Calculate Lift from a full-blown-circulation airfoil?

[leviterande]

Hello everyone!

I am interested in roughly calculating the lift from an airfoil that has air circulating around its full circumference, (can be done by a treadmill skin but probably best done in practice by blowing air from slots around the entire circumference as seen in Fig.1)

As we know, 2D lift from rotating cylinders or ordinary airplane airfoils in a free-stream airspeed can be roughly calculated using the Kutto-Joukowski Theorem circulation equation where the force per unit span is:

L = p v r

Where p is the air density, the free-stream velocity and r is the total amount of air vorticity around the object.

A rotating cylinder:
The r value here is simply the vortex strength which is the rotational speed times the circumference of the cylinder.

A fixed wing inclined/cambered airfoil
The r value or circulation value can be computed from the different air speeds resulting on top and bottom sides of the airfoil.

What about a circulation airfoil in Fig.1?
How do I obtain the circulation value r
My question is, how would you go about in approximately calculating the lift from the proposed air-circulating airfoil in FIG.1, where the air is circulating around the ENTIRE airfoil by for example blowing slots? That is, if I know the blown net average rotating air speed around the airfoil circumference and if I also know the free-stream velocity, how do I go about calculating the lift here?

My attempt:
I Do I just go about calculating just as if it was an ordinary fixed airfoil, and for the r value I just use the resulting different speeds on top and bottom of the circulating airfoil(as the p airfoil is going through the stream)? But, I am not sure really.

Thank you for your time and help
Regards
Levi

(NOTE: I am aware of the current state of art trailing edge Circulation Control WIng blowing at the round training edge. I just can't see why they don't make a full blowing : upper and lower side + trailing edge,+ leading edge blowing instead of only trailing edge blowing. But that's perhaps a different subject.)

 

[leviterande]

I am reading as usual as much as I can to learn more about this specific subject. I have found lots of papers but not specifically related to the problem.
However, recently among my research I have discovered,-- if I am not wrong-- that upper + lower blowing wouldn´t affect lift, in other words, entire full circulation seems unnecessary. This conclusion may settle the issue I have with why all information about CCW I find is about EXCLUSIVELY purely trailing edge circulation. It seems that all that is necessary to create lift is to deflect airflow and for this it is only the trailing edge modification that is necessary. This trailing edge treatment calls for either jet blowing or mechanically rotating a trailing edge cylinder/ biconvex flap

Almost solved
This required trailing edge configuration to enhance lift is thus as I understand this far and as is used in the state of the art technology, is seen in fig.2 where either a trailing edge jet is blowing tangentially or a mechanically rotating cylinder is used to deflect the air(yellow circle is rotating cylinder/jet plenum).Perhaps a little different subject but still related
I am still wondering however; for the same u/v (tip speed/velocity ratio )what the difference is in generating lift between rotating a smooth cylinder and a "vaned cross fan". IMHO the latter should arguable produce much larger coefficients of lift for the same u/v when all other things being equal ha?, yet the few sources I found indicate that its the u/v ratio that decide the CL regardless if its a jet flow, smooth cylidner or a flap, which I still can't understand exactly how. I drew this conclusion from the diagram below: Cl of rotating Cylinder vs rotating flap

You can clearly see in the diagram that Coefficient of lift is mainly depending on u/v. In fact , the cylinder+endplates achieves higher Cl at the same u/v which seems odd!. But considering the fact that this endplated cylinder has an AR of 12 could be the major role in it having the bigger Cl compared to the rotating biconvex flap of AR 5. So, if the rotating biconvex flap would have an AR of 12, then both the cylinder and rotating flap would have comparable CLs.
I just find that interesting as I originally thought the opposite, that is that a vaned or biconvex flap when rotating would achieve a far higher Cl for the same u/v compared to a smooth cylinder, my reasoning was, you know, that the crossfan/biconvex flap/wing when rotating spanwise would ofcourse move a whole lot more airmass with it, but that doesn't seem the case at all. Either that or I am lost in the abyss

 

[rcgldr]

It seems that all that is necessary to create lift is to deflect airflow and for this it is only the trailing edge modification that is necessary.

Lift can be created with a blunt trailing edge, the drag will be higher, but such an airfoil will still generate lift as seen in this old photo of a lifting body prototype M2-F2 glider (in this case the blunt trailing edge is there because that is where the rocket engine nozzles were located in the later rocket powered M2-F3).

A sharp trailing edge will reduce drag.

For most airfoils, at the trailing edge, there's a vortice with upwash just behind the trailing edge where the higher pressure stream from below a wing merges with the lower pressure stream from above, where the combined streams result in a net downwash (assuming the wing is generating lift). I"m not sure of the cost versus benefits of trying to reduce the trailing edge vortice.

Correction, the flow at the trailing edge is downwards relative to the original flow, but the angle of the downwards deflected flow may be less than the downwards angle of the chord line of the trailing edge, and this could result in a squirrel caget type cylinder spinning the wrong way.

 

[leviterande]

Yep,a blunt trailing edge alone w/o blowing allows more lift at cost of drag. I was headed towards the various options to get that even extra lift with the various active circulation methods. I am currently trying to finding out information about lift calculations of rotating centrifugal fan cylinder/crossflowfan/spanwise rotating flap instead of plain smooth cylinder. I was thinking that the spanwise rotating wing/flap/crossflow fan would produce much more lift at lower rpm and lower U/V. So far however that doesn't seem to be the case as shown in the earlier diagram1 from my previous post. In diagram-9 here in this pdf http://www.icrepq.com/icrepq'14/354.14-Massaguer.pdf you can also get the same conclusion.
One gets the picture of (from diagram1 in my earlier post and diagram 9 in the pdf here) that a much fewer bladed fan produces lower rpm and correspondingly lower lift, and thus the performance would be exactly equal if not worse overall than its smooth cylinder sister. However I am not absolutely sure about all this and would love to have confirmation. Best would to have a definitve lift equation for say a spanwise rotating airfoil immersed in a stream. It is not the same equation as the cylinder. The circulation in the equation is different I believe.

Thanks

 

[rcgldr]

WIndmills operate with a free and external energy source, the wind, so efficiency of power output versus power extracted from the wind isn't that important. I'm wondering how the article's magnus effect windmills handle high wind speeds without damage.

I'm not sure that a squirrel cage type cylinder placed at the tail of an airfoil would spin in the "correct' direction, or if it would spin at all, since the flows near the trailing edge of an efficient airfoil generally have the same speed and pressure. The angle of the downwash at the trailing edge could be less than the angle of the chord line of the trailing edge. (This might be a side effect of watching a smoke wind tunnel where vertical movement of air is restricted). It also seems that such a cylinder would be relatively thick compared to a sharp trailing edge, increasing drag more than any relative lift gained by the cylinder.

 

[leviterande]

WIndmills operate with a free and external energy source, the wind, so efficiency of power output versus power extracted from the wind isn't that important. I'm wondering how the article's magnus effect windmills handle high wind speeds without damage.

I'm not sure that a squirrel cage type cylinder placed at the tail of an airfoil would spin in the "correct' direction, taking into account that there's typically upwash at the tail. If the squirrel cage is force to spin in the "correct" direction by drawing power from the aircrafts engine, it's probably more efficient to use a conventional airfoil.

With all due respect I am not sure how much you are aware about the vast and various research and results concerning CCW, active Upper surface blowing and-or trailing edge blowing/circulating, basically active circulation. (Pardon me if in any case got you wrong). CCW research indicates actually real conservative 40% more efficiency. My starting point of asking here was a specific question why a different kind/more complete of circulation is never or very rarely ever mentioned or investigated among all the papers I came across:). I realized very recently that perhaps it would be better to actually come in contact with one of the specializing authors of the papers themselves as they are the only ones involved in this research area. From my reading I however think I got a better idea of an answer to my question as I said in earlier posts.
Thanks

 

[rcgldr]

My starting point of asking here was a specific question why a different kind/more complete of circulation is never or very rarely ever mentioned or investigated among all the papers I came across.

It's not clear to me what you mean by circulation based on the diagram above that shows forward flow under a wing.

As to circulation control wings, they would only be useful for slow speed flight like takes off and landings, and the main issue is the power source for the high energy air is the aircrafts engine, reducing engine power, which is an issue during takeoff. Wiki article:

http://en.wikipedia.org/wiki/Circulation_control_wing

 

[boneh3ad]

WIndmills operate with a free and external energy source, the wind, so efficiency of power output versus power extracted from the wind isn't that important.

It certainly is important. Higher efficiency means fewer windmills and less maintenance. If wind energy is going to be cheaper than, say, fossil fuels, then efficiency is important.

I'm not sure that a squirrel cage type cylinder placed at the tail of an airfoil would spin in the "correct' direction, taking into account that there's typically upwash at the tail.

There is absolutely not upwash at the trailing edge of an airfoil except in the case of some supersonic geometries. If there was upwash, that would signify negative lift, which is obviously not desirable. In the supersonic case, there is initially upwash at certain Mach numbers due to the wave structure near the trailing edge, but those waves continue to interact downstream and the net result is still downwash.

It's not clear to me what you mean by circulation.

Sure it is. In fluid dynamics and things like the Kutta-Joukowski theorem, circulation is pretty intuitively a measure of the net rotation of a flow field. If you draw some closed curve around your region of interest (such as your airfoil), the circulation, ##\Gamma##, is defined using a line integral as
[tex]\Gamma = \oint_C \vec{v}\cdot d\vec{\ell},[/tex]
where ##C## is your closed curve, ##\vec{v}## is the velocity vector, and ##d\vec{\ell}## is the differential length along the contour and points in the direction tangent to the contour (in the counter-clockwise sense). A positive circulation indicates a counter-clockwise rotation and a negative circulation indicates the opposite (in a right-handed coordinate system).

Using the wing as a frame of reference, there's upwash at the front, a net downwash sufficiently behind the trailing edge, and the air above flows backwards across a wing faster than the air below a wing. The air below a wing never travels forwards, so it's not true circulation in that since.

But according to the definition of circulation there absolutely is a circulation there, as evidenced by the fact that you just cited about the air moving over the airfoil being faster than under the airfoil. This exactly the effect exploited by the Kutta-Joukowski theorem and is one way to calculate lift.

From the air's frame of reference, there's still upwash at the leading edge, a forwards and downwards wake left behind the wing, and part of the air above a wing is momentarily displaced backwars, while the air below is displaced forwards. The net combined flows from a wing are downwards (corresponding to lift), and a bit forwards (corresponding to drag).

Since these sorts of calculation are generally done using potential flow theory, changing the frame here doesn't matter, as the only difference is subtracting off the free-stream velocity. Potential flow is still valid for getting accurate lift calculations on an airfoil assuming you use the displacement thickness to account for the effect of the boundary layer (though it still neglects viscous drag).

So, if you split the velocity into its free-stream component and the perturbation component caused by the airfoil, ##\vec{v} = \vec{V} + \vec{v}^{\prime}##, and use that in the circulation equation, you can separate it into two integrals (integration and dot products are both linear).

[tex]\Gamma = \oint_C \vec{v}\cdot d\vec{\ell} = \oint_C \left( \vec{V} + \vec{v}^{\prime} \right)\cdot d\vec{\ell} = \oint_C \vec{V}\cdot d\vec{\ell} + \oint_C\vec{v}^{\prime}\cdot d\vec{\ell}.[/tex]

Since the line integral on a closed curve of a vector field with constant magnitude and direction such as ##\vec{V}## is identically zero, you end up with
[tex]\Gamma = \oint_C\vec{v}\cdot d\vec{\ell} = \oint_C\vec{v}^{\prime}\cdot d\vec{\ell}.[/tex]

So in cases like this, frame of reference doesn't matter.

 

[rcgldr]

There is absolutely not upwash at the trailing edge of an airfoil except in the case of some supersonic geometries.

I corrected (struck out) my previous post. Some of the wind tunnel videos I've seen show some upwash relative to the trailing edge chord angle, but not relative to the original flow, so the trailing edge flow is downwards, but in some cases at less downwards angle than the trailing edge chordline, and this is the situation where it would seem that a squirrel cage type cylinder placed at the trailing edge would spin in the "wrong" direction. This might be a side effect of watching a smoke wind tunnel where vertical movement of air is restricted.