What is the modified 2D relation between circulation and angle of attack including induced velocity?
Γ(y) = πUc(y)[α + α₀ + wᵢ/U], where wᵢ is the induced velocity from trailing vortices.
What does each term in Γ(y) = πUc(y)[α + α₀ + wᵢ/U] represent?
α = angle of attack, α₀ = zero-lift angle (due to camber), wᵢ = induced velocity, c(y) = chord length at spanwise position y.
Why is Equation (1.1) justified?
Because when the wingspan b ≫ chord c, the local flow behaves approximately as 2D, with distant vortices inducing nearly uniform velocity.
What is the horseshoe vortex model?
A simplified model with constant circulation along the wing and two trailing vortices at the wingtips extending downstream.
Why is the horseshoe vortex model inconsistent?
It assumes constant Γ along the span, but solving the induced velocity equations shows Γ must vary with y.
How does Prandtl’s lifting line theory correct the horseshoe model?
It allows circulation to vary with spanwise position and accounts for an infinite series of infinitesimal trailing vortices.
What is the induced velocity at spanwise position y in lifting line theory?
wᵢ(y) = (1/4π) ∫[dΓ(y₀)/dy₀] / (y - y₀) dy₀, integrated over the wingspan.
Why is the integral for wᵢ(y) taken as a Cauchy principal value?
Because the integrand is singular at y₀ = y, so a symmetric limit excluding this point ensures convergence.
Write Prandtl’s integro-differential equation for the circulation distribution.
Γ(y) = πUc(y)[α + α₀ + (1/4πU) ∫[dΓ(y₀)/dy₀]/(y - y₀) dy₀].
What is the physical meaning of the circulation varying with y?
It reflects how lift changes along the wingspan, producing trailing vortices that induce downwash and drag.
What is the local lift per unit span?
dL = ρUΓ dy.
What is the induced drag per unit span?
dD = -(wᵢ/U) dL, arising from the backward tilt of the lift vector.
Why does inviscid flow produce drag in lifting line theory?
Because the lift vector is inclined backward by the induced angle αᵢ = wᵢ/U, creating an induced drag component.
What change of variable simplifies the lifting line equation?
y = (b/2)cosθ, mapping spanwise positions to 0 < θ < π.
How is the circulation expanded in terms of Fourier coefficients?
Γ(y) = 2bU ∑ Aₙ sin(nθ), where n = 1, 3, 5, … (odd terms for symmetry).
Why are only odd n terms used in the circulation expansion?
Because Γ(y) must be symmetric about the wing center; only odd sine terms satisfy this condition.
What is Glauert’s result used in lifting line theory?
It removes the singularity in the integral for wᵢ, allowing an algebraic relation between Aₙ coefficients.
Write the algebraic form of Prandtl’s equation using Glauert’s transformation.
2bAₙ sin(nθ) = πc(y)[α + α₀ - nAₙ sin(nθ)/sinθ].
How is the infinite series in the lifting line equation solved numerically?
By truncating it to N terms and solving the resulting N×N linear system for A₁, A₃, A₅, etc.
What is the total lift on the wing according to lifting line theory?
L = ½ πρU²b²A₁.
What is the total drag according to lifting line theory?
D = ½ πρU²b² ∑ nAₙ², where each term adds induced drag.
Define the lift and drag coefficients in terms of Aₙ.
CL = A₁(πb²/S), CD = (πb²/S)∑ nAₙ².
What is the aspect ratio and how is it defined?
Aspect ratio AR = b²/S, where b is the wingspan and S is the wing area.
What circulation distribution gives minimum induced drag?
An elliptical distribution: Γ = 2UbA₁(1 - 4y²/b²)¹/².
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Midterm Prep - Ch 217
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Midterm Prep - Ch323
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Midterm Prep - Ch430
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Midterm Prep - Ch513
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Midterm Prep - Ch640
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Exam 2 - Chapter 724
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Exam 2 - Chapter 834
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Exam 2 - Chapter 933
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Exercises Ch88
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Exercises Ch98
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Chapter 11 - Shocks12
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Exercises Ch115
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Exercises Ch125
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Chapter 1316
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Chapter 1045
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Exercises 17
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Exercises 26
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Exercises 36
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Exercises 42
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Exercises 67
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Midterm 1 theory4
