This set of Aerodynamics written test Questions & Answers focuses on “Prandtl’s Classical Lifting-Line Theory – 2”.

1. The induced angle of attack for an elliptical lift distribution increase when ______

a) Lift decreased

b) Aspect ratio decreased

c) Decrease in planform area

d) Increase in span length

View Answer

Explanation: If we decrease the planform area or increase the span length, the aspect ratio increases. Since the induced angle of attack is indirectly proportional to the aspect ratio, these options are wrong. We need to decrease the aspect ratio to increase the induced angle of attack. Increasing the lift can also increase the induced angle of attack.

2. For a finite wing, induced drag increases with an increase in aspect ratio.

a) False

b) True

View Answer

Explanation: The induced drag coefficient for a finite wing is inversely dependent on the aspect ratio. Thus, the induced drag increases with the decrease in aspect ratio and vice versa for a finite wing with the same lift.

3. The incorrect statement for a finite wing with elliptical lift distribution out of the following is_____

a) Induced drag coefficient ∝ coefficient of lift

b) Induced drag coefficient increases rapidly with lift

c) Induced drag coefficient is around 25%

d) To reduce induced drag, we want wing with the lowest aspect ratio

View Answer

Explanation: The induced drag coefficient varies directly with the square of lift coefficient. Thus, it increases rapidly with the lift. At high speeds, it is around 52% of total drag. For least induced drag, we want high aspect ratio for the wing.

4. For an elliptical lift distribution, the planform area is also elliptical for a wing.

a) True

b) False

View Answer

Explanation: The chord distribution is elliptical for an elliptical lift distribution. This is observed from the formula. Thus, the planform area is elliptical for an elliptical lift distribution for a finite wing.

5. The incorrect choice for the general lift distribution for a finite wing is ________

a) Circulation is assumed a Fourier series

b) 0 ≤ θ ≤ 2π

c) Coordinate transformation is for spanwise direction to θ

d) Coefficient of lift depends on aspect ratio directly

View Answer

Explanation: For a general lift distribution of a finite wing, the circulation is assumed a Fourier series from the expression obtained for elliptic lift distribution. The transformation involves spanwise coordinate transformed to θ, where 0 ≤ θ ≤ π. Using this the lift coefficient obtained depends directly on the aspect ratio.

6. Select the incorrect equation for the induced drag coefficient for a finite wing.

a) C_{D,i}=\(\frac {C_l^2}{\pi AR}\)(1+δ)

b) C_{D,i}=\(\frac {C_l^2}{\pi AR}\)

c) C_{D,i}=\(\frac {C_l^2s}{\pi eb^2 }\)

d) C_{D,i}=\(\frac {C_l^2}{\pi eAR}\)

View Answer

Explanation: The correct formula for induced drag coefficient for a general lift distribution is C

_{D,i}=\(\frac {C_l^2}{\pi eAR}\) where e=\(\frac {1}{1+\delta}\) and AR=\(\frac {b^2}{S’}\) where δ is related to the Fourier coefficients and e is the span efficiency factor. Thus, the option C

_{D,i}=\(\frac {C_l^2}{\pi AR}\) is wrong.

7. The elliptical lift distribution has a major interest in making wings because_____

a) Minimum induced drag

b) e = 0

c) δ = 1

d) Aspect ratio lowest

View Answer

Explanation: The coefficient of induced drag is C

_{D,i}=\(\frac {C_l^2}{\pi eAR}\). For an elliptic wing, e = 1 and/or δ = 0 i.e. the induced drag is minimum. The aspect ratio for the elliptic wing is not lowest. Moreover, we need a higher aspect ratio for lower induced drag.

8. Tapered wings are used more rather than elliptical wing planforms practically. The reason being______

a) Easy to manufacture than elliptic wings

b) Aspect ratio more important for low induced drag

c) Tapered wing resembles elliptic wing highly

d) Tapered wing is another name for elliptic wing

View Answer

Explanation: In reality, tapered wings are preferred over elliptic wings because they are easier to manufacture (they have straight edges). But the resemblance to elliptic wings (variation of δ) is secondary to reduce wing induced drag compared to the aspect ratio. Here we are only asked why tapered wings used; the answer is easy to manufacture.

9. The incorrect difference between airfoil and wings is_______

a) Presence of induced drag in wings

b) Different effective angles of attack seen by

c) Difference in the lift-curve slope

d) Lift slope of the wing is higher than of airfoil

View Answer

Explanation: The finite wing has a lower lift slope than infinite wing (airfoil). This is another difference than the presence of induced drag between wings and airfoil. Also, the effective angles of attack are different for both, being the same only when the lift is zero (at zero lift angle of attack).

10. For a finite wing and an infinite wing (airfoil), zero lift angle of attack is zero.

a) True

b) False

View Answer

Explanation: When the lift is zero, there are no induced effects i.e. induced drag and induced angle of attack are zero. Also, since the induced angle of attack is zero, the geometric angle of attack is the same as the effective angle of attack. This means zero lift angle of attack is the same for wing and airfoil.

11. The values of lift slope for elliptic wing (a_{e}) and general planform wing (a) is less than that for airfoil (a_{0}).

Mark the false statement.

a) a_{e}=\(\frac {a_0}{1+\frac {a_0}{\pi AR}}\)

b) a=\(\frac {a_0}{1+\frac {a_0}{\pi AR}(1+\tau)}\)

c) a_{0} is always 2π

d) τ is a function of Fourier coefficients

View Answer

Explanation: The lift curve slope of thin airfoils is 2π for small angles of attack and that also changes after stalling occurs. For the elliptical planform area, lift slope is a

_{e}=\(\frac {a_0}{1+\frac {a_0}{\pi AR}}\) and for general wing planform lift slope is a=\(\frac {a_0}{1+\frac {a_0}{\pi AR}(1+\tau)}\) where τ is a function of the Fourier coefficients used to describe the planform area.

12. For very high aspect ratio wings, the lift curve slope resembles that of the airfoil.

a) False

b) True

View Answer

Explanation: For any general wing planform lift slope is given by a=\(\frac {a_0}{1+\frac {a_0}{\pi AR}(1+\tau)}\) where τ is a function of the Fourier coefficients used to describe the planform area. As aspect ratio becomes very large (i.e. AR tends to ∞), we find a becomes a

_{0}i.e. for very high aspect ratio wings, lift slope for wing and airfoil is the same. Prandtl verified this through experiments also.

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