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National Advisory Committee for Aeronautics, Report - Minimum Drag Ducted Pointed Bodies of Revolution Based on Linearized Supersonic Theory

The linearized drag integral for bodies of revolution at super-
sonic speeds is presented in a double-integral form which is not
based on slender-body approximations but which reduces to the
usual slender—body expression in the proper limit. With‘the aid
of a suitably chosen auxiliary condition, the minimum—Mental-
waoe—drag problem is solved for a transition section connecting
two semi-infinite cylinders. The projectile tip is a special case
and is compared with the Von Kdrmdn projectile tip. Calcula-
tions are presented which indicate that the method of analysis
gives good first-order results in the moderate supersonic range.

In making the slender-body approximation to the linear-
ized supersonic-flow theory for bodies of revolution, 3. basic
approximation leads to replacing the axial source distribu—
tion with the cross-sectional-area derivative. Slender-body
theory, therefore, becomes linear in the superposition sense
for cross-sectional areas as well as for sources or fields in
contrast to linear supersonic-flow theory which is linear in
the superposition sense for sources or fields but not for areas.
Making the slender-body approximation, however, eliminates
a large part of the Mach number dependence of the results.
Lighthill (ref. 1) has shown that this basic approximation,
for sufficiently smooth bodies, has the same mathematical
order of accuracy as the linearized supersonic-flow equation.
Ward (ref. 2) has extended the generality of slender—body
theory by presenting a drag expression which is valid for a
body with a finite slope at the base. Lighthill (ref. 3) has,
at the price of a large increase in complexity, modified
slender-body theory to include area-derivative discontinu-
ities at a finite number of points.

In 1935 Von Karmfin (ref. 4) determined the minimum—
wave-drag projectile tip. Later Sears (ref. 5) and Haack
(ref. 6) determined minimum—wave—drag shapes for projectile
tips and closed bodies of revolution subject to various com~
binations of auxiliary conditions of constant length, constant
caliber, and constant volume. The Von Karman tip and
the Sears-Haack bodies are based on slender-body theory.