * CL = 0.5 * 1.2 kg/m^3 * (30 m/s)^2 * 1 m * 0.349 = 628.4 N

3. Net circulation:

Gamma = 2 * pi * CL = 2 * pi * 0.349 = 2.19

4. Moment coefficient about the leading edge:

From airfoil tables, the center of pressure location xcp/c for a NACA 0012 airfoil at 10 deg angle of attack is 0.25. Thus,

CmLE = -CL * (0.25 - c/4 * (xcp/c - 0.5)) = -0.349 * (0.25 - 1/4 * (0.25 - 0.5)) = -0.052

5. Moment about the leading edge:

MLE = 0.5 * rho * V^2 * c^2 * CmLE = 0.5 * 1.2 kg/m^3 * (30 m/s)^2 * (1 m)^2 * (-0.052) = -56.4 Nm

The center of pressure (CP) and the aerodynamic center (AC) of an airfoil depend on its geometry and angle of attack. For a symmetric airfoil like the NACA 0012, the CP is located at 0.25 chord length from the leading edge and the AC is located at 0.25 chord length from the trailing edge. The moment across the CP is zero, while the moment across the AC is approximately constant with changes in angle of attack.

Therefore, for this NACA 0012 airfoil at 10 deg angle of attack, the CP is located at 0.25 chord length from the leading edge, and the AC is located at 0.75 chord length from the leading edge. The moment across the CP is zero, while the moment about the AC can be calculated using the moment coefficient and the lift force:

CmAC = -0.052 + CL * (0.25 - 0.75) = -0.204

MAC = 0.5 * rho * V^2 * c^2 * CmAC = 0.5 * 1.2 kg/m^3 * (30 m/s)^2 * (1 m)^2 * (-0.204) = -221.8 Nm

Therefore, the moment about the AC is -221.8 Nm.