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f07_sp - Fluids Lecture 7 Notes 1 Elliptical Lift...

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Fluids – Lecture 7 Notes 1. Elliptical Lift Distribution Reading: Anderson 5.3.1 Elliptical Lift Distribution Definition and lift calculation Consider an elliptical spanwise circulation distribution given by Γ Γ 0 2 y 2 Γ( y ) = Γ 0 1 b y −b/2 0 b/2 where Γ 0 is the circulation at the wing center at y = 0. The overall lift on the wing is the integral of the corresponding lift/span distribution L ( y ) = ρ V Γ( y ). b/ 2 b/ 2 2 y 2 π L = L ( y ) dy = ρ V Γ 0 1 dy = ρ V Γ 0 b b/ 2 b/ 2 b 4 The integral can be evaluated via integral tables, or by inspection by noting that the area under an ellipse is π/ 4 times the area of the enclosing rectangle. Downwash calculation Computation of the downwash first requires knowing the trailing vortex sheet strength, which is minus the derivative of the circulation. d Γ 4 y γ ( y ) = = dy b 2 1 (2 y/b ) 2 The downwash at some location y o is then b/ 2 y 1 b/ 2 γ ( y ) dy Γ 0 dy w ( y o ) = = 4 π b/ 2 y o y πb 2 b/ 2 1 (2 y/b ) 2 y o y As in thin airfoil theory, the mathematical problem is considerably simplified by making the trigonometric substitution θ ο b y o = cos θ o 2 b y = cos

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f07_sp - Fluids Lecture 7 Notes 1 Elliptical Lift...

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