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lectures-chapter6 - 650:460 Aerodynamics Chapter 6 Prof...

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650:460 Aerodynamics Chapter 6 Prof. Doyle Knight Tel: 732 445 4464, Email: [email protected] Office hours: Tues and Thur, 4:30 pm - 6:00 pm and by appointment Fall 2009 1
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Circulation and Generation of Lift Consider superposition of uniform flow and a circulating flow We seek the appropriate combination of elementary solutions of Laplace’s equation which will yield a streamline coinciding with the desired airfoil shape Fig. 6.1 2
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Starting Vortex Consider an airfoil at rest in a quiescent fluid. The circulation contintegraltext vector v · vector ds = 0 The flow is accelerated to U A starting vortex forms and is swept downstream. The circulation about the airfoil becomes finite (and hence lift is generated). The circulation around the airfoil is equal and opposite to the circulation around the starting vortex (Kelvin’s Theorem) Fig. 6.2 3
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Circulation Potential flow past a 2-D airfoil has an infinite number of possible solutions that satisfy the boundary conditions Experiments indicate that the flow leaves the trailing edge as shown at the right provided the airfoil is not stalled This is the Kutta condition and only one potential flow solution satisfies this condition Fig. 6.3 4
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General Thin Airfoil Theory Assumptions Two-dimensional Incompressible Irrotational Steady Neglect airfoil thickness Maximum camber is small compared to chord Represent camber line by line vortices Fig. 6.4 5
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General Thin Airfoil Theory Strength of vortex at x = ξ is d Γ = γ ds Normal component of velocity induced by vortex at x = ξ on point P dV s , n = γ ds 2 π r bracehtipupleftbracehtipdownrightbracehtipdownleftbracehtipupright vortex cos δ 3 where the minus sign implies clockwise circulation Fig. 6.5 6
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General Thin Airfoil Theory From geometry cos δ 2 = ( x ξ ) r ds = d ξ cos δ 1 The resultant velocity at any point P normal to the airfoil is V s , n = 1 2 π integraldisplay c o γ ( ξ ) cos δ 2 cos δ 3 d ξ ( x ξ ) cos δ 1 Fig. 6.5 7
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General Thin Airfoil Theory The component of the freestream velocity normal to the airfoil at P is U , n = U sin( α δ P ) where δ P is the slope δ P = tan - 1 dz dx and thus U , n = U sin( α tan - 1 dz dx ) Fig. 6.5 8
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General Thin Airfoil Theory Thus, the boundary condition vector v · vector n = 0 on the airfoil is U sin( α tan - 1 dz dx ) + V s , n = 0 where V s , n = 1 2 π integraldisplay c o γ ( ξ ) cos δ 2 cos δ 3 d ξ ( x ξ ) cos δ 1 Fig. 6.5 9
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General Thin Airfoil Theory Thus U sin( α tan - 1 dz dx ) = 1 2 π integraldisplay c o γ ( ξ ) cos δ 2 cos δ 3 d ξ ( x ξ ) cos δ 1 Assuming δ i are small U parenleftbigg α dz dx parenrightbigg = 1 2 π integraldisplay c o γ ( ξ ) d ξ x ξ where α is in radians. This is the governing equation for two-dimensional thin airfoil theory.
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