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National Advisory Committee for Aeronautics, Technical Notes - Approximate Methods for Calculating the Flow About Bodies of Revolution at High Supersonic Airspeeds

Flow at high supersonic speeds about a body of revolution is
investigated analytically. With the assumption that the flow at the
vertex is conical, it is found that algebraic solutions can be obtained
which yield the Mach numbers and pressures at the surface over a con-
siderable range of free-stream Mach numbers and apex angles. In the
special case of cones, these solutions define the entire flow field with
good accuracy, and may therefore provide a useful adjunct to the well-
known M.I.T. tables.

The investigation of flow dowustream of the vertex reveals that
when the value of the hypersonic similarity parameter for the flow
(i.e., the ratio of the free-stream mach number to the slenderness ratio
of the body) is large compared to l, the Mach number along a streamline
(downstream of the nose shock) varies with flow inclination angle in
approximately the same manner as fer two-dimensional (Prandtl-Meyer)
flew. In the special case of streamlines near the surface, it is sug-
gested that this parameter may approach 1. This result and the solutions
obtained for flow at the vertex are combined to yield what might be
called a conical-Shock-expansion method for calculating the mach number
and pressure distributions at the surface of a body. These calculations
are shown to be particularly simple in the case of slender bodies.

Surface MaCh number and pressure distributions calculated with the
simplified methods of this paper for a number of ogives are found to be
in good agreement with those Obtained.with the method of characteristics
at values of the hypersonic similarity parameter greater than 1. In the
case of the conical-shock-expansion calculations, the agreement is within
the order of accuracy of the characteristics solutions When the hyper-
sonic similarity parameter has a value of only 2. Because of the rela-
tive simplicity of these calculations, the methods for determining the
flow at the surface of a body may prove useful for engineering purposes.

Determination of supersonic flow fields about nonlifting bodies of
revolution by means of the method of characteristics (see, e.g., refer-
ence l) is generally accepted as an accurate but tedious and time-
consuming operation. Because of the latter features of this method,
recourse is often made to simpler methods which, although less accurate,
can be applied with relative simplicity and rapidity.