<DOC>
<DOCNO>
1201
</DOCNO>
<TITLE>
a study of slender shapes of minimum drag using the
newton-busemann pressure coefficient law .
</TITLE>
<AUTHOR>
miele,a.a.
</AUTHOR>
<BIBLIO>
aiaa jnl. 1, 1963, 168.
</BIBLIO>
<TEXT>
  the problem of minimizing the drag of a slender, two-dimensional
or axisymmetric body in hypersonic flow at zero angle of
attack is considered under the assumption that the pressure coefficient
law is newton's impact law as modified by busemann in
order to include centripetal acceleration effects .  after the condition
that the pressure coefficient be nonnegative is accounted
for and after arbitrary conditions are imposed on, in addition
to the thickness and the length, the enclosed area and the moment
of inertia of the contour in the two-dimensional case and the
wetted area and the volume in the axisymmetric case, the minimal
problem is formulated as a problem of the mayer type and
solved by the combined use of the euler-lagrange equations,
the transversality condition, the erdmann-weierstrass corner
condition, and the properties of the switching function .
  particular attention is devoted to the class of problems such
that, among the four quantities being considered, two are prescribed
while the remaining are free .  for these problems, the
extremal arc is composed of two subarcs ..  one is characterized by
a positive pressure coefficient and is called the regular shape,. the
other is characterized by a zero pressure coefficient and is called
the free layer .  in this connection, the analysis shows the existence
of two different types of solutions depending on whether the
thickness is given or free .
  if the thickness is given, the expression for the regular shape is
a power law, and the transition from the regular shape to the free
layer occurs in the second half of the body .  in the two-dimensional
case, the exponent of the power law is 1 if the length is given
if the enclosed area is given, and 3 if the moment of inertia
of the contour is given,. the transition point from the power
body to the free layer is located at 50 percent of the length if the
length is given, at 66 percent if the enclosed area is given, and at
the axisymmetric case, the exponent of the power law is if
the length is given, 1 if the wetted area is given, and if the
volume is given,. the transition point from the power body to the
free layer is located at 60 percent of the length if the length is
given, at 70 percent if the wetted area is given, and at 80 percent
if the volume is given .
  on the other hand, for problems where the thickness is free,
the equation governing the regular shape is not that of a power
body, and the point of transition to the free layer is located in the
first half of the body .  in the two-dimensional case, the transition
point is at 28 percent of the length if the length and the enclosed
area are given, at 32 percent if the length and the moment
of inertia of the contour are given, and at 45 percent if the enclosed
area and the moment of inertia of the contour are given .
in the axisymmetric case, the transition point is located at 35
percent of the length if the length and the wetted area are given,
at 39 percent if the length and the volume are given, and at 46
percent if the wetted area and the volume are given .
  for all of the cases considered, analytical expressions are obtained
for the optimum shapes, the thickness ratios, and the drag
coefficients .
</TEXT>
</DOC>
