Vortices

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Vortices Lift

• Potential Flow

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Vortex
A vortex has no radial velocity, only tangential
velocity
Is there an irrotational vortex?
small order
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Vortex Strength
K
us
ds
K positive is an anti-clockwise vortex, K
negative a clockwise vortex.
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Velocities in an irrotational vortex
The cartesian velocities are given by
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Vortex strength K-20, giving clockwise rotation.
Streamline separation decreases with radius
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Vortex strength K-30, giving clockwise rotation.
Streamlines are closer together.
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VortexDoubletLinear gives flow over a rotating
cylinder
s
s
s
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Lift from a rotating cylinder
It has been found experimentally that a rotating
cylinder produces a lift force. This is known as
the Magnus effect. We can reproduce lift on a
rotating cylinder by adding a free vortex to a
doublet and linear flow.
The stagnation points are then on the same side
of the cylinder
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Stagnation Points
From the earlier section on doublets, we could
relate the doublet strength to the incoming
velocity
The stagnation points are not on y0 in this
case. To find them we note that the velocity at
the stagnation points is zero
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Stagnation Points
From the first equation, the stagnation points
are on the cylinder, so ra as expected, and from
the second equation
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Force on a rotating body
Kutta-Joukowski showed that there is a net
upwards force on the cylinder.
Note that the first result ignores the effects of
viscosity. The real circulation is about 50 of
the predicted circulation (and therefore the
force is too) due to boundary layer slip.
This is true for any cylinder cross section
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Magnus effect (Flettner Rotor)
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The rotor ship Barbara
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Force on a wing
For a streamline flow around a flat plate, the
stagnation points are equally spaced
The Kutta condition states that any body with a
sharp trailing edge will maintain the stagnation
point at the edge.
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Lift force on a wing
Take the simplest example of force on a flat
plate, represented by one vortex and a horizontal
flow.
K
a/2
u0
a/2
?
Kutta condition Velocity at the sharp tail of
the wing must be parallel to the wing, hence
velocity perpendicular to the wing is zero.
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Lift force on a wing
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Example simple flat plate
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Force on a wing
This implies a clockwise rotation has been appied
to the flow field, in other words the wing is
carrying a bound vortex.
The lift is generated by this bound vortex.
There is a special spatial tansformation which
can transform a circle to a wing shape, known as
the Joukowski transformation.
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Force on a wing
Note that the drag values are not predicted by
this analysis.
Since, from a standing start, the total
circulation is zero, there must be an equal and
opposite vortex produced. This is known as the
starting vortex, and is formed every time the
wing undergoes a change in lift.
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Multi-flap
Choose appropriate control points, in this case
at the end of the wing and the end of the flap.
K2
K1
a/2
a/2
b/2
b/2
?
eg b0.5a
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http//www.flaviir.com/
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Mirror images - flow near a wall