3241 Lecture 11

# 3241 Lecture 11 - MAE 3241 AERODYNAMICS AND FLIGHT...

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MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS Introduction to Lifting Line Theory April 11, 2011 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

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NECESSARY TOOL Return to vortex filament, which in general maybe curved General treatment accomplished with Biot-Savart Law 3 4 r r dl dV × Γ = π Electromechanical Analogy: Think of vortex filament as a wire carrying an electrical current I The magnetic field strength, dB, induced at point P by segment dl is: 3 4 r r dl I dB × = μ
EXAMPLE APPLICATIONS h V π 4 Γ = h V 2 Γ = Case 1: Biot-Savart Law applied between ± ∞ Case 2: Biot-Savart Law applied between fixed point A and ∞ 3 4 r r dl dV × Γ = Case 1 Case 2

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BIOT-SAVART LAW
EXAMPLE APPLICATIONS Case 1:

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HELMHOLTZ’S VORTEX THEOREMS 1. The strength of a vortex filament is constant along its length 2. A vortex filament cannot end in a fluid; it must extend to boundaries of fluid (which can be ± ∞) or form a closed path Note: Statement that “vortex lines do not end in the fluid” is kinematic, due to definition of vorticity, ϖ , (or ξ in Anderson) and totally general We will use Helmholtz’s vortex theorems for calculation of lift distribution which will provide expressions for induced drag L’=L’(y)= ρ V Γ (y)
CONSEQUENCE: ENGINE INLET VORTEX

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CHAPTER 4: AIRFOIL Each is a vortex line One each vortex line Γ 1 =constant Strength can vary from line to line Along airfoil, γ = γ (s) Integrations done: Leading edge to Trailing edge z/c x/c Side view Entire airfoil has Γ Γ 1 Γ 4 Γ 7
CHAPTER 5: WINGS

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PRANDTL’S LIFTING LINE THEORY Replace finite wing (span = b) with bound vortex filament extending from y = -b/2 to y = b/2 and origin located at center of bound vortex (center of wing) Helmholtz’s vorticity theorem: A vortex filament cannot end in a fluid Filament continues as two free vorticies trailing from wing tips to infinity This is called a ‘ Horseshoe Vortex
PRANDTL’S LIFTING LINE THEORY Trailing vorticies induce velocity along bound vortex with both contributions in downward direction (w is in negative z-direction) ( 29 ( 29 2 2 2 4 2 4 2 4 4 y b b y w y b y b y w h V - Γ - = - Γ - + Γ - = Γ = π Contribution from left trailing vortex (trailing from –b/2) Contribution from right trailing vortex (trailing from b/2) This has problems: It does not simulate downwash distribution of a real finite wing Problem is that as y → ±b/2, w → ∞ Physical basis for solution: Finite wing is not represented by uniform single bound vortex filament, but rather has a distribution of Γ (y)

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3241 Lecture 11 - MAE 3241 AERODYNAMICS AND FLIGHT...

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