Question

Consider a finite wing of aspect ratio 10 with span effectiveness factor 0.95. Its airfoil section has a lift slope of 0.106 per degree and a zero-lift angle of attack of -1.5\(^{\circ}\). The lift coefficient of the wing at an angle of attack of 3.5\(^{\circ}\) is ________ (rounded off to 2 decimal places).

Hint:

Use the finite-wing lift-slope correction a = a0/(1 + a0/(pi e AR)), convert a0 to per radian, then multiply by (alpha - alpha_L0).

Answer 1

Correct Answer: 0.44

Step 1: Recall the finite-wing lift-slope correction formula.
Finite-span effects reduce the lift-curve slope compared with the 2D airfoil (section) value \(a_0\). From lifting-line theory, the finite-wing lift-curve slope \(a\) (per radian) is
\[ a = \frac{a_0}{1+\dfrac{a_0}{\pi\,e\,AR}} \]
where \(e\) is the span effectiveness factor and \(AR\) is the aspect ratio, with \(a_0\) and \(a\) both expressed per radian.

Step 2: Convert the given section lift slope to per radian.
\[ a_0 = 0.106\ \text{per degree} \times \frac{180}{\pi} = 0.106 \times 57.296 = 6.073\ \text{per radian} \]

Step 3: Compute the denominator term.
\[ \pi\,e\,AR = \pi \times 0.95 \times 10 = 29.845 \]
\[ \frac{a_0}{\pi e AR} = \frac{6.073}{29.845} = 0.2035 \]

Step 4: Compute the finite-wing lift slope.
\[ a = \frac{6.073}{1+0.2035} = \frac{6.073}{1.2035} = 5.046\ \text{per radian} \]
Converting back to per degree, \(a = 5.046/57.296 = 0.0881\) per degree.

Step 5: Apply the lift equation with the zero-lift angle.
\[ c_l = a\,(\alpha-\alpha_{L0}) \]
\[ \Delta\alpha = 3.5-(-1.5) = 5^{\circ} \]
\[ c_l = 0.0881 \times 5 = 0.440 \]

Final Answer:
Rounded to 2 decimal places, the wing lift coefficient is about 0.44.
\[ \boxed{c_l \approx 0.44} \]