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National Advisory Committee for Aeronautics, Report - On Similarity Rules for Transonic Flows

A method used by Tsien to derire similarity rules for hyper—
sonic flows is utilized to derive Von Kdrmdn’s similarity rules
for transonic flows. A slight generalization is introduced by
the inclusion of 7, the ratio of specific heats, as a parameter.
At the lower limit of the transonic region of flow the theory
yields a formula for the critical stream .Mach numbers of a
gicen family of symmetrical profiles. It is further shown that
this formula can also be obtained by means of the Prandtl—
Glaue-rt small-perturbation method. Incestigation of the be-
havior of the similarity parameter in the region where the
thickness coefiicient approaches zero and the critical stream
M'ach number approaches unity shows that it possesses a
limiting value' characteristic of the prescribed family of shapes.

The rigorous solution of the subsonic flow of a compressible
fluid past a prescribed closed body thus far has proved to
be of insurmountable difficulty. As a. consequence of this
difliculty the emphasis has been placed on the establishment
of a correspondence betWeen the flow past a given body in an
incompressible fluid and the same body in a compressible
fluid. Among the best known results of this mode of attack
are the Prandtl-Glauert rule and the Von Karmén-Tsien
velocity or pressure correction factor—both based on some
form of linearization of the fundamental nonlinear flow equa-
tions. None of the methods based on the linearization of the
flow equations, however, can yield correct results in the
transom'c range where the flow is partly subsonic and partly
supersonic. For this case a certain amount of the feature of
nonlinearity of the flow equations must be retained in order
to obtain useful and nontrivial results. In the present paper
a detailed derivation is given of the transomjc similarity
rules recently given by Von Karman (reference 1).