Originally posted by Guest
...The Helmhold vortex theorem states that "vorticies end on boundries or form a closed path". This theory is just plain wrong. A funnel cloud is a vortex that does not end on a boundry or form a closed loop. The vortex I create with a boat paddle does not extend to the bottom of the lake. ...
I agree that any description of fluid mechanics has to agree with what we all observe in practice. However, the vortex theories with which you take issue have been well substantiated by experiment and form the basis of most practical fluid dynamic engineering calculations. I respect your legitimate questions, and my appologies to the rest of the board if I'm responding to an obvious troll.
When I was in college, one of the professors, Dr. C. T. Tsu, was doing research into tornadoes. He had made an experimental rig with a blower that exhausted through a duct with a spinning honeycomb disk inside, followed by an exquistely machined plexiglass nozzle. Below the vertically oriented nozzle was a ground board that could be moved up and down to vary the distance to the nozzle. To make a long story short, he found that he didn't need the nozzle, and he didn't even need the blower - all that was necessary was the rotating disk. It turns out tornadoes are a boundary layer phenomenon, and all you need is a rotating air mass in close proximity to a surface. That's why so much of the damage in hurricanes is actually due to tornadoes spawned by the hurricane's rotation. His rig produced tiny little tornadoes that had all the characteristics of the real thing, including the sheath at the bottom where debris (ground walnut shells, for the model) is picked up. So it's not correct to say that the tornado doesn't obey Helmholz' laws. One end does end at the boundary to the fluid and the other end is the concentration of the vorticity already contained in the airmass. You need both - the surface and the rotation. If the surface is too far away from the rotating airmass you don't get tornadoes. If this still seems to contradict the Helmholtz theorem, remember that the theorem specifically assumed an irrotational fluid in which the net vorticity had to add up to zero. That assumption is violated by the conditions that spawn tornadoes, just as it is violated inside the boundary layer of the wind flowing around a sail.
As for the paddle, the two vortices you see at the surface are in fact the ends of one vortex that goes down the side of the paddle, wraps around the tip, and comes up the other side. Prandtl even used the analogy of the flow about a downward-moving paddle to describe the vortex wake left by a wing:
"Instead of a downward acceleration of the wing itself in its various consecutive positions we shall consider an instantenous acceleration (impulse) of the whole surface of separation [meaning the depressed wake behind the wing - TS]. In other words, this surface of separation is momentarily solidified into a 'board,' and this board is given a downward impulse (In order to take care of the variation of
w1 with
x, the 'board' may be considered elastic)." (Prandtl, L.and Tietjens, O. G., "Applied Hydro- and Aeromechanics," Dover Publications, Inc., NY, 1934.Section 111, pp. 192)
So, you're right - the vortices don't go all the way to the bottom of the lake. They are a continuous loop and only have ends at the boundary of the fluid, just as the Helmholtz theory says they must.
...There is no real circulation around a wing. I know that by subtracting the average velocity vector it then looks like there is circulation, but that same trick can be used to argue that the cars in the right lane of the interstate are going backwards. The cars in the right lane aren't really going backwards and there isn't really any circulation just upwash down wash and a slight difference in velocity above and below the wing...
I think it would be perfectly valid to describe automobile traffic by a mean velocity and a perturbation, both positive and negative, with respect to that mean velocity. So relative to the mean flow, yes, some cars would be moving backward and some moving forward. That doesn't mean that their total speed is negative any more than it does for the flow on the windward side of a sail.
The vortices used in the Helmholtz theorem are one solution to the approximation of the flow phyics that result from assuming an inviscid fluid. It's one component used to describe the flowfield by superposition, in addition to a uniform flow and sources/sinks. The circulation about a wing and the Kutta condition used to determine its value are really just approximations used to model the vorticity generated in the boundary layer. By combining all three of these elements, one can put together a very realistic approximation to the flow that accounts quite well for lift and induced drag.
Presumably your argument about interrupting the starting vortex means that the entire vortex structure would have to fall apart. But that belies a misunderstanding of the starting vortex. First of all, there isn't a large, distinct starting vortex in most cases because the lift comes on gradually, generating a diffuse vortex sheet instead of the single large vortex of an idealized impulsive startup. And this diffuse sheet decays away because of the viscosity in the air. Large vortices also spawn smaller ones through instabilities in the flow. As Lewis Richardson put it,
Big whorls have little whorls,
Which feed on their velocity,
And little whorls have lesser whorls,
And so on to viscosity.
(see, for example,
http://mixing.oce.orst.edu/people/jmoum/courseinfo/3Dturbulence.pdf)
Second, under the assumption of an inviscid irrotational fluid, the flow about your proverbial seagull would be the sum of both the flow about the seagull in the absence of the vortex and the vortex itself, so the seagull would not destroy the vortex just by flying through it. The seagull would experience quite a jolt going through the core, but the lifting surface that generated the vortex would not be affected.
...Does anyone out there also have trouble believing the earth is flat or vortices are infinite.
The infinite vortices are just an approximation that simplify the flow physics so to be able to make decent engineering calculations with a reasonable effort. Just like most of us function quite well using the flat earth approximation for our maps and just about everything we do in everyday life. It's only if we have to concern ourselves with more extreme situations like orbital mechanics or the difference between the rhumb line and great circle route of a trans-oceanic crossing that the flat earth assumption proves inadequate and we have to assume a round earth. Which is itself an approximation that doesn't give good enough results in even more demanding situations.
