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National Advisory Committee for Aeronautics, Technical Notes - The Similarity Law for Nonsteady Hypersonic Flows and Requirements for the Dynamical Similarity of Related Bodies in Free Flight

The similarity law for steady hypersonic flow about slender shapes ‘
is extended to nonsteady flows. Similitude for nonsteady flows is found
to depend on the same conditions as for steady flows plus additional
conditions derived from the nonsteady motions of the bodies under con—
sideration. The aerodynamic forces and moments are correlated for
related shapes by means of this law.

Requirements for dynamical 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,
dynamical similarity depends upon conditions derived from the inertial
properties of the bodies and the immersing fluids. In order to haVe
dynamical similarity, however, rolling motions in combination with other
motions must be eliminated.

The law of similarity for steady hypersonic flows has been studied
in some detail. Tsien, in reference 1, derived the law for potential
flow about related slender bodies for the two-dimensional and axially
symmetrical cases. It was found that similarity of flow exists when
the bodies have the same thickness distributions and when the ratio of
the free—stream.Mach number to the fineness ratio is the same in each
flow system. Hayes, in reference 2, showed that the law should remain
valid even when the flow includes shock waves and vorticity, and indi-
cated that it should apply to more general three-dimensional flows. In'
reference 3 the law was derived for flows about slender three-dimensional
shapes in terms of parameters relating the Mach number to the fineness
ratio, the aspect ratio, and the angles-of-flight attitude of the
bodies. The correlation of force and moment parameters in terms of
the similarity parameters was also made for similar flow fields.

The consideration of the problems of free flight suggests the
desirability of extending the hypersonic similarity law to cover the
case of nonsteady flows. Lin, Reissner, and Tsien, in reference 1L,
developed necessary conditions for similarity of flow about oscillating
two—dimensional bodies in compressible fluids, including flow at hyper-
sonic speeds. An analysis for slender three-dimensional shapes in
hypersonic flow is apparently not available, and has therefore been
undertaken in the present report following methods similar to those
employed in reference 3.