Figure 1
If an obstruction of a given
diameter (d) is
placed
in a flowing fluid, the boundary layer will separate from the surface
at
some point and there will be a turbulent wake. I think modelers
are
generally aware of the turbulent wake downstream of such a
bluff
body but what is not commonly known is that hidden in the chaos is
a limit cycle called a "Kármán
vortex street " *
. Figure 1 is a schematic of the the inviscid calculations
of Dr.
Theodore Von Kármán and does not represent the
usual
appearance of vortex streets, although at extremely low Reynolds
numbers (less than 400) it is possible to produce a street that dose
look
surprisingly like the diagram and, if the axial
flow is blocked at both ends, a well defined vortex street can be
observed
at an extremely
large scale *.
It's presented here to show that there is order on a small scale.
And where there is order we may be able to
exploit
it. When discussing vortex induced lift we're not
interested
in the whole street, just the first eddy or two. As
an
aid to visualization it may be useful to create a simplified model such
as figure 2. In this model the parcel of air that will become the
vortex is represented by a length of steel pipe and the free-stream is
a conveyor belt. As long as the power is off nothing happens (the
system is in equilibrium). Now turn the power on. The
belt transfers energy (thrust) to the pipe at the point of
contact
and if the pipe weren't free to rotate it would move downstream at the
same speed as the belt. Since the pipe is free to rotate and the
thrust is applied at some distance from the center of gravity a turning
moment is imparted to the pipe and it tries to roll up-stream.
Due
to part of the originally linear energy being converted to rotational
motion
the pipe's progress downstream is reduced by a percentage which is
determined
by the acceleration of the belt when you turned the power on. For
the vortex street of figure 1 the eddys drift downstream at
86 percent of the free stream velocity (eddy speed=0.86v).
Which direction are the belt's drive rollers rotating?
Figure 2
Of course air is not a solid
and the truth is a lot more complex than a pipe on a conveyor
belt.
This call's for another experiment, this time with air.
Since
we're only interested in the first vortex, or two, lets see what we can
do to prevent the others from forming. Lets take the
cylinder
from figure 1 and cut it lengthwise (we'll call it the "bubble
generator").
Now affix it to a flat plat as in figure 3 thus creating a crude
airfoil. When the airflow hits the half cylinder it forms a
boundary layer which
flows around the leading edge until it reaches the top, after
that
point it encounters the adverse pressure gradient and starts losing
energy. Some distance downstream from the minimum pressure point
the boundary layer
no longer has enough energy to follow the curvature of the surface
and it separates. Normally this would lead to a turbulent wake
but
we have created a void for the separated wake to fall into. As
the
wake falls into this space it tumbles and starts spinning.
As more materiel separates from the bubble generator it becomes
entrained
in the vortex and the wake bubble expands to fill all available
space. Notice that the fluid that dose not become part of the
vortex follows a
nice streamlined curve. Now we're getting somewhere!
Figure 3 
Now that we have a two
dimensional
picture of what might be happening on a wing with vorticality we need
to
stretch it out into the third dimension. By extruding the vortex
from figure 3 along the Z axis and adding the 3D vector information we
get a model that looks something like figure 4. Due to
centrifugal
force the vortex develops a partial vacuum at it's core, it is this
vacuum
that prevents the vortex from being pulled apart by it's own
centrifugal
force.
Referring to figure
4, lets follow a particle that enters the system at (A). Since
this
particle is free to move in any direction it's velocity at any given
moment
can be expressed as vectors in a 3D Cartesian coordinate system.
The axes of this coordinate grid are labeled, Tangential, Radial and
Axial.
At first the particle's motion is entirely tangential but after a
while it starts moving radialy inward due to the pressure
differential
between the core and the outside atmosphere. As it moves toward
the
core the particle's tangential velocity increases and it begins to pick
up some axial velocity, it also looses heat during this inward
spiral.
At some time during it's inward travel our particle crosses a threshold
where the centrifugal force pushing out is equal to the pressure
pushing
in , this threshold is called the inner radius. By
the
time our particle reaches the inner radius boundary it has circled the
core dozens of times and has given up quite a bit of heat. If the
relative humidity was high enough before the air became entrained
in the vortex, it has cooled to the dew point and a white condensation
trail will be visible near the core. If you have an opportunity
to
see a condensation trail up close notice that it appears to be a hollow
tube. That's because the relative humidity inside the core is
lower
than outside, so there is usualy not mist in the core.
Figure 4
