Student #: Duestion 3 (20 Points): In low speed, incompressible flow, the following experimental data are obtained for an NACA 4412 airfoil section at an angle of attack of 4^circ ,C_(1)=0.85 and c_(m,c/4)=-0.09 Calculate the location of center of pressure.

To calculate the location of the center of pressure for an airfoil section in low-speed, incompressible flow, we can use the following formula:<br /><br />\[ x_{cp} = \frac{c_1 \cdot x_{c_1} + c_{m,c/4} \cdot \frac{c_1^2}{\pi} \cdot \sqrt{AR}}{c_1} \]<br /><br />Where:<br />- \( x_{cp} \) is the location of the center of pressure<br />- \( c_1 \) is the coefficient of lift<br />- \( x_{c_1} \) is the position of the aerodynamic center (usually given or can be found in tables for the specific airfoil)<br />- \( c_{m,c/4} \) is the coefficient of the moment about the quarter-chord axis<br />- \( AR \) is the aspect ratio of the wing<br /><br />Given:<br />- \( c_1 = 0.85 \)<br />- \( c_{m,c/4} = -0.09 \)<br /><br />Assuming typical values for an NACA 4412 airfoil at an angle of attack of \( 4^\circ \):<br />- \( x_{c_1} \approx 0.3 \cdot c \) (where \( c \) is the chord length)<br />- \( AR \approx 10 \)<br /><br />Let's assume the chord length \( c \) is 1 (for simplicity, you can adjust this based on the actual chord length).<br /><br />\[ x_{c_1} = 0.3 \cdot 1 = 0.3 \]<br /><br />Now, plug in the values:<br /><br />\[ x_{cp} = \frac{0.85 \cdot 0.3 + (-0.09) \cdot \frac{0.85^2}{\pi} \cdot \sqrt{10}}{0.85} \]<br /><br />Calculate the numerator:<br /><br />\[ 0.85 \cdot 0.3 = 0.255 \]<br />\[ \frac{0.85^2}{\pi} \cdot \sqrt{10} = \frac{0.7225}{\pi} \cdot 3.162 = \frac{0.7225 \cdot 3.162}{3.1416} = 0.730 \]<br />\[ -0.09 \cdot 0.730 = -0.0657 \]<br /><br />So,<br /><br />\[ 0.255 - 0.0657 = 0.1893 \]<br /><br />Finally,<br /><br />\[ x_{cp} = \frac{0.1893}{0.85} \approx 0.222 \]<br /><br />Therefore, the location of the center of pressure is approximately \( x_{cp} \approx 0.222 \) chord lengths from the leading edge.