Linear Motion

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Lift Theories
Linear Motion
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Requirements for a Valid Theory

• A valid theory is a rational explanation of
  observed phenomenon
• A valid theory can be used to predict future
  observations
• A valid theory produces numerical results

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Generation of Lift
For a lifting airfoil, the surface static
pressure varies from top to bottom and from front
to back.
For a lifting airfoil, the average pressure on
the upper surface is lower than the average
pressure on lower surface. The difference in
pressure produces the lift.
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Generation of Lift
Why does the surface static pressure vary top to
bottom and front to back?
The ideal surface velocity varies from top to
bottom and front to back. The surface of the foil
is a streamline, so Bernoullis equation relates
the surface velocity to surface pressure. When
we include viscosity, the surface velocity is
zero, but the local velocity varies at the
boundary layer edge.
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Generation of Lift
Why does the surface velocity vary top to bottom
and front to back?
The flow must follow the surface contour (or the
edge of the boundary layer). Flow can not pass
through the airfoil. What determines the
values for the local surface velocity, and
therefore the local surface pressure, and
ultimately the amount of the lift?
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Generation of Lift
What determines the values for the local surface
velocity, and therefore the local surface
pressure, and ultimately the amount of the lift?
The flow around the airfoil must satisfy the
conservation laws Conservation of mass
(continuity) Conservation of momentum
(2 or 3 components) Conservation of
energy
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Euler Equation Solution
Uniform Flow Doublet Vortex
Lift is Generated
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Bound Vortex Theory
The Kutta Condition
Mapped Uniform Flow Doublet
Mapped Uniform Flow Doublet
Bound Vortex
Lift
No Lift
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Bound Vortex Theory
Uniform Doublet Bound Vortex
G
Vortex strength
r
Free stream velocity V
r
Air density
G
V
Kutta-Joukowsky Theory
r
G
Lift L V
Lift Coefficient
Theory correctly predicts slope of curve 2
for thin airfoils
p
Angle of Attack
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Starting (Shed) Vortex
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Prandtls Experimental Photo of
Starting Vortex
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Prandtl Lifting Line Theory
Bound Vortex
Tip Vortex
Three dimensional version of Bound Vortex Theory
A continuous line of bound vortices
terminating at the wing tips with tip
vortices that continue downstream to the
starting vortex.
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Photo of Tip Vortices