Hi All,

A few days ago, Ben posted the following as an aside in News Picks [link|http://z.iwethey.org/forums/render/content/show?contentid=9439|here]:

My favorite math mistake is the flying wing design.

In the 50's they asked the question, "What plane design would get the maximum range?" They set up their equations, solved for all of the partials being 0, and came up with the flying wing. Then they built it at the cost of billions, tried it, and it sucked.

Turns out that if they had checked the second derivative, they would have found it was the absolute worst possible design. (Basic max-min problem, the first derivative tells you where to look but doesn't tell you if you got a max, min, or inflection point.) But by then billions were invested, careers on the line...and they still build them.

I'm not an Aeronautical Engineer, so take this with a grain of salt.

Flying Wing designs are quite old. They didn't originate in the 1950s. (I'm not saying that Ben said they did.) E.g. the [link|http://www.danford.net/horten.htm|Horten Brothers] were doing work on flying wings in the 1920s in Germany. Northrop was working on Flying Wings in the 1920s and had a Flying Wing built in 1940 (same page).

Now why would people work on them when a simple second derivative shows that they would have minimum range? Maybe because aircraft design is more complicated than that.

Remember that these planes were first developed when propellers were needed for propulsion. These engine housings, propellers, etc., introduce parasitic drag which reduce the range of the plane. As Northrop says in his famous lecture, [link|http://www.danford.net/northrop.htm|here]:

Our work, therefore, through the years has been directed solely to all-wing aircraft, by which I mean a type of airplane in which all of the functions of a satisfactory flying machine are disposed and accommodated within the outline of the airfoil itself. Of course, we have not as yet built any pure all-wing aircraft. All have had some excrescences, such as propellers, propeller drive shaft housings, jet nozzles, gun turrets and the like. We have, however, built a number of airplanes in which the minimum parasite drag coefficient has been reduced to approximately half that ordinarily attained in the best conventional aircraft of like size and purpose, and in some of the designs completed and tested the excrescences and variations from the airfoil contour have been responsible for less than 20 percent of the minimum airplane drag.

So there is a good reason to get lots of things out of the airstream. The ultimate approach is to eliminate everything except the wing. Note that he talks about measurements.

Northrop discusses issues with building such a craft of appropriate stiffness, controlling the craft, aerodynamics, etc.

Also, from what I've read on other pages, the efficiency of a wing depends on what speed the airflow is. For maximum efficiency, (lift vs. drag) you want a large wing with a large cross-sectional area. But you can't use such a wing at high speeds because it has too much drag. It's difficult to compare these things in a simple back-of-the-envelope calculation.

Another issue to consider is that in the 1920s-1950s the availability and understanding of high-strength materials necessary to build large stressed wings was in its infancy. It's much easier to build a large strong cylinder (fuselage) than a large strong wing.

In addition, [link|http://home.att.net/~jbaugher2/b35.html|this] page argues:

The advantages of a flying wing format were perceived as providing both low drag and high lift, which meant that the XB-35 could carry any weight faster, farther, and cheaper than conventional aircraft. In addition, the use of a flying wing meant that simpler construction methods could be used with fewer structural complications. A flying wing should cost less to build since it was built as a single unit with no added tail or fuselage. A flying wing provides a better weight distribution for the offensive load, since compartments along the entire span could distribute the weight of the bomb load much more evenly. Finally, a flying wing presented a smaller target when seen from fore, aft, or from the side when engaged in either offensive or defensive operations.

So there's much more potential advantage to a Flying Wing design than simply aerodynamics. The B-2 has (a modified version of) such a configuration because, among other things, it's more stealthy.

But what of the infamous second derivative? [link|http://www.aoe.vt.edu/Mason/ACiFlyWngs.html|The Controversy] says:

Is a flying wing the best configuration? Sears and Ashkenas wrote an appendix to a secret government report showing that a flying wing is best. The report, Toward New Horizons, was eventually issued in a commemorative edition in 1992. However, the appendix was not included. I have not yet seen the appendix. Foa studied the result and claimed that although Sears and Ashkenas had found the result for a slope of zero, they missed a sign on the second derivative and actually proved that the flying wing was the worst! Foa demanded that Sears and Ashkenas issue a revision. The result was a paper by Ashkenas which contains another analysis. Foa claimed that the paper was inadequate, and, with the emergence of the B-2, the controversy continues. Essentially, the question was where to distribute volume. As such, the actual size of the required volume changes the answer, sometimes it is best to put the volume in a distinct fuselage, sometimes not. Thus this has become an interesting configuration issue. As I see it, none of the papers to date have provided the answer in a simple, easy to understand analysis.

- Irving L. Ashkenas, "Range Performance of Turbojet Airplanes," Journal of Aeronautical Sciences, Vol. 15, No. 2, Feb. 1948, pp. 97-101.
- J.V. Foa, "Suitability of Flying Wings as Jet Airplanes, " Journal of Aeronautical Sciences, Vol. 16, No. 4, April 1949, pp. 253-254.
- I.L. Ashkenas, "Author's Reply," Journal of Aeronautical Sciences, Vol. 16, No. 4, April 1949, pp. 254-255
- J.V. Foa, "The Flying Wing Reconsidered," Canadian Aeronautics and Space Journal, Vol. 30, No. 1, March 1984, pp. 62-65..
- Wayne Biddle, "Skeleton Alleged in the Stealth Bomber's Closet," Science, Vol. 244, May 12, 1989, pp. 650-651.
- Wayne Biddle, "B-2 Comes Up Short," Science, Vol. 246, October 20, 1989, pg 322.
- E. Torenbeek, "Aerodynamic Performance of Wing-Body Configurations and the Flying Wing," SAE Paper 911019, April, 1991. This paper comes very close to providing the type of analysis needed to clarify the issue. It may in fact be the definitive paper, but I don't understand the conclusions well enough and need to derive his results for myself. When I do this, I'll report on my conclusions.
- Wayne Biddle, Barons of the Sky, Simon & Schuster, New York, 1991, pp. 312- 313. This book discusses the controversy, although only from Foa's viewpoint.
- Peter J. Torvik, "On the Maximum Range of Flying Wings," AIAA Paper 92- 4223, August 1992. An Air Force Officer tries to do an independent examination of the problem.
- William R. Sears, Stories from a 20th-Century Life, Parabolic Press, Stanford, 1994, pp. 119. Sears responds indirectly to the controversy.

(Emphasis added.)

So, I'd say that there's evidence that even if a simple back-of-the-envelope calculation indicates that a Flying Wing has minimum range, a real life aircraft design is more complicated so one shouldn't take that calculation as a veto.

The B-35 program (and its successors) had problems with non-wing related aspects of the design - gearboxes, propellers, etc. And it had control issues which were difficult to resolve with technology of that era.

For the conspiracy buffs out there, Northrop [link|http://www.nurflugel.com/Nurflugel/Northrop/xb-35/xb-35_blurb/conspiracy/conspiracy.html|claimed] in 1980 that the Flying Wing program was killed as punishment for not agreeing to merge his company with Convair in 1948.

Just some grist for the mill...

Cheers,
Scott.