Explain this and win the prize!

[gripen]

In the "Draining E in turns" thread rose a question if the Oswald's efficiency factor (know also as e factor) can be estimated from the unknown wing by knowing only the aspect ratio of that wing. In other words, is it possible to estimate the lift distribution of the wing by knowing only the aspect ratio of it?

So I open a little contest: The first one who can explain if this is possible or not with sensible logic will receive a pdf copy of the RAE report called: "Notes on the Dog Fight" from me. As a bonus prize the winner will also get a flight tested German report on the Bf 109G/AS for performance analysis.

gripen

[Angus]

I'd say NO.
Aspect ratio is only a function of chord and span.
You'd be missing shape, wingtip shape, thickness and dihedral.

What's an E factor anyway

[Schutt]

I dont think you can.

proof:

use 2 wings,
A got a profile 1cm thik 400cm deep, square.

B one got a profile 400cm thik, 1cm deep

now with same aspec ratio these 2 wings have diffrent factors.

You say second one is no wing? well but then you define a profile for what is a wing and what not.

So that is the proof you need to know more than the aspect ratio.

[joeblogs]

Here's the problem - the efficiency of a wing depends on it's planform and crosssection.

The aspect ratio tells you something about the planform, but it is not even a sufficient statistic for that-compare a straight taper vs. an elliptical wing for example.

Applied physicists figured out most of the effects of planform on the efficiency of wings very early (say by the 1920s at least). It took many decades to work out the effects of different cross-sections and even today some of the results are approximations.

That is why NACA spent two decades conducting wind tunnel tests of airfoil cross sections and publishing these results. The principal was called parameter variation-you don't know the exact formula but you try to approximate it by gathering data generated by small changes in the airfoil.

Consider the P51 wing with and without a laminar flow cross section. The planform does not change, so the aspect ratio is the same. But, in principal at least, the laminar cross section is more efficient.

-Blogs

[frank3]

Joeblogs winns the cooky!

[Crumpp]

Here is what a guy that teaches aeronuatics told him:

------------------------------------------------------------------------------------


quote:
--------------------------------------------------------------------------------
Originally posted by gripen
This is the only one I have in the hand of those you mentioned and the graphs in the page 73 actually give correction factors of the rectangular wing against aspect ratio.

--------------------------------------------------------------------------------



Correct, Figure 2-44 (b) shows the relationship of those factors to Aspect Ratio for a rectangular wing, and that illustrates the dependency of e to Aspect Ratio. In the previous graph you see the influence that taper ratio has for an aspect ratio of 6. There are other formulae that also include the taper ratio so that values for non rectangular wings can be estimated also.


quote:
--------------------------------------------------------------------------------
 
In the previous page the writers actually say that:

"For an untwisted wing with elliptical planform shape these correction factors are zero"

--------------------------------------------------------------------------------



Correct, and that’s why the elliptical planform theoretically has the least induced drag, in that case the efficiency factor e would equal one.


quote:
--------------------------------------------------------------------------------
 
If I understand this correctly those formulas you gave above work only for the rectangular wings and we can't determine the shape of the wing from the aspect ratio only.

--------------------------------------------------------------------------------



Partly correct… It is true that formulae that only include aspect ratio only apply to a single taper ratio, it just happens that the graph in Perkins & Hage is for zero taper. The previous graph on the same page illustrates the relationship for both taper ratio and aspect ratio but even so the dependency upon aspect ratio is clear.

However, the formulae presented by Professor Wood is for a taper ratio of 0.57 which yields an almost elliptical lift distribution, which is why it fits the experimental data so closely over a wide range of examples. Most WWII aircraft had a taper ratio close to that value (Me109 was approx 0.52) because the designers knew about the benefits of elliptical lift distribution and that it could be achieved quite closely with a wing of that taper, that’s why a formulae that only includes aspect ratio could still be of such good practical use. However, other graphs and formulae that include both taper ratio and aspect ratio were common, today it is just as easy to include everything, including twist, camber, and sweep etc.

Hope that helps

Badboy
------------------------------------------------------------------------------------

Crumpp

[joeblogs]

I did not actually see the discussion of this e variable, but the wider discussion sure sounded to me like it is an application of the mathematics for lift and drag most commonly characterized in terms of the "wetted" portion of a flat plate exposed to the slip stream.

If that is an accurate characterization of the discussion, this is all about measuring the properties of the planform. The effects of cross-section (and other things) are typically subsumed in a constant. In other words, the effect of the shape of the cross section has to be explained by an additional set of equations.

-Blogs

Originally posted by Crumpp
Here is what a guy that teaches aeronuatics told him:

------------------------------------------------------------------------------------


quote:
--------------------------------------------------------------------------------
Originally posted by gripen
This is the only one I have in the hand of those you mentioned and the graphs in the page 73 actually give correction factors of the rectangular wing against aspect ratio.

--------------------------------------------------------------------------------



Correct, Figure 2-44 (b) shows the relationship of those factors to Aspect Ratio for a rectangular wing, and that illustrates the dependency of e to Aspect Ratio. In the previous graph you see the influence that taper ratio has for an aspect ratio of 6. There are other formulae that also include the taper ratio so that values for non rectangular wings can be estimated also.


quote:
--------------------------------------------------------------------------------
 
In the previous page the writers actually say that:

"For an untwisted wing with elliptical planform shape these correction factors are zero"

--------------------------------------------------------------------------------



Correct, and that’s why the elliptical planform theoretically has the least induced drag, in that case the efficiency factor e would equal one.


quote:
--------------------------------------------------------------------------------
 
If I understand this correctly those formulas you gave above work only for the rectangular wings and we can't determine the shape of the wing from the aspect ratio only.

--------------------------------------------------------------------------------



Partly correct… It is true that formulae that only include aspect ratio only apply to a single taper ratio, it just happens that the graph in Perkins & Hage is for zero taper. The previous graph on the same page illustrates the relationship for both taper ratio and aspect ratio but even so the dependency upon aspect ratio is clear.

However, the formulae presented by Professor Wood is for a taper ratio of 0.57 which yields an almost elliptical lift distribution, which is why it fits the experimental data so closely over a wide range of examples. Most WWII aircraft had a taper ratio close to that value (Me109 was approx 0.52) because the designers knew about the benefits of elliptical lift distribution and that it could be achieved quite closely with a wing of that taper, that’s why a formulae that only includes aspect ratio could still be of such good practical use. However, other graphs and formulae that include both taper ratio and aspect ratio were common, today it is just as easy to include everything, including twist, camber, and sweep etc.

Hope that helps

Badboy
------------------------------------------------------------------------------------

Crumpp

[Crumpp]

I did not actually see the discussion of this e variable, but the wider discussion sure sounded to me like it is an application of the mathematics for lift and drag most commonly characterized in terms of the "wetted" portion of a flat plate exposed to the slip stream.

You are correct.  Badboy simply points out that Oswalds Efficiency factor is a much more accurate way of determining the efficiency of a wingtip than just estimating it.  

Gripen contended that any formula (Oswalds) that includes AR has nothing to do with the wingtip efficiency as he was estimating the value.

The discussion came up after this article was presented:

http://www.thetongsweb.net/AH/EAAjanuary1999.pdf

Using Oswalds Formula, the FW-190 and the Spitfire are very close in their wingtip efficiency with the Spitfire having a slightly better value as is confirmed by this article.  It is not nearly as large as many of the estimates.  Reason being the physical shape of the Spitfire's wingtip is not quite the same as the wet area lifting surface.  In Short, the more detailed your calculations the more Lednicers conclusions are confirmed.  

This leads to the FW-190 having less drag than the Spitfire in many portions of the flight envelope.  The Spitfire gains advantage in low speed manuvering and pulling hard turns.

In level flight the faster you go the more the 190 moves to advantage, in the zoom climb, and in turning below around 4 G's and high speed.

You can see the whole the thread here:

http://www.hitechcreations.com/forums/showthread.php?s=&threadid=128143&perpage=50&pagenumber=5

Great discussion with lots of good information in it.

Crumpp

[bozon]

aspect ratio is not the whole story. consider a triangular wing, swept wing and rectangular wing.

The "e" factor is where all the errors of the models are burried. It is a fuzziness factor that can be estimated in the design stage but better be tested when the wing is actually built.

Bozon

[Crumpp]

Definately agree Bozon.

Crumpp

[joeblogs]

An interesting application of the e factor is found at

http://www.aerospaceweb.org/question/aerodynamics/q0184.shtml

I was wrong - Oswald's e takes into account efficiency losses that result from departures from a perfect elliptical planform.

The equation for total drag contains two unknowns. The first is a proportionality constant and the second is a "minimum 3d drag coefficient." What's interesting about the latter is that the author resorts to numbers derived from actual wings. In other words, he needs data (or an equation) that is outside of the lifting line equation he is working with.

-Blogs

Originally posted by joeblogs
Here's the problem - the efficiency of a wing depends on it's planform and crosssection.

The aspect ratio tells you something about the planform, but it is not even a sufficient statistic for that-compare a straight taper vs. an elliptical wing for example.

Applied physicists figured out most of the effects of planform on the efficiency of wings very early (say by the 1920s at least). It took many decades to work out the effects of different cross-sections and even today some of the results are approximations.

That is why NACA spent two decades conducting wind tunnel tests of airfoil cross sections and publishing these results. The principal was called parameter variation-you don't know the exact formula but you try to approximate it by gathering data generated by small changes in the airfoil.

Consider the P51 wing with and without a laminar flow cross section. The planform does not change, so the aspect ratio is the same. But, in principal at least, the laminar cross section is more efficient.

-Blogs

[justin_g]

If you read Oswalds paper where he introduces the "e" factor, he was deriving it from flight tests and it is supposed to consider the efficiency of the whole aircraft, not just the wing. Link: http://naca.larc.nasa.gov/reports/1933/naca-report-408/index.cgi?thumbnail1#start

[joeblogs]

That's true.

The efficiency factor in that article appears in a formula that tries to correct for calculations of parasitic drag that assume an elliptical planform. When the actual wing does not take that shape, an adjustment is made, which essentially reallocates some drag from parasitic to induced drag. The efficiency factor is introduced in that calculation.

Getting back to Grippen's point, though, Oswald's e is used to correct for departures from eliptical planform and is therefore not just a function of the aspect ratio.

In Oswald's paper, he's backing actual numbers out from data on aircraft weight, span loading, and span. This is the equivalent of calculating an ideal for a given amount of power (controlling for propeller efficiency) and calculating the distance from the ideal from flight test data.

-Blogs

Originally posted by justin_g
If you read Oswalds paper where he introduces the "e" factor, he was deriving it from flight tests and it is supposed to consider the efficiency of the whole aircraft, not just the wing. Link: http://naca.larc.nasa.gov/reports/1933/naca-report-408/index.cgi?thumbnail1#start

[Crumpp]

Getting back to Grippen's point, though, Oswald's e is used to correct for departures from eliptical planform and is therefore not just a function of the aspect ratio.

Gripen is not correct.  An instructor that teachs aeronautics explained it to him.

Read the third paragraph down and it explains Oswald's e factor and it's use.

http://naca.larc.nasa.gov/reports/1933/naca-report-408/index.cgi?page0001.gif

A correction is made for the variation of parasitic resistance with angle of attack and nonelliptical wingloading by including in the induced drag term a factor e, called the "airplane efficiency factor".  The correction is thus assumed to be proportional to CL^2.  

Now check out the induced drag formula.

As Badboy explained:

 

Correct, Figure 2-44 (b) shows the relationship of those factors to Aspect Ratio for a rectangular wing, and that illustrates the dependency of e to Aspect Ratio. In the previous graph you see the influence that taper ratio has for an aspect ratio of 6. There are other formulae that also include the taper ratio so that values for non rectangular wings can be estimated also.

And for using AR as part of the formula to solve for "e":

Most WWII aircraft had a taper ratio close to that value (Me109 was approx 0.52) because the designers knew about the benefits of elliptical lift distribution and that it could be achieved quite closely with a wing of that taper, that’s why a formulae that only includes aspect ratio could still be of such good practical use.

Crumpp

[gripen]

Well, if we have no idea about taper ratio, there is no way we can estimate the e factor with the aspect ratio only. However Badboy's poin't is quite sensible; most rectangular winged WWII fighters had quite similar taper ratio so in many cases generalized formula do quite well.

I have calculated some e factor data from the drag polars I have. These are e factors for whole airplane as Oswald and Perkin&Hage handle it:

Bf 109G, this is from the VL (Finnish State aircraft factory) drag polar:

Cl 0,1 => 0,741658161
Cl 0,2 => 0,798708788
Cl 0,4 => 0,769126981
Cl 0,6 => 0,759747384
Cl 0,8 => 0,787352739
Cl 1,0 => 0,769126981
Cl 1,2 => 0,713350597
Cl 1,4 => 0,619327448

The average is about 0,75

I have also a polar from Messerschmitt AG (I had to approximate Cd0):

Cl 0,1 => 1,038321425 (not in average)
Cl 0,2 => 0,83065714
Cl 0,3 => 0,778741069
Cl 0,4 => 0,755142854
Cl 0,5 => 0,70156853
Cl 0,6 => 0,692214283
Cl 0,7 => 0,696955477
Cl 0,8 => 0,692214283
Cl 0,9 => 0,7067566
Cl 1,0 => 0,711179058
 
The average 0,73

Mustang, RAE wind tunnel data:

Cl 0,6 => 0,789174119
Cl 0,4 => 0,814631349
Cl 0,2 => 0,717431018

The average is about  0,77

I also tried to calculate from the NACA 916 but there is so much varition that sensible analysis seems to be quite impossible; the e factors varied from 0,5 to 1,3 or something.

Regarding prizes; well I think the answer is not possible but in certain conditions it might be possible as Badboy pointed out.

So I think I'll send the prize to Angus and Blogs and Schutt will receive the Bonus, so post your adress and I'll send the docs. If Badboy is listening I can send both documents to you too.

gripen