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National Advisory Committee for Aeronautics, Report - The Similarity for Hypersonic Flow and Requirements for Dynamic Similarity of Related Bodies in Free Flight

The similarity law for nonsteady, inviscid, hypersonic flow
about slender three-dimensional shapes is derived in terms of
customary aerodynamic parameters. The conclusions drawn
from the potential analysis used in the development of the law
are shown to be valid for rotationatflow. A direct consequence
of the hypersonic similarity law is that the ratio of the local
static pressure to the free-stream static pressure is the same at
corresponding points in similar flow fields.

Requirements for dynamic similarity of related shapes in free
flight, including the correlation of their flight paths, are obtained
using the aerodynamic forces and moments as correlated by the
hypersonic similarity law. In addition to the conditions of
hypersonic similarity, dynamic similarity depends upon con
ditions derived from the inertial properties of the bodies and the
immersing fluids. In order to have dynamic similarity, how-
ever, rolling motions must not occur in combination with other.
motions.

The law is examined for steady flow about related three-
dimensional shapes. The results of a computational investiga-
tion showed that the similarity law as applied to nonlifting
cones and ogives is applicable over a wide range of Mach num-
bers and fineness ratios. .' In the special case of inclined bodies
of revolution, the law is extended to include some significant
efi'ects of the viscous cross force. Results of a limited 33712917;-
mental investigation of the pressures acting on two inclined
cones are found to check the law as it applies to bodies of
revolution.

The hypersonic similarity law for steady potential flows
about thin airfoil sections and slender nonlifting bodies of
revolution was first developed by Tsien in reference 1.
Hayes (ref. 2) investigated this law from the standpoint of
analogous nonsteady flows and concluded that it would also
apply to nonpotential flows containing shock waves and
vorticity, provided the local Mach number was everywhere
large with respect to 1. He also reasoned that similitude
could be obtained in hypersonic flows about slender three-
dimensional bodies of arbitrary shape; however, the form of
the similarity law in terms of customary aerodynamic
parameters was not determined.