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National Advisory Committee for Aeronautics, Technical Notes - Supersonic Flow Around Circular Cones at Angle of Attack

The flow around cones without axial symmetry and moving at super-
sonic velocity is analyzed. Singular points are shown to exist in the
flow around the cone if no axial symmetry exists. The results of the
analysis are applied to the determination of flow around circular cones
at small angles of attack. The concept of a vortical layer around the
cone at small angles of attack is introduced, and the correct values of
the first—order terms of the velocity components are determined.

The method used is applied to cones at finite angles of attack, and
it is shown that good agreement with experimental results can be- obtained
from the first-order theory if the complete equation for the pressure
distribution is used.

The flow around a cone having a circular cross section and moving at
supersonic speed has been determined by means of the assumption of small
disturbances or by means of more rigorous methods that consider the
existence of the shocks. The. latter methods can be applied for any Mach
number larger than unity and have been developed by several authors, at
first by assuming all the flow as potential flow (references 1 and 2) and
later by also considering the variation of entropy due to the change in
angle of attack (reference 3). By means of the development given in
reference 3, values of flow properties around circular cones at an angle
of attack have been tabulated in reference I4. The method has been
extended in reference 5 to larger angles of attack.

In the method given in references 3, I4, and S, the flow properties
were considered continu'ous and were developed in Fourier series in terms
of the angle of attack; however, the existence of a singular point at the
surface of the cone was neglected. The derivatives of the flow properties
were obtained by differentiating the Fourier ‘series term by term, and the
terms of the series that represent the derivatives were assumed to be of
the same- order as the corresponding terms of the integral quantities.
For this reason, an erroneous distribution of the entropy at the surface
of the cone was obtained.

In this paper, the flow around the cone in the general case is dis-
cussed, the existence of singular points in the flow is proved, and a
different procedure for determining the flow around cones at small, but
finite, angles of attack is developed. This procedure shows the way in
which the values tabulated in reference )4 can be used if a simple cor-
rection is introduced. The values obtained in this way are compared
with experimental results at several values of angle of attack.