16100lectre32_cj

16100lectre32_cj - Single Horseshoe Vortex Wing Model S b b...

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Single Horseshoe Vortex Wing Model Lift due to a horseshoe vortex Kutta-Joukowsky Theorem b V dy V L b b Γ = Γ = ρ 2 2 S b b V C S V b V S V L C L L 2 2 2 2 2 1 2 1 Γ = Γ = = A b V C L Γ = 2 b S b trailing vortices bound vortex Γ Γ Γ
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Single Horseshoe Vortex Wing Model 16.100 2002 2 Induced Drag To estimate the induced drag using this simple model, we will assume that the 3-D lift is tilted by the downwash occurring at the wing root ( 0 = y ). i i eff i eff i eff i eff i eff L L L D L L L α = = = = sin cos For small i & V w i i To calculate downwash , we apply Biot-Savart: × Γ = filament r r d z y x w 3 4 ) , , ( v l v v π L eff L i i D V i w i eff x b y x Γ Γ Γ r v +∞ x +∞ x ) 0 , 2 , ( b x l d i D L V L
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Single Horseshoe Vortex Wing Model 16.100 2002 3 At 0 = = = z y x (i.e. wire root), the bound vortex does not produce a downwash, so, we only have the 2 trailing vortices at 2 b y ± = . + Γ = + = = 2 2 4 ) 0 , 0 , 0 ( b y b y w π v Let’s do the 2 b y = integral first: ∫∫ + = +∞ =+ =
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16100lectre32_cj - Single Horseshoe Vortex Wing Model S b b...

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