Using Helmbold’s relation for low-aspect-ratio wings, calculate the lift coefficient of a finite wing of aspect ratio 1.5 with an NACA 2412 airfoil section. The wing is at an angle of attack of 5°.
Using Helmbold’s relation for low-aspect-ratio wings, calculate the lift coefficient of a finite wing of aspect ratio 1.5 with an NACA 2412 airfoil section. The wing is at an angle of attack of 5°. Compare this result with that obtained from Prandtl ‘s lifting line theory for high-aspect-ratio wings. Comment on the difference between the two results. Assume a span efficiency factor \(e_{1} = 1.0\).
Using Helmold’s relation \(a_{0}=0.105/deg = 6.016/radian;\alpha _{L=0} = -2.2^{\circ}\)
Therefore,
\[a = \frac{a_{0}}{\sqrt{1+\left ( \frac{a_{0}}{\pi AR} \right )^{2}}+\left ( \frac{a_{0}}{\pi AR} \right )}=\frac{6.016}{\sqrt{1+\left ( \frac{6.016}{15 \pi} \right )^{2}}+\frac{6.016}{15\pi}} = 2.0757/ radian = 0.0362 /deg\]
Therefore,
\[C_{L} = 0.0362[5-(-2.2)]=0.261\]
On using Prandtl’s lifting line theory,
\[a =\frac{a_{0}}{1+\frac{a_{0}}{\pi AR}}= \frac{6.016}{1+1.2766}=2.643 /rad = 0.0461/deg\]
\[C_{L}=0.0461(7.2)=0.332\]
Prandtl’s lifting line theory gives a higher result.
