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National Advisory Committee for Aeronautics, Report - On Slender Body Theory at Transonic Speeds

The basic ideas of the slender-body approximation have been
applied to the nonlinear transonic—flow equation for the velocity
potential in order to obtain some of the essential features of
slender—body theory at transonic speeds. The results of the
investigation are presented from a unified point of view
which demonstrates the similarity of slender-body solutions in
the various Mach number ranges. The primary difference
between the results in the diferent flow regimes is represented
by a certain function which is dependent upon the body area
distribution and the stream Mach number. The transonic
area rule and some conditions concerning its validity follow
from the analysis.

Slender—body theory originated with Munk’s report (ref. 1)
in 1924 in which the forces on slender airships were calculated
for low—speed flight. In 1938 Tsien (ref. 2) pointed out that
Munk’s airship theory also applied to the flow past-inclined
pointed bodies at supersonic speeds. The subject gained
new importance in 1946 with the appearance of Jones’s re-
port (ref. 3) in which it was shown that the basic ideas of the
slender-body approximation could be used to calculate the
forces on slender lifting wings at both subsonic and super-
sonic speeds provided that proper account was taken of
trailing-vortex sheets.

Since Jones’s report, the subject has
received wide treatment. In an important paper in 1949,
Ward (ref. 4) developed a general unifying theory for the
flow past smooth slender pointed bodies at supersonic speeds.
This theory contains as special cases the lifting planar Wings
of Jones and the slender nonlifting bodies treated by Von
Karmfin (ref. 5). The corresponding problem at subsonic
speeds has been examined by Adams and Sears (ref. 6) who
also extended the slender-body concepts to shapes which are
“not so slender.” Lighthill (ref. 7) has given a method for
calculating the flow past bodies with discontinuities in slope.
Keune (ref. 8) has developed solutions for slender wings with
thickness, and various lifting configurations have been
treated by Heaslet, Spreiter, Lomax, Ribner, and others.