Aero II

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Aero II Fundamental Theory of Finite Wings
Lecture 35
J. E. Brandenburg EAS 3010 Spring 2007
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Agenda

• Kutta Jakowski Theorem
• Kelvins Theorem and Starting Vortex
• Wing tip Vortices
• Finite Wing Effects

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Vorticity and Circulation
Vorticity is important
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Circulation
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Bernoulli Equation
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Kutta-Jackowski Theorem
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Approximating Flows with Vortices
Using Vortex solutions , airflow can be modeled
continuously
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Trailing Vortex Cont.
Trailing vortex creates tube of rotating air of
approximate circulation ? and approximate Kinetic
energy per unit length Drag Force ½ ?V2???R2 ?
1/8? ? (2?VR)2 1/8? ? ?2 This leads to Drag
relationship D??2?(Lift)2
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(No Transcript)
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Kelvins Theorem
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Kelvins Theorem
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Kelvins Theorem
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Vortex Ring
Mt. Etna Italy
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Trailing Vortex
At end of wings vorticity does not end but spills
off end of wing
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Aircraft Have Turbulent Wakes
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Smoke Angel
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Finite Wings

• Wing tip Vortices distort flow over finite wing-
  leading to downwash
• Downwash behind wing gives downward component to
  flow over wing leading to induced angle of attack
  ?i
• Induced angle of atttack leaded to effective
  angle of attack ?eff ?-?i

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Downwash
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Downwash
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Downwash
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Induced angle of attack
Induced angle of attack
Geometric angle of attack ?
Effective angle of attack ?eff ? - ?i
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Infinite versus Finite wings
CL is reduced by finite wing effects
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Pressure Drag
Backwards lift component
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Lift Distribution
lift
wing
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Space Shuttle Landing
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Horseshoe Vortex
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Horseshoe Vortex
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Lift Distribution
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Sail plane
One way to get rid of finite wing effects
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U-2
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Lifting Line Theory
We can define ?(x) on wing surface
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Lifting line theory
Loop velocities cancel except at edges
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Lifting result