Blades slowing the air down

[TomG]

On a turbine, a blade hits the slower air propagated upwind by the previous blade, with the upwind distance equal to

Chopping_Distance = dia*pi/(blades*tsr)

My question is: how much slower will that air be? I'm guessing

disturbance ∝ lift ∝ apparent_wind_speed² *chord*Coeff_of_lift

And I'd also guess that the disturbance dies away with 1/Chopping_Distance rather than 1/Chopping_Distance², because of the length of the blade...

So

Disturbance_seen = apparent_wind_speed² *chord*Coeff_of_lift/(dia*pi/(blades*tsr))

         = apparent_wind_speed² *chord*Coeff_of_lift*blades*tsr/(dia*pi)

         = windspeed²*(1+tsr²)*tsr *chord*Coeff_of_lift*blades/(dia*pi)

Hence doubling the diameter requires doubling the chord to keep the same
Disturbance_seen level, which makes sense.

But this implies that doubling the windspeed quadruples the Disturbance_seen??

And also that halving the windspeed would allow us to quadruple the number of blades while keeping the Disturbance_seen constant...

Comments? Suggestions? I haven't seen the equations for this sort of thing around in any of the helpful documents, so I'm unsure of how these values are calculated...

[finnsawyer]

What makes you think there is any air propagated upwind by a blade?  The leading edge of a blade can be modeled by a cylinder (well, at least airplane wings have a rounded leading edge).  When the air flows around a cylinder it speeds up and causes a region of disturbance on either side that dies away with distance from the cylinder.  If you cut the cylinder in half there will be no disturbance of the air on the flat underside.  The flat or cambered side of a wind mill blade is always toward the incident wind.  This flat side will cause very little disturbance of the effective wind, so there is essentially no wake on that side to carried away by the incident wind before the next blade comes around.  There will be a wake on the curved side, but it has already had a head start due to the angle of the blade, and hopefully is gone by the time the next blade comes around.  

[TomG]

The _air_ doesn't propagate upwind, but a pressure wave does - otherwise how does the wind upstream of the turbine know to slow down?





See that blue bulge? High pressure, upwind (down the screen) from the blade.

[SparWeb]

Tom,

I think you're trying to visualize the streamlines passing through the plane of the propeller disk, but you're getting it mixed up because you lack terminology: the effect you're describing is actually called "advance ratio".

Advance ratio, efficiency, blade angle of attack, and the power coefficient are all inimitely related to each other, and are REALLY hard to describe unless you're willing to sit down for a few weeks as I stumble through it all.  I definitely don't understand it all myself, and it was one of my best subjects in school.

You're probably looking in the wrong place, if you want to discuss the fine points of aerodynamics.  Just to get the terminology right takes a fair bit of reading, preferably from textbooks on the subject, not websites.

It looks like you're good with the math, so this shouldn't be too daunting.  There are two ways to study this: new or old.  You can either go to the NREL and dig up reports and manuals on aerodynamic analysis models, or you can to to NASA and find the reports with propellor performance charts developed in the 1920's, which explain very clearly the function and aerodynamics of aircraft props.

[Ungrounded Lightning Rod]

... or you can to to NASA and find the reports with propellor performance charts developed in the 1920's, which explain very clearly the function and aerodynamics of aircraft props.

Then you can interpolate to turbines by remembering that a propeller's job is to:

 - Gain thrust by

 - Accelerating the air stream and

 - Consuming shaft horsepower

while a turbine's job is to:

 - Gain shaft horsepower by

 - Decelerating the air stream and

 - Being thrust upon (dragged).

which means the hump is on the other side.  B-)

[TomG]

"I think you're trying to visualize the streamlines passing through the plane of the propeller disk"

Are you talking about the streamlines shown at the top of this image?

"the effect you're describing is actually called "advance ratio""

Sounds about right - that's the distance "forward" (through the wind, in our case) relative to the size of the prop? Although I'm dividing it by the number of blades, because the more blades the prop has, the more finely it slices the air.

"...REALLY hard to describe..."

 Dang, I was hoping someone would pop up and point me to a webpage with a nice simple equation.

"You're probably looking in the wrong place, if you want to discuss the fine points of aerodynamics."

Yeah, I suppose. Although a couple of people here seem to be pretty technical in their approach. I wonder if SamoaPower is around...

"Just to get the terminology right takes a fair bit of reading, preferably from textbooks on the subject, not websites."

True. I might have to see if the library has any aerodynamics books. Wish I was still at uni - good libraries!

Re: studying - the NASA site looks better (particularly the older papers), but it still all looks pretty daunting! Still, at least it makes me feel better about my prototype simulation not working.

[Gordy]

TomG,

Another good place to look might be " sandia national labs - wind turbine " Paste this in your google search Plenty of reading ;-)

Gordy

[wdyasq]

I'll agree with Steven here - and some of the best manuals may be those of the teens and twenties, when they were trying to explain this new 'science' to 'old school' engineers and new young engineers.

MANY of the best manuals out there are of that age when they took kids from the farms and ranches and developed them into the folks who built the weapons that won WWII.

The UIUC site is informative and is well worth going through. Try to learn about Reynolds numbers, AOA and the various other 'minor details before you attempt to improve on the old standards.

Ron

[thefinis]

Here is a thread I keep hotlisted so I can find it and reread it every once in a while. It covers blade numbers incident wind turblence etc. called "Re-thinking blades"

http://www.fieldlines.com/story/2006/2/9/152426/8102

Finis

[finnsawyer]

The figure doesn't show the direction of air flow, so one actually doesn't know the direction of the incident air and the attack angle.  In any case, the pressure build up is slight.  You need to relate the situation to that of the wind turbine.  One could envision the area of high pressure as a wake that tends to move down wind between the blades.  Then it affects the following blade.  If some higher pressure is left on the windward side of the blade assembly, on average it would act as a whole to cause air to spill around the entire turbine.  It doesn't cause air flow up wind.  It would rob power.

   

[finnsawyer]

There's another factor here.  You are trying to use a static situation to model a dynamic one.  In the static case no work is done or power put out.  In the dynamic case or wind turbine, power is presumed generated at the "sweet spot", which affects the apparent wind and will affect the pressures around the blades, as well as how the wakes are removed.  So, you really need wind tunnel data for the dynamic case.  Which brings me to your curves.  It would be nice if you would tell everyone what they purport to represent.

[TomG]

Sorry, I should have added more details. The apparent angle of the airflow is 10° (roughly deriving from a TSR of 6), and the angle of attack of the blade is 6°, giving an airflow in the image of 4° from the chordline of the blade.

The black lines are streamlines. The white lines are iso Cp lines. They help to clearly show the high pressure zone on the flat underside of the blade, which is pointing upwind (in the free air flow).

"If some higher pressure is left on the windward side of the blade assembly, on average it would act as a whole to cause air to spill around the entire turbine.  It doesn't cause air flow up wind.  It would rob power."

Yes, exactly. As I understand it, this is what causes Betz' limit. The more lift our blades derive from the wind, the more they slow the wind, causing more air to flow around the turbine instead of through it.

[SamoaPower]

Tom is correct. It may be clearer if you think in terms of air mass velocity rather than flow. The pressure front upwind of the rotor disk slows the velocity at the rotor to less than the free-stream velocity.

In the ideal rotor (Betz limit), the velocity far downwind should be one-third the far upwind (free-stream) velocity. The velocity at the rotor should be the average of the free-stream velocity and the far downwind velocity, or two-thirds of the free-stream velocity. Hence, the slowing of the upwind velocity should be one-third of the free-stream velocity for maximum power extraction.

I don't believe I've seen Lift to Drag ratio (L/D) mentioned in this discussion. This is the single, most significant factor effecting rotor efficiency and I find it astounding that it is generally ignored by the DIY blade carvers. Arbitrary airfoils seem to be the order of the day.

I really don't understand why using a published, documented airfoil section seems to be considered too difficult to do. An improvement in rotor Cp from 0.3 to 0.4 represents a rotor power output improvement of 33%. Why don't more people want this?

[TomG]

GeoM - "There's another factor here.  You are trying to use a static situation to model a dynamic one."

Am I? In the image above, I was modelling TSR=6, hence the blade moving at about 6 times the windspeed. How is this a static situation?

GeoM - "So, you really need wind tunnel data for the dynamic case.  Which brings me to your curves.  It would be nice if you would tell everyone what they purport to represent."

The image above is from the JavaFoil simulation, which is about as close to a windtunnel as I can afford to get. It's supposed to be fairly accurate, though.

What curves are you talking about? I'm a bit confused there...

SamoaPower - "I don't believe I've seen Lift to Drag ratio (L/D) mentioned in this discussion."

Ah! Maths! OK, my program does take account of drag, but when I see Lift/Drag rations quoted, is the drag the Parasitic drag, the drag due to lift-in-the-wrong-direction (lift counteracting the required torque on the blade) or both combined?

Also, a very newbie question which I've been unable to find the answer for (I need an "aerodynamics for dummies" book): I've been assuming that drag (Cd) acts in the direction of the incoming airflow, but which direction does the lift (Cl) act in? At right angles to the incoming airflow, or at right angles to the chord line, or some other direction?

SamoaPower: given that the "Ideal rotor" is just a model and doesn't actually _have_ a TSR, why does its theoretical maximum efficiency drop to zero at zero TSR?

[SamoaPower]

I haven't been associated with wind tunnel measurements so am guessing somewhat to answer your question. I believe that Cd represents total drag in the airflow direction. Of course, in wind tunnel measurements, the airfoil is held stationary and pressure measurements are taken at various points along it's surfaces. The coefficients are then calculated from these. Measurements are not taken from a rotor. I also believe that Cl is perpendicular to the chord line.

I think the answer to your TSR question is obvious, so I probably don't understand the question.

TSR = Tip Speed/Windspeed. For TSR = 0, Tip Speed = 0 (non-rotation). No rotation, Cp = 0.

I haven't seen the derivation of the ideal model but I do accept it. I think it must be based on the basic physics and doesn't include a physical rotor.

[TomG]

Regarding Cl and Cd: that makes sense. Thanks.

"I haven't seen the derivation of the ideal model but I do accept it. I think it must be based on the basic physics and doesn't include a physical rotor."

I have seen it (and also accept it, for the record - good maths), and indeed it makes no mention of an actual rotor, just a disk capable of maintaining a pressure gradient. TSR doesn't appear in it at all. Which is what makes me think there should be _some_ design capable of approaching Betz for low TSR. Nothing in his derivation says otherwise (since it doesn't mention TSR at all). All of which leaves me wondering what model was used for the "Ideal Rotor"...

[TomG]

ULR - "Then you can interpolate to turbines by remembering that a propeller's job is to..."

Yeah, it's that translating that is proving a problem. Props are designed to take torque of the shaft and turn it into lift pointing forward along the shaft, ie in the direction of flight. Rotors need to develop lift around the shaft to produce torque. Any thrust along the shaft axis is just pushing the tower over. We have to have a bit of that to stop the whole arrangement flying away, but it's not the point of the exercise. So blade design, although very informative, doesn't translate directly

[TomW]

Tom;

I have had a "gut" feeling about this for a long time. Used to argue it with Bobn [R.I.P.] over on IRC. Just seemed wrong that the difference would not be factored into the translation. Math and aerodynamics not my strong areas but I always thought something was missing. Your comment tends to explain and support that there is a, perhaps, not so subtle difference.

Cheers.

TomW

[TomG]

I think I've worked it out! I found this very helpful page:

Froude's Momentum Theory for an Actuator Disk


which derives the normal, actuator-disk model (which makes no reference to rotation or TSR). Then, at the bottom, it goes into the advanced model, which considers rotation in the flow. And, although the maths is scary, I _think_ I gathered that as the TSR drops, the turbine (regardless of its detailed design) needs to "spin" the air more to extract the energy from it. This leaves the wake behind the turbine spinning, which is an energy "leak".

So, in two words: rotational losses.  Odd that a faster-spinning rotor leaves less rotation in its wake, but that seems to be the case...

All of which suggests that a twin-rotor, contra-rotating design might be effective at very low TSR? Has _anyone_ ever bothered to build one of those beasts? I know 'planes have been built with contraprops, but I've never seen a wind turbine like that.

[SamoaPower]

Tom,

Yes, I think you've got it. Thank you very much for that link. I had written the velocity relationships in my notes some thirty years ago but had neglected to include the derivation. Your reference ties it all back together for me. I think equations 28/29 say it.

I was aware of the rotational losses (swirl effect) but had not really considered the magnitude at low TSR since such low RPM isn't very useful for electrical generation. I guess it's possible to reduce rotational losses but I'm not sure it's worth the effort for the same reasons.

Sorry for the rant earlier about L/D. I guess it falls in the same catagory as why so many continue to use cast-in-resin stators when it's contra-indicated for larger machines. Just some of my pet peeves.

Keep up the good work.

[finnsawyer]

I'm glad you decided to show the air streams.  That is more useful than the pressure per se.  If you rotate the blade counter-clockwise by ten degrees, you more closely show the moving blade.  Note that the air on the windward side moves smoothly without any wake along the blade in spite of the slight region of pressure build up.  So, it is possible that each blade is always operating in undisturbed air.  It's too bad we do not have actual velocity values.

As far Betz' Limit is concerned my understanding is that it has to do with the simple fact that the air leaving the turbine must be removed.  Any air going past the turbine due to pressure build up would be sped up, resulting in a drop in pressure behind the turbine like in the solid disk case.  This would actually help to move on the air behind the disk.  More farther down.  

[finnsawyer]

Were you also having the blade do work as part of your modeling?  Oh well, the result is only as good as the model.  I thought these were wind tunnel results, which is where the action is.  I suppose in the wind tunnel one could put a weight on the blade and allow it to rise.  That's kind of what a wind mill blade is doing.

The curve I was wondering about was the one that showed a sharp rise at the front of the turbine and a sharp drop at the back of it.  This looked like the pressure distribution one would expect at the center of a solid disk, but the label looked like a D.  It's a good idea to always tell what a curve shows.  Still more farther down.

[finnsawyer]

I took a look at the link and my reaction is: Bull Sh**.  Well, maybe that's a bit strong.  I have no prejudice against theoretical analysis, but one should be skeptical about its application.

Let's take a look at the assumptions as they might apply to a solid disk or a wind turbine:

  1. 1-Dimensional analysis and disk is essentially a discontinuity moving    through the fluid

  2. Infinitesimally thin disk of area A which offers no resistance to fluid passing through it as frictional forces are negligible compared with momentum flux and pressure changes (hence can make assumption 5)

  3. Thrust loading and velocity is uniform over disk

  4. Far-field is at free-stream pressure but far up and downstream

  5. Inviscid (thus irrotational), incompressible and isentropic flow

One dimensional analysis?  Forget that.  That's wrong to start with since the air flow is deflected and flows around the disk.  The turbine as a whole isn't moving.

Only the blades are.  Two is obviously wrong.  Three is wrong because one is wrong.  Four may be nearly correct for a solid disk but is manifestly wrong for a turbine producing power.  Five may be a useful set of constraints.

The question of the turbine spinning the air behind it came up before in a discussion with Anthony.  I don't believe it happens, but it is something that could be tested in a wind tunnel using smoke in the air stream.  So could this idea of a pressure build up in front of the turbine.

 

[finnsawyer]

Some people espouse maximizing the ratio of the lift coefficient to the drag coefficient.  While this might seem the way to go, it has the effect of setting the value of the attack angle, which might not prove to be ideal for an actual operating turbine, which has to deal with a wide range of wind conditions.  A similar issue will exist for the ratio L/D.  So, the question becomes, "What values of L/D work well for an actual operating wind turbine?"  The situation is made more complicated by the fact that people really give up trying to match the blade characteristics to the apparent wind toward the root.  Since you are interested in this issue, perhaps you should write a computer program for some of the common air foil profiles to see what attack angles would give the best ratios of L/D over the entire blade length for say a TSR of 7.  The community could then give feed back as to whether they think they would result in buildable turbines.

[TomG]

I'm not going to go into those points one by one. Suffice it to say that those assumptions are what the VERY SIMPLISTIC model was based on. My whole point was that the very simple model doesn't make sense, so I agree with you on that.

Hence the need for the more complex analysis, which takes into account many more effects, including rotation.

"The question of the turbine spinning the air behind it came up before in a discussion with Anthony.  I don't believe it happens, but it is something that could be tested in a wind tunnel using smoke in the air stream.  So could this idea of a pressure build up in front of the turbine."

They both have been. See:

http://www.risoe.dk/vea-aed/numwind/results.htm - especially the first and second images.

http://www.aero.gla.ac.uk/Research/LowSpeedAero/images/lsawake.gif

And particularly clearly in

http://www.fluent.com/about/news/newsletters/02v11i1/img/a1i6_lg.gif

where the upstream flow lines (left) are straight and parallel, and the downstream lines (right) spiral around each other.

http://allentech.net/about/images/windtunnel-flow-280w.gif

Similar processes for an aircraft rotor:

http://rotorcraft.arc.nasa.gov/cfd/CFD4/New_Page/Pics/v22_rwake.gif

Small image, but you can _see_ the trailing vortices:

http://www.sbe.hw.ac.uk/research/flic/pics/fl01a.jpg

Anyway, if all that doesn't convince you, you're going to have to stand upwind of a turbine with a smoke-stick...

I'm not sure how to convince you that there's a high-pressure zone upwind of a turbine. Too tired, late at night. But if there isn't, then what causes 1/3rd of the wind to go around the turbine instead of through it?

[TomG]

Yes, the blade was doing work in the model. Much as I'd love a wind tunnel to play with, I have neither the funds nor the space! None of these images come from my research, by the way, they're all from aerodynamics labs and such, garnered through web-research while trying to understand...

"The curve I was wondering about was the one that showed a sharp rise at the front of the turbine and a sharp drop at the back of it.  This looked like the pressure distribution one would expect at the center of a solid disk, but the label looked like a D."

Oh, that one! That's a p, as in pressure. It's also the distribution for a disk which is partially-permeable to airflow - it's just a bit smaller, depending upon the permeability. (The dotted horizontal line is p infinity, ie the free-flow pressure.

[TomG]

Hmm, that's a pretty interesting idea, actually. So, a graph of L/D on the vertical axis, against radius (root-to-tip) on the horizontal. Have the AOA vary from root to tip in some fashion (I think there's a cotan function for this). But the chord normally varies as well, so I guess I'd have to make that variable too...

Either I'd end up posting a LOT of graphs on here, or I'd be better off posting the program and letting prople play with it themselves.

Would anyone be the slightest bit interested, or would I be wasting my time?

[TomG]

"It's too bad we do not have actual velocity values."

I think I was using 10m/s for the windspeed. Although interestingly, the axial distance between blades in the airflow doesn't vary with windspeed (provided the TSR remains constant). The same "distance" of air has always moved past by the time the next blade gets there.

"Any air going past the turbine due to pressure build up would be sped up, resulting in a drop in pressure behind the turbine like in the solid disk case."

And indeed, we _do_ see a drop in pressure behind the turbine.

[SamoaPower]

GeoM,

I'll tackle this one since I brought it up. I'll let Tom respond to your difficulty in linking the value of theoretical work (necessary for understanding) to the real world.

"Some people espouse maximizing the ratio of the lift coefficient to the drag coefficient."

and

"A similar issue will exist for the ratio L/D."

What?

The second IS the first.

"... it has the effect of setting the value of the attack angle, which might not prove to be ideal for an actual operating turbine, which has to deal with a wide range of wind conditions."

Of course it establishes the correct Angle of Attack (AOA) (typically, 3-8 degrees). The polars derived from wind tunnel tests (hopefully done at low Reynolds numbers applicable to wind turbines) of a given airfoil, relate Cd to Cl to AOA which is what is need to properly design a blade.

There is an additional correction factor needed to account for the blade aspect ratio but you need the AOA to calculate it.

What constitutes a less than ideal AOA for an actual turbine and what are the wind conditions you are concerned about?

"What values of L/D work well for an actual operating wind turbine?"

Refer to the chart in an above comment. It depends on what value of Cp you are willing to settle for. If the average DIY rotor has a Cp of 0.3 and if there are practical ways to improve that to say 0.45, wouldn't you want to do it? An L/D of 100 is good to shoot for.

Higher L/D isn't the only way to improve Cp, but it is the most significant parameter and is not difficult to implement.

"The situation is made more complicated by the fact that people really give up trying to match the blade characteristics to the apparent wind toward the root."

And rightly so. The area of the blade inside the 20% radius contributes little (4%)to the rotor power output. This region should be more given over to the structural requirements of the blade.

"... perhaps you should write a computer program for some of the common air foil profiles to see what attack angles would give the best ratios of L/D over the entire blade length for say a TSR of 7."

Why reinvent the wheel? You seem to be approaching the problem backwards. The first step should be to research the published, documented airfoils suitable to wind turbines and obtain the polars for low Reynolds numbers. From these, the AOA for best L/D pops out.

I hope you don't mean by "common airfoil profiles", the arbitrary airfoil-like shapes that seem to be used by the majority of the DIY rotor makers. There is no way I'm aware of to predict the performance of an arbitrary shape with any degree of accuracy. Wind tunnel testing is required.

I think that perhaps, you may be confusing AOA with blade angle when you talk about "over the entire blade length". Blade angle and AOA are different animals. There is only one AOA for best L/D, for a given airfoil, and that is true regardless of the blade radius station.

[SamoaPower]

Careful Tom, don't confuse AOA with blade angle. See below.

[SamoaPower]

To further clarify, blade angle is that angle between the airfoil chord line and the plane of rotation of the rotor.

In a well designed blade, the blade angle will vary with the radius because the speed ratio varies with radius. This is the blade twist often referred to. The blade angle is not equal to the AOA except coincidentally. The blade angle will be both greater and less than the AOA depending on the radius and TSR.

The AOA is the angle between the chord line and the apparent wind. Best L/D is obtained at a particular AOA, for a given airfoil, and does not change with blade radius as long as the correct AOA is maintained.

If the AOA is increased to about 12 degrees, the blade will stall and the Cp will drop considerably.

This confusion of angles happens often judging by what I read here.

[TomG]

Ah, very good. I've been calling blade angle "angle of attack" and AOA "apparent angle of attack". Thanks for straightening out terminology.

[finnsawyer]

I think we need to get back to fundamentals.  Certain force measurements were made of air profiles in  wind tunnels.  If memory serves me right the forces were measured in the direction of air flow and at right angles to it.  I don't know how that relates to your "polars", but for the airplane designer those are two of the the forces that he is interested in.  The blade designer, on the other hand, is interested in the net force on the blade in the direction of rotation.  TomG asked about the direction of lift.  In going from raw data to a model such as that involving the lift coefficient and the drag coefficient things can happen.  I don't know if that's the case, but, if the lift as calculated by the lift coefficient is at right angles to the apparent wind, then the component of lift in the direction of rotation will increase right up to saturation, even though the blade angle relative to the plane of rotation can become negative.  The fraction of lift in the direction of rotation stays constant and becomes independent of the AOA.  A similar situation exists for the drag, as the fraction of drag in the direction of rotation also would stay constant.  The net force driving the blade is the lift force minus the drag force.  So, really, when you talk about maximizing L/D, you really mean maximizing the ratio of the component of lift in the direction of rotation divided by the component of drag in the direction of rotation.  And you do this for all points along the blade.  Finally, Flxv = .59xPr - Fdxv, where Fl is the component of the lift force in the direction of rotation, v is the speed of the blade section, Pr is the power available in the air stream, and Fd is the component of the drag force, all measured at radius r.

"The area of the blade inside the 20% radius contributes little (4%)to the rotor power output. This region should be more given over to the structural requirements of the blade."

I disagree with this.  As I see it the problem begins more likely at 50% radius by failure to keep the proper twist.  Once the AOA exceeds the saturation angle, drag increases greatly.  So, not only do you not get any power from the air, but you put power in due to the drag.  One other thing.  A 33% increase in power implies a 10% increase in wind speed.  So, this whole discussion is pretty academic.