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National Advisory Committee for Aeronautics, Technical Notes - On Supersonic and Partially Supersonic Flows

The present paper is one of a series of papers to ex-
tend the analytical methods used so successfully in the
theory of an incompressible fluid to the case of a compress-
ible fluid.

A stream function of an irrotational flow of an incom—
pressible fluid satisfies the Laplace equation_and, con-
versely, the.imaginary part of an arbitrary analytic func—
tion of a complex variable can be considered as the stream
function of a possible flow; many results of the highly de-
velOped theory of analytic functions can thus be interpret—
ed as theorems on the motion of an incompressible fluid.

By using the hodograph method (introduced in the theory
of compressible fluids by Chaplygin) a formula had previous—
ly been obtained for the stream function of a possible sub—
sonic compressible flow in terms of an arbitrary analytic
function of a complex variable; procedures for using some
methods and techni ues of the theory of analytic functions
in the theory of subsonic flOWS have likewise been indicated
and, as a consequence, new flow patterns ha.ve been obtained.
These flow patterns include examples of flows around_ symmet—
ric and nonsymmetric obstacles, underk the assumption of the
true pressure~density relation, p: opk c and k being '
constants.

In this paper the foregoing results are improved and
completed. A formula (analogous to that for subsonic flows)
is derived, which represents a stream function of a pos_sible_
supersonic flow in terms of two arbitrary differentiable __
functions of one real variable. Finally, some instances are
discussed in which flow pattern defined in two neighboring
parts of the plane can be combined into one flow pattern de-
fined in the combined domain. This last method leads, in
some instances, to partially supersonic flows.

The development of research in compressible fluid the— _
ory has made it desirable to have adequate mathematical
tools for dealing with problems of compressible flows. One L
of the reasons for the success of mathematical methods in
the study of two—dimensional irrotational steady flows of
an incompressible fluid is based on the fact that it is
possible to represent the stream function of such a flow as _
the imaginary part of an analytic function of a complex
variable. As a consequence, various results in the highly
developed theory of analytic functions can be applied to
yield solutions of problems in hydrodynamics.