eqnsheetEXAM2 - AE311 Exam 2 Equation Sheet Ψ Uniform Flow...

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Unformatted text preview: AE311 Exam 2 Equation Sheet Ψ Uniform Flow from left to right Source Flow Vortex flow Doublet − φ V∞rcosθ Λ ln r 2π − Γ θ 2π − Vr V∞cosθ Λ 2π r Vθ - V∞sinθ V∞rsinθ Λ θ 2π Γ ln r 2π K sin θ 2π r −Γ 2π r K cosθ 2π r 2 − K sin θ 2π r 2 K cosθ 2π r Ψ Uniform Flow from left to right Source Flow Vortex flow Doublet V∞y Λ y arctan 2π x Γ 2 2 1/ 2 ln( + y ) x 2π φ V∞x u V∞ v 0 Λy 2π (x 2 + y 2 ) − Γx 2π ( x 2 + y 2 ) Λx Λ ln( x 2 + y 2 )1 / 2 2π (x 2 + y 2 ) 2π − Γ y arctan 2π x Γy 2π ( x 2 + y 2 ) − K x 2 2π (x + y 2 )3 / 2 − y K 2 2π (x + y 2 ) K x 2 2π ( x + y 2 ) − y K 2 2π (x + y 2 )3 / 2 Cylinder K = 2 πV∞R2 Conservation of mass ∂ρ + ∇. ρV = 0 − ∂t − ∂ρ u ∂p + ∇ . ρu V = − + ρf x + Fvis , x − − ∂t ∂x () Conservation of momentum in x D ∂ = +V ⋅∇ Dt ∂t () ξ = 2ω = ∇ × V ⎛∂ ∂⎞ ⎛∂ 1 ∂⎞ ∇=⎜ , ⎟=⎜ , ⎟ ⎜ ∂x ∂y ⎟ ⎝ ⎠ ⎝ ∂r r ∂θ ⎠ wdy − vdz = 0 udz − wdx = 0 vdx − udy = 0 Thin airfoil theory 1 γ (ξ ) dξ dz ⎞ ⎛ ∫ x − ξ = V∞ ⎜ α − dx ⎟ 2π 0 ⎝ ⎠ c C L ' = π ( 2 A0 + A1 ) ∞ 1 + cos θ + 2V∞ ∑ An sin nθ γ (θ ) = 2 AoV∞ sin θ n =1 1 dz A0 = α − ∫ dθ π 0 dx 2 dz An = ∫ cos nθdθ π 0 dx π π α L0 x= 1 dz = − ∫ (cos θ − 1)dθ π 0 dx c (1 − cos θ ) 2 π ...
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This note was uploaded on 03/07/2011 for the course AE 311 taught by Professor Dutton,j during the Spring '08 term at University of Illinois, Urbana Champaign.

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eqnsheetEXAM2 - AE311 Exam 2 Equation Sheet Ψ Uniform Flow...

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