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3241 Lecture 12

3241 Lecture 12 - MAE 3241 AERODYNAMICS AND FLIGHT...

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1 MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS Finite Wings: General Lift Distribution Summary April 18, 2011 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk
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2 SUMMARY: PRANDTL’S LIFTING LINE THEORY (1/2) ( 29 ( 29 ( 29 ( 29 ( 29 - - - = - Γ = - Γ = - Γ + + Γ = 2 2 0 0 2 2 0 0 2 2 0 0 0 0 0 4 1 4 1 4 1 b b i b b b b L dy y y dy d V y dy y y dy d y w dy y y dy d V y c V y y π α π π α π α Fundamental Equation of Prandtl’s Lifting Line Theory Geometric angle of attack, α , is equal to sum of effective angle of attack, α eff , plus induced angle of attack, α i Equation gives value of Downwash, w, at y 0 Equation for induced angle of attack, α i , along finite wing
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3 SUMMARY: PRANDTL’S LIFTING LINE THEORY (2/2) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 dy y y S V C dy y y V D dy y S V S q L C dy y V L y V y L b b i i D i b b i b b L b b - - - - Γ = Γ = Γ = = Γ = Γ = 2 2 , 2 2 2 2 2 2 0 0 2 2 α α ρ ρ ρ Lift distribution per unit span given by Kutta-Joukowski theorem Total lift, L Lift coefficient, C L Induced drag, D i Induced drag coefficient, C D,i
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4 PRANDTL’S LIFTING LINE EQUATION Fundamental Equation of Prandtl’s Lifting Line Theory In Words: Geometric angle of attack is equal to sum of effective angle of attack plus induced angle of attack – Mathematically: α = α eff + α i Only unknown is Γ (y) – V , c, α , α L=0 are known for a finite wing of given design at a given a Solution gives Γ (y 0 ), where –b/2 ≤ y0 ≤ b/2 along span ( 29 ( 29 ( 29 ( 29 - = - Γ + + Γ = 2 2 0 0 0 0 0 0 4 1 b b L dy y y dy d V y y c V y y π α π α
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5 WHAT DO WE GET OUT OF THIS EQUATION? 1. Lift distribution 2. Total Lift and Lift Coefficient 3. Induced Drag ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 dy y y S V S q D C dy y y V dy y y L D L D dy y S V S q L C dy y V L dy y L L y V y L b b i i i D i b b i b b i i i i b b L b b b b - - - - - - Γ = = Γ = = 2245 Γ = = Γ = = Γ = 2 2 , 2 2 2 2 2 2 2 2 2 2 0 0 2 2 α α ρ α α ρ ρ
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6 GENERAL LIFT DISTRIBUTION (§5.3.2) Circulation distribution Transformation At θ =0, y=-b/2 At θ = π , y=b/2 Circulation distribution in terms of θ suggests a Fourier sine series for general
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