Aircraft wings - Kelvins Circulation Theorem and the conservation of vorticity

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This is what I understand about Kelvin's Circulation Theorem

1)for inviscid (where the viscous forces are much LESS than inertial forces) AND uniform density flow, the circulation is conserved.

2)This implies (by some arduos vector calculus manipulations) that the vorticity of each fluid blob is conserved following the motion of each blob (ie 0 in this)...

NOW, this is what my notes say:

"When an aircraft wing starts to move, it sheds a vortex. But the total vorticity in the region must remain zero, by Kelvin's Circulation Theorem, so an opposite vortex is generated round the wing (this provides the swirl required to generate lift)."

So according to my notes, there are two opposite vortexes (whose vorticity sum equals zero) SEPARATED in space. So infact, vorticity is not locally conserved which is at odds with (2)...

Can anyone explain what I am not understanding?

Many thanks:)

 

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Fao Mods

MODS can you please MOVE this thread to the

Mechanical & Aerospace Engineering

section?

 

[rcgldr]

I don't get any of this:

From Wiki:

In fluid mechanics, Kelvin's Circulation Theorem states "In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant with time"[1]. The theorem was developed by William Thomson, 1st Baron Kelvin. Stated mathematically:

http://en.wikipedia.org/wiki/Kelvin's_circulation_theorem

"When an aircraft wing starts to move, it sheds a vortex. But the total vorticity in the region must remain zero, by Kelvin's Circulation Theorem, so an opposite vortex is generated round the wing (this provides the swirl required to generate lift)."

A wing peforms work on the air, increasing the total energy of the air. A wing passing through the air is in an virtually open system. This would seem to violate the "conservative body forces" and "closed curved" aspect of the Kevlin theroem. "Circulation" around a wing deceases as air speed increases. As air speeds approach mach 1, for example a 747 flying at mach .85 to mach .90, there is no significant forward (relative to a wing) flow of air anywhere on a wing. The axis of wing tip vortices is close to parallel to the direction of travel. The axis of turbulent vortices above and below a wing are close to perpendicular to the direcion of travel. The low pressure area above a wing will divert some air from below a wing, but I personally doubt that there's any significant forward movment (realtive to a wing) of air under a wing all the way from the trailing edge of a wing to the leading edge of a wing and back over the top.

 

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I don't get any of this:
A wing peforms work on the air, increasing the total energy of the air. A wing passing through the air is in an virtually open system. This would seem to violate the "conservative body forces" and "closed curved" aspect of the Kevlin theroem. "Circulation" around a wing deceases as air speed increases. As air speeds approach mach 1, for example a 747 flying at mach .85 to mach .90, there is no significant forward (relative to a wing) flow of air anywhere on a wing. The axis of wing tip vortices is close to parallel to the direction of travel. The axis of turbulent vortices above and below a wing are close to perpendicular to the direcion of travel. The low pressure area above a wing will divert some air from below a wing, but I personally doubt that there's any significant forward movment (realtive to a wing) of air under a wing all the way from the trailing edge of a wing to the leading edge of a wing and back over the top.

thanks for the reply jeff. Yes it would seem to violate Kelvin's theorem...

You've explained things well but can you point to any graphical or pictorial representation of what you're saying, which shows the streamlines and vortices of the air flow around the wing?

 

[rcgldr]

You've explained things well but can you point to any graphical or pictorial representation of what you're saying, which shows the streamlines and vortices of the air flow around the wing?

A summary of posts I've made in previous threads:

After visiting a large number of web sites, my conclusion is that lift is a combination of Coanda effect, "void effect" and simple deflection.. Coanda effect explains how laminar flow follows a convex suface. "Void effect" explains how turbulent flow follows a convex surface. Concave surfaces simply deflect airflow. The curvature of air flow accelerates the air and generates lift. "Void effect" explains how drag is developed "behind" a wing, while direct forward deflection of air accounts for the drag in front of a wing, along with friction along the surface of a wing.

For most wings, the flow transitions from laminar to turbulent flow above and below a wing, detaching during the transition, but reattaching after the transition. This happens in the first 30% of the chord length or sooner on a "normal" airfoil, and between the first 30% to 70% of chord lengh for a "laminar" airfoil (by definition). In some cases, rough surfaces and/or turbalators are used to cause the transition to occur at a specific position on an air foil. In the case of gliders, an "oil flow test" is done to visualise this transition. A bead of oil is placed on the leading edge of the wings, the glider is flown for a while at a fixed speed, then landed and the oil pattern observed. It's common practice to do this in glider magazine reviews.

http://www.standardcirrus.org/Turbulators.html

Oil flow testing is also done in wind tunnels:

http://www.hisacproject.com/news.html

A link to John Dreese's web site, page 4 and 5 discuss how little air flow is laminar over many wings, and how "laminar" air foils increase laminar flow to 30% or more over the chord length of a wing. In the case of gliders, laminar "bubbles" result in either more drag or less lift so the laminar air flow is deliberately broken up sooner than it normally would via rougher surfaces or turbulators (this is mentioned in the article). The laminar section starts mid way down page 4:

http://www.dreesecode.com/primer/airfoil4.html

"All airfoils must have adverse pressure gradients on their aft end. The usual definition of a laminar flow airfoil is that the favorable pressure gradient ends somewhere between 30% and 75% of chord."

http://www.aviation-history.com/theory/lam-flow.htm

The next website does a descent job of explaining lift, but with a bit too much emphasis on Coanda effect, ignoring void effect and turbulent flow, but towards the end of this web page, there's a diagram of a wind blowing over a roof, and although the air downwind of the roof is turbulent, it's also at lower pressure, due to void effect. Since both laminar and tubulent air flows contribute to lift, both cases should have been covered better than it was at this web page:

"The physical cause of low or high pressure is the forced normal (perpendicular) acceleration of streaming air caused by obstacles or curved planes in combination with the Coanda-effect.":

http://user.uni-frankfurt.de/~weltner/Mis6/mis6.html

videos

Assuming this next video isn't GGI, it appears to be a series of pictures of a flame aimed at various angles over an glowing (from the heat) airfoil at a fixed angle of about 45 degrees. As the flame angle is made more horizontal, the effective angle of attack becomes higher. What I call "void" effect is more evident here, as the flame flow is detaches from the aft end of the airfoil at low effective angle of attack. At higher effective angle of attack, the flame flow detaches from the "upper" surface of the airfoil, but it's still accelerated (curved) "downwards", while below the airfoil there is significant direct deflection. About 28 seconds into this video (you can hold it at this position), the downwards curvature of the flame over the "top" of the wing is still evident, in spite of the large amount of apparent detachment.

http://www.youtube.com/watch?v=hkJaTTIiXSc&fmt=18

Next is a link to a narrow wind tunnel video, considered a "2d" airflow (equivalent to a 3d wing with infinite wingspan). Air speed is slow, chord length is small, so the Reynolds number is quite low, and the air flow is much more laminar and the angle of attack before stall is much higher than it would be if everything were scaled up to a faster speed and a larger size. The transition into the stalled condition is very abrupt. In the segment annotated as "stall", there's virtually no lift, but near the end of the video, that starts off "flow attached", then "stall", there's still significant lift although there is a stall.

http://www.youtube.com/watch?v=6UlsArvbTeo&fmt=18

For this model, the stalling angle of attack is fairly small:
http://www.youtube.com/watch?v=5wIq75_BzOQ&fmt=18

Another wind tunnel, slow air speed, short chord, but not as much as the first video. Again the nature of the wing tunnel (proably drawing air inwards from the right), prevents the air flow from remaining deflected, and skews what would happened in an open environment:

http://www.youtube.com/watch?v=TGUSmdFmXDg&fmt=18

I've been trying to find videos of wind tunnels that are much larger than the models being tested, which better simulate an actual open environment, such as the one I posted a picture of above with an oil flow test, but I've haven't had much luck with this search yet.

 

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wow thanks Jeff:)

very informative post... videos are interesting too.

It would seem my lecturer had been oversimplifying things somewhat!

 

[rcgldr]

Although I'm not sure how this relates to "circulation", here are a couple of pictures of a pre-shuttle lifting body prototype, which are unusual in that the top is flat and the bottom is curved (basically 1/2 of a cone, with the flat part on top, and tapered at the back). Mostly I just think these old prototypes look cool:

M2-F2 glider with F104 chase plane:
m2-f2.jpg

M2-F3 rocket powered model (reached a speed of Mach 1.6) with B52:
m2-f3.jpg