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# Aerodynamics – Prandtl’s Classical Lifting-Line Theory – 1

1 - Question

According to the reasoning given by Prandtl for his lifting line theory, finite wing is like a ________

a) Bound vortex

b) Horseshoe vortex

c) Free vortex

d) Trailing vortex

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Answer: aExplanation: Prandtl reasoned that the finite wing can be thought like a bound vortex, which is opposite to the free vortex. Since a vortex filament cannot end in the fluid, it is assumed that there are two trailing vortices which gives the appearance like that of a horseshoe vortex.

2 - Question

Circulation varies along the lifting line.

a) False

b) True

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Answer: bExplanation: In lifting line theory, the lifting line along the span is a superimposition of many bound vortices with different lengths. This gives different strengths for each of the horseshoe vortex i.e. the circulation varies along the lifting line.

3 - Question

The downwash along the wing in the downward direction in Prandtl’s lifting line theory comes from______

a) Bound vortex

b) Horseshoe vortex

c) Free vortex

d) Trailing vortices

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Answer: dExplanation: The bound vortex does not induce any velocity along itself. The downwash along the bound vortex (wing) comes from the two trailing vortices. Thus, horseshoe is also an incorrect option.

4 - Question

The incorrect statement regarding the downwash for a single horseshoe vortex in Prandtl’s lifting line theory is____

a) Downwash has contribution from trailing vortices

b) Downwash becomes infinite at the tips

c) Downwash is given by w=−Γ4πb(b2)2−y2

d) The contribution of two semi-infinite trailing vortices is same as an infinite vortex

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Answer: dExplanation: The downwash along the bound vortex is induced by the two semi-infinite trailing vortices, which is not same as that of an infinite trailing vortex. It acts downwards on the wing. When calculated using the Biot-Savart law, it comes equal to w=−Γ4πb(b2)2−y2, which becomes infinite at the tips (±b/2) if we consider only a single horseshoe vortex.

5 - Question

The infinite downwash at the wing tips for a single horseshoe vortex in Prandtls lifting line theory was a wrong result. Which of the following does not relate to the correction made?

a) Superimposition of horseshoe vortices

b) Lifting line along the span

c) Trailing vortices only at the tip

d) Different length of bound vortices

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Answer: cExplanation: The downwash due to a single horseshoe vortex in Prandtls lifting line theory gave a wrong result as infinite downwash at the wing tips. This was corrected by assuming a superimposition of many horseshoe vortices, each with different length of bound vortices lying along the span on a line, called lifting line. This gave many trailing vortices distributed along the span as well.

6 - Question

For the case of infinite horseshoe vortices along the lifting line, a vortex sheet exists which_________

a) Is formed by continuous trailing vortices

b) Is perpendicular to free-stream velocity

c) Total strength is zero

d) Equal and opposite trailing vortices

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Answer: bExplanation: The continuous trailing vortices gives a vortex sheet in the direction parallel to the free-stream velocity. The total strength of the vortex sheet is zero since it consists of trailing vortices with equal and opposite strength.

7 - Question

The induced angle of attack in terms of flow parameters for a wing is_____

a) αi=wV∞

b) αi=2wV∞

c) αi=V∞w

d) αi=wV∞

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Answer: aExplanation: The induced angle of attack at an arbitrary position along span, for a finite wing can be given as αi=wV∞ where V∞ is the free-stream velocity and w is the downwash (modulus only).

8 - Question

The lift coefficient for the airfoil section on a wing ______

a) Depends on local lift slope for airfoil

b) Is same for the entire wing

c) Is equal to the aerodynamic twist

d) Is constant always

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Answer: aExplanation: In wings, the lift coefficient for local airfoil section is equal to the local airfoil lift curve slope (2π for thin airfoils) into effective angle of attack minus zero lift angle of attack. Thus, it may or may not be same for the entire wing and is not equal to the aerodynamic twist.

9 - Question

The incorrect statement in relation to the fundamental equation of Prandtls lifting line theory is_____

a) Geometric AoA is sum of effective AoA and induced AoA

b) Integro-differential equation

c) Γ is known

d) Finite wing design, geometric AoA and free-stream velocity is known

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Answer: cExplanation: The fundamental equation of Prandtls lifting line theory simply states that the geometric angle of attack is the sum of effective angle of attack and induced angle of attack. In mathematical form, it is an integro-differential equation where Γ is unknown and wing design, geometric AoA and free-stream velocity is known.

10 - Question

Downwash for an elliptic wing circulation distribution is constant.

a) True

b) False

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Answer: aExplanation: For an elliptic distribution of lift over the span of a wing, downwash is a constant. This can be calculated from Biot-Savart law and using substitution to reveal that downwash is a constant along the span.

11 - Question

The induced drag (Di’) in terms of lift per unit span (L’) for a finite wing is_____

a) Di’=L’sin sinαi

b) Di’=L’αi

c) Di’=πL’αi

d) Di’=L’cos cosαi

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Answer: aExplanation: For a finite wing flat plate airfoil the induced drag is obtained by the component of lift in the direction of free-stream velocity i.e. Di’=L’sin sinαi. For a thin airfoil, we can make the small angle approximation in sine but not for all general airfoils.

12 - Question

Which is the wrong implication of the elliptical lift distribution for a wing?

a) Circulation varies elliptically along span

b) Span length does not affect circulation

c) Lift is zero at the tips

d) Maximum lift is at the center

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Answer: bExplanation: The elliptical lift distribution comes from the elliptical distribution of circulation along the span. The lift (or circulation) involves span length in the governing equations and hence the wrong statement. Lift is zero at the wing tips and maximum at the center.

13 - Question

Which of the following is not implied by the elliptical lift distribution?

a) Chord length is constant along the span

b) Induced angle of attack is zero for infinite wing span

c) Downwash is zero for infinite wing span

d) Constant downwash along the span

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Answer: aExplanation: The elliptical lift distribution gives an elliptical chord length distribution along the span. The induced angle of attack and downwash becomes zero for infinite wing span, proving the infinite wing (airfoil) theory. The downwash is a constant value along the span.

14 - Question

The circulation at the center for an elliptical wing distribution does not depend upon ________

a) Lift distribution

b) Density of fluid at center

c) Free-stream velocity

d) Total span length

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Answer: bExplanation: The circulation at the center for an elliptical wing distribution is dependent directly on the total lift which can be found from the lift distribution. Also, it depends upon the free-stream velocity, total span length and free-stream fluid density.