Gripen,
I’ve only just realized where we appear to have our lines crossed and what has caused some of the confusion that has crept into this thread, and I think I can clear it up. Throughout this thread you have been questioning why the formulae posted seem to give values of the e factor that are too high, and that don’t correspond to empirical data… You said this in a previous message…
Originally posted by gripen
All I say above is that generalized formulas by Wood and NADC seem to give too high value of the e factor for WWII fighters and this conclusion is backed up by empirical data.
I didn’t realize what was happening until I saw your recent posts, and I think I can explain what you are seeing. We have been referring to Oswald’s efficiency factor, (Airplane Efficiency factor) and we have been using the character e to represent it, which is what normally happens, but there is a catch... There are two values associated with it and they are not the same, and they don’t measure the same things. Take another look at the explanation I posted previously:
Originally posted by Badboy
The number normally represented by the character e in induced drag calculations was originally known as Oswald’s efficiency factor, and his original paper is available for download from the NACA report server. More commonly it has a component of parasite drag lumped in with it and is just called the airplane efficiency factor and can be estimated depending on the aspect ratio, taper ratio, sweep angle and twist. Theoretically an elliptical wing would have an efficiency factor of 1, meaning that it will have a coefficient of induced drag close to the theoretical maximum. Even though it is a function of aspect ratio, sweepback angle, taper ratio, camber, Mach number and twist, the largest influence on the wing of a WWII fighter with very little sweep or twist comes from aspect ratio and taper ratio and so there are approximate formulae for estimating e that only include aspect and taper ratio, and even more approximate methods that only include aspect ratio. The important thing you must appreciate is that they are only approximate. But better than just assuming a constant value for every aircraft..
You might remember that, but the important point is where I explain that the airplane efficiency factor has a component of parasite drag lumped in with it. In your copy of Wood’s book you can see this, and I’ll point out the references to Wood and Perkins & Hage in a moment. Firstly, let me explain what happens… If you plot a graph of parasite drag and induced drag you can see that they both vary with the coefficient of lift. This is expressed mathematically in 2-82 at the top of page 93 of Perkins & Hage. Normally the two terms that vary with lift are lumped together, and when you do that, you are combining the induced drag from the wing, and induced drag due to a component of the parasite drag resulting from the lifting capability of the fuselage and the tail, in fact from the aircraft as a whole. In the normal parabolic drag polar, the accounting is done that way so that all the lift related components can be lumped together in a single term like the one in 2-83 on the same page. The problem is that they are not always combined in the same way, and when the wing is being treated in isolation, during airfoil theory for example, you may see the induced drag for the wing expressed using a value for e that does not include the effects of the lift dependant component of the parasite drag for the fuselage or tail, or anything other than the induced drag of the wing. Wood does both in his book and he uses the term ew with the subscript w to indicate a value for the wing only, and he uses e by itself for the airplane efficiency factor for the aircraft as a whole. If you look at empirical data, such as the drag polar from a flight test, you will see the later value, if you see a drag polar for an airfoil you will see the former value. I believe this has caused much of the confusion in this thread, because we have been discussing two different things, and of course the formulae posted were for the wing, which is why you have noticed that those values are too high. I hope that clears up that misunderstanding.
Now I would like to address your following point:
Originally posted by gripen
The formula is not for tapered wings with certain taper ratio but simply for rectangular wings and here is a quote from the page 36 (above the part Badboy quoted in another thread):
"In the practical airplane calculations the graphs for tapered wings may be assumed to lie between the graphs for rectangular and elliptical wings. For ratios of tip chord to root chord between 0.2 and 0.6, the tapered wing has practically the same characteristics as the elliptical wing."
What Wood is saying in your quote is that the value for all wings will fall between the two graphs shown below, with elliptical wings at one extreme and rectangular at the other.
You can see from this graph, that the value for wings with various other taper ratios can be determined simply by choosing a line between them. So in effect you can have a generalized formula for any taper. But a formulae that only yields a value for the wing alone. However, Wood goes on to say in the part of the quote that you appear to have left off:
"
Glauert gives corrections for tapered wings as a function of the amount of taper, but this refinement is believed to be not justified in practice."
When this thread started, you were only asking for someone to explain how e could be estimated from aspect ratio, however, if you wish to include taper ratio as well, you can do it using a slightly different formulae, the formulae is:
e = (2083T – 1083)/(33R + 970) - 2.083T + 2.083
Where R is the aspect ratio and T is the taper ratio between 1 for a rectangular wing and 0.52 equivalent to an elliptical wing (
you can only use values between 1 and 0.52 in this equation). So for example, if you calculate for an aspect ratio of 6 and a taper ratio of 0.8 you get a value for e = 0.92. For an aircraft with a particular fuselage shape and size that could end up being closer to 0.83 for the whole aircraft.
Glauert also published a very nice graph that allows you to determine e from both the aspect ratio and from the taper ratio, I can post it if that would be helpful?
But it is better to work from drag polars derived from flight test data.
Hope that helps…
Badboy
