[From Bill Powers (931220.0710 MST)]
Martin Taylor (931219) --
On further consideration, I believe your analysis of how a vortex
forms is incomplete. It is not the Bernoulli effect, but
conservation of angular momentum that gives rise to a bathtub
vortex.
In the laminar flow case, fluid approaches the center radially;
there must be a radial height gradient to sustain the radial
flow. If there is a slight deflection of any part of this flow so
it is not radial, a packet of water will follow a path tangential
to some small circle around the center. The initial angular
momentum will be (save for friction) preserved, so as the packet
approaches the center, it will have greater and greater angular
velocity as the radius decreases. Shear friction will then
accelerate packets of water with a greater radius, ultimately
giving the entire body of water a spin that decreases in angular
velocity as radius increases. Equilibrium will be reached when
the distribution of angular velocity is sufficient to make the
shear forces times speed of rotation everywhere equal to the work
done by the descending flow. The work done by the vertical
component of velocity is then equal to the energy dissipated by
shear friction over the whole volume. There must, of course, be
sufficient head to produce enough energy to overcome frictional
losses, which are a nonlinear function of shear rate.
The rotation sets up a counterpressure due to centrifugal force.
As the vortex develops, the flow through the drain hole
decreases, because of the increasing counterpressure. I
discovered as a child that the last of the bathwater will drain
much faster if you keep the vortex from forming. Equilibrium
occurs when the vortex has a shape such that the gravitational
force minus the centrifugal force, both projected into lines of
flow, is just balanced by frictional forces and acceleration due
to the converging flow, everywhere within the affected volume.
This gives the typical curved funnel shape, with the greatest
angular velocity at the surface of the funnel nearest the drain
hole, and thus the greatest counterpressure at that height.
The angular momentum of the stable vortex is, I suspect, zero if
there is no continuing tangential disturbance. Assume that the
vortex as a whole is rotating clockwise. If you follow any
orbiting packet, you find that a packet just inside that radius
is moving faster, and one just outside it is moving slower.
Drawing a line between the inside and outside packets, you would
see that relative to the center packet there is a
counterclockwise rotation superimposed on the general clockwise
rotation. Conservation of momentum suggests that the sum of all
these angular momenta is zero.
While a vortex involves a complex set of actions and reactions in
three dimensions, it is still an equilibrium system in which the
steady state involves a balance of forces acting in both
directions at once. The forward forces are everywhere identical
to the reverse forces when friction and acceleration are taken
into account. A perturbation of the vortex forces it into a
nonequilibrium state, which is then restored as the excess energy
(or deficiency of energy) is rebalanced. All of the restoring
forces come from the deviation caused by the perturbation, and
the interaction of all parts of the flow. While I don't know how
to calculate the power gain involved, I strongly suspect that it
is less than 1.
You are incorrect, by the way, in saying that a tendency to
return to equilibrium after a disturbance is removed requires a
loop gain greater than 1. It requires only a loop gain greater
than zero. A control system with a loop gain of 0.5 will produce
an output that is 0.5/1.5 or 1/3 the magnitude of a disturbance.
When the disturbing influence is removed, the output will restore
the system to equilbrium.
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I really object to calling a vortex a "self-organizing" system in
the context of any quantitative discussion. Organization is a
perception; who is to say whether a vortex impresses everyone as
more organized than a laminar flow, or less? The "organized-ness"
is based on the familiarity of the vortex shape. Equating
organization to a function of entropy is gratuitous; One might
as well just compute entropy and leave "organization" out of it
because that, in effect, is what is done.
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Best,
Bill P.