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National Advisory Committee for Aeronautics, Report - A Unified Two Dimensional Approach to the Calculation of Three Dimensional Hypersonic Flows, with Application to Bodies of Revolution

A simplified two—dimensional method for calculating three-
dimensional steady and nonsteady hypersonic flows of an
inviscid (non-heat-conducting) gas is deduced from character-
istics theory. This method is appropriately termed a generalized
shock—expansion method. It is demonstrated that the method is
applicable when disturbances associated with the divergence of
streamlines in planes tangent to a surface are of secondary
importance compared to those associated with the curvature of
streamlines in planes normal to the surface. When this con-
dition is met, surface streamlines may be treated as geodesics,
which, in turn, may be related to the geometry of the surface.

It is inquired further if the two-dimensionality o_f inviscid
hypersonic flows has a counterpart in hypersonic boundary—
layer flows. This question is answered in the ajfirmative,
thereby permitting a unified two-dimensional approach to
three-dimensional hypersonic flows.

This concept is applied to bodies of revolution in steady flight
and, with the assumption that flow at the vertex is conical,
approximate solutions for the flow field are obtained for values
of the hypersonic similarity parameter (i. e., the ratio of the
free-stream Mach number to the fineness ratio of the body)
greater than about 1 and for small angles of attack. Surface
streamlines are approximated by meridian lines and the flow
field is calculated in meridian planes. Simple explicit espres-
sions are obtained for the surface Mach numbers and pressures
in the special case of slender bodies.

The validity of theory is checked by comparison with surface
pressures and shock-wave shapes obtained experimentally at
Mach numbers from 3.00 to-6.30 and angles of attach: up to
15° for two ogives having fineness ratios of?! and 5. At the lower
angles of attack, theory and experiment approach agreement
when the hypersonic similarity parameter is in the neighborhood
of 1 or greater. At the larger angles of attack, theory tends to
break down noticeably on the leeward sides of the bodies.