The Minimum Induced Drag of Aerofoils Page: 1 of 16
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REPORT No. 121.
THE MINIMUM INDUCED DRAG OF AEROFOILS.
By AMAX M. MUNK.
The following paper is a dissertation originally presented(l by the author to the University
of Goettingen. It was intended principally for the use of mathematicians andl physicists. The
author is pleased to note that the paper has aroused interest in other circles, to the end that
the National Advisory Committee for Aeronautics will make it available to a larger circle in
America. The following introduction has been adlde(l in order to first acquaint the reader
with the essence of the paper.
In the following development all results are obtained by integrating some simple expressions
or relations. For our purposes it is sufficient, indeed, to prove the results for a pair of small
elements. The qualities dealt with are integrable, since, under the assumptions we are allowed
to make, they can not be affected by integrating. We haveT to consider only the relations
between any two lifting elements and to add the effects. That is to say, in the process of inte-
grating each element occurs twice -first, as an element producing an effect, and, second, as an
element experiencing an effect. In consequence of this the symbols expressing the integration
look somewhat confusing, and they require so much space in the mathematical expression that
they are apt to divert the reader's attention from their real meaning. We have to proceed up
to- three dimensional problems. ' ach element has to be (lenoted twice (by a Latin letter and
by a Greek letter), occurring twice in a different connection. The integral, therefore, is sixfold,
six symbols of integration standing together and, accordingly, six differentials (always the same)
standing at the end of the expression, requiring almost the fourth part of the line. The meaning
of this voluminous group of symbols, however, is not more complicated and not less elementary
than a single integral or even than a simple addition.
In section 1 we consider one aerofoil shaped like a straight line and ask how all lifting
elements, which we assume to be of equal intensity, must be arranged on this line in order to
offer the least drag.
If the distribution is.the best one, the drag can not be decreased or increased by transferring
one lifting element from its old position ( ) to some new position (h). For then either the
resulting distribution would le improved by this transfer, and therefore was not best before, or
the transfer of an element from (b) to (a) would have this effect. Now, the share of one element
in the drag is composed of two parts. It takes share in producing a downwash in the neighbor-
hood of the other lifting elements and, in consequence, a change in their drag. It has itself a
drag, being situated in the downwash produced by the other elements.
Considering only two elements, Fig. 1 shows that in the case of the lifting straight line the
two downwashes, each produced by one element in the neighborhood of the other, are equal.
For this reason the two drags of the two elements each produced by the other are equal, too,
and hence the two parts of the entire drag of the wings due to one element. The entire drag
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Munk, Max M. The Minimum Induced Drag of Aerofoils, report, 1979-12?; (https://digital.library.unt.edu/ark:/67531/metadc53402/m1/1/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.