lec30 - Aerodynamics Lecture 30 Prandtls Lifting Line...

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Aerodynamics Prandtl’s Lifting Line Theory 30.1 Lecture 30 Aerodynamics Wing: Prandtl’s Lifting Line Theory AE311 Aerodynamics Manoj T. Nair IIST
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Aerodynamics Prandtl’s Lifting Line Theory 30.2 Agenda 1 Prandtl’s Lifting Line Theory
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Aerodynamics Prandtl’s Lifting Line Theory 30.3 Prandtl’s Lifting Line Theory I Fundamental equation of Prandtl’s lifting line theory α = Γ( y 0 ) π V c ( y 0 ) + α L = 0 + 1 4 π V Z b / 2 - b / 2 ( d Γ / dy ) dy ( y 0 - y ) The solution yields Γ = Γ( y 0 ) y 0 ranges along the span from - b / 2 to b / 2 The aerodynamic characteristics of a finite wing are then obtained The lift distribution from Kutta-Joukowski theorem L 0 ( y 0 ) = ρ V Γ( y 0 ) The total lift is L = Z b / 2 - b / 2 L 0 ( y ) dy L = ρ V Z b / 2 - b / 2 Γ( y ) dy
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Aerodynamics Prandtl’s Lifting Line Theory 30.4 Prandtl’s Lifting Line Theory II The lift coefficient is C L = L q S = 2 V S Z b / 2 - b / 2 Γ( y ) dy Induced drag D 0 i = L 0 i sin α i D 0 i = L 0 i α i The total induced drag D i = Z b / 2 - b / 2 L 0 ( y ) α i ( y ) dy D i = ρ V Z b / 2 - b / 2 Γ( y ) α i ( y ) dy The induced drag coefficient is C D , i = D i q S = 2 V S Z b / 2 - b / 2 Γ( y ) α i ( y ) dy
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Aerodynamics Prandtl’s Lifting Line Theory 30.5 Prandtl’s Lifting Line Theory III Elliptic Lift Distribution Before discussing the general solution, lets discuss a special case Consider the circulation distribution given by Γ( y ) = Γ 0 s 1 - 2 y b 2 Γ 0 is the circulation at the origin Circulation varies elliptically with y along the span
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Aerodynamics Prandtl’s Lifting Line Theory 30.6 Prandtl’s Lifting Line Theory IV
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