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National Advisory Committee for Aeronautics, Technical Notes - On a Method of Constructing Two Dimensional Subsonic Compressible Flows Around Closed Profiles

It is shoWn that under‘certain conditions a two—dimen—_
sional snbsonic compressible flow around an airfoil profile
can be derived from an incompressible flow around another ' ’
profile.f The connec-tion between these two "conjugate flows“'
is given bv a simple conformal transformation of- the respec—--
tive hodograph planes.

The transformation of a given in-compressible flow into
a compressible flow around a slightly' distorted- profile re— a
duces to the integratio‘n of a linear' partial differential.
equation in the phvsical plane of th'e incompressible flOWp
An approximate solution of this equation is indicatei. - Fur~
ther research is necessary in order to extend the applies—j
bility of the method and in order to reduce the computational
work involved in the rigorous solution to an acceptable. min—
imum.

The transformation of an incompressible flow into a
compressible one can be carried-oat completely'and'in a'
closed form un6.er the assumption of the linearized'preSsure-
density- relation. The final formulas" represent an extension
of the re_eult of- von Karman and Tsien, to which thev reduce._
in the special case of a flow without circulation. It_ is-
shown' that essentiallv all compressible flows can be obtained
by this method.

The high level which has beenfattmined by the theory of
two—dimensional incompressible flows is due to the fact that
this theory is based upon a highly developed mathematical
theory, that of analytic functions of a‘complex variable.
Every analytic function vields a.possible flow pattern and
vice versa. Furthermore, the use of transformatid‘ns pere
formed by means of analytic functions (conformal transforma—
tions) permits t1;e derivation of all possible flows from a
few simple standard forms. It seems obvious that the theory
of two-dimensional compressible flows (at least as far as
subsonic flows are concerned) requires the development of a
similar mathematical background.

The theory of sigma monogenic functions (reference's l
an_d 2_) is an .attempt to study a class of. complex functions
the role of which in gas dynamics is cor .parable to that of
analyt.ic functions in the theory of incompressible flows.
Gelbhrt (reference 5) has outlined the application of this
method to the study of comnressible flows. Further applica~
tions depend upon the investigation of singularities of
sigma—monogenic functions. (Such an investigation is being
conducted 3 Reference also is made to a recent report bv
Garrick and Kaplan (reference 4) The investigation of
transformations which for the case of compressible flows take
the place of conformal transformations is the main theoreti—
cal aim of the present report.