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National Advisory Committee for Aeronautics, Technical Notes - Determination of Shapes of Boattail Bodies of Revolution for Minimum Wave Drag

By use of an approximate equation for the wave drag of slender
bodies of revolution in a supersonic flow field, the optimum shapes of
certain boattail bodies are determined for minimum wave drag. The prop—
erties of three specific families of bodies are determined, the first
family consisting of bodies having a given length and base area-and a
contour passing through a prescribed point between the nose and base,
the second family having fixed length, base area, and maximum area, and
the third family having given length, volume, and base area. The method
presented is easily generalized to determine minimum-wave—drag profile
shapes which have contours that must pass through any prescribed number
of points.

According to linearized theory, the optimum profiles are found to
have infinite slope at the nose but zero radius of curvature so that the
bodies appear to have pointed noses, a zero slope at the body base, and
no variation of wave drag with Mach number. For those bodies having a
specified intermediate diameter (that is, location and magnitude given),
the maximum body diameter is shown to be larger, in general, than the
specified diameter. It is also shown that, for bodies having a specified
maximum diameter, the location of the maximum diameter is not arbitrary
but is determined from the ratio of base diameter to maximum diameter.

The wave drag of slender bodies of revolution having cross— sectional
areas with a zero slope at the base was shown by Von Karman (reference 1)
to be given approximately by a double integral dependent only on the body
shape and independent of Mach number. By use of Von Karman' s integral
and the calculus of.variations, several authors (references 1 to A) have
treated the problem of determining optimum body shapes to give minimum
wave drag. All these investigations have been concerned with either
closed bodies or a body, such as a shell, having its maximum thickness
at the base; however, none have treated bodies having boattails, and
this problem is considered herein.

Ward (reference 5) has shown that bodies having a finite slope at
the base give rise to drag terms in addition to Von Karman' s integral
and these additional terms include Mach number effects. The present
paper shows, however, that the optnmum bodies must have a zero slope at
the base and consequently the additional terms vanish. The determination
of the minimum-wave—drag bodies of revolution with boattailing then
resolves itself to minimizing the same integral as used by the reference
papers but with a more general treatment of the body profile.