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National Advisory Committee for Aeronautics, Report - General Potential Theory of Arbitrary Wing Sections

This report gives an exact treatment of the problem of
determining the 2-dimensional potential flow around
wing sections of any shape. The treatment is based
directly on the solution of this problem as advanced by
Theodorsen in N. A. C. A. Technical Report No. 411.
The problem condenses into the compact form of an inte-
gral equation capable of yielding numerical solutions by
a direct process.
An attempt has been made to analer and coordinate
the results of earlier studies relating to properties of wing
sections. The existing approximate theory of thin wing
sections and the Joulcowslcy theory with its numerous
generalizations are reduced to special cases of the general
theory of arbitrary sections, permitting a clearer perspec-
tive of the entire field. The method not only permits the
determination of the velocity at any point of an arbitrary
section and the associated lift and moments, but furnishes
also a scheme for developing new shapes of preassigned
aerodynamical properties. The theory applies also to
bodies that are not airfoils, and is of importance in other
branches of physics involving potential theory.
The solution of the problem of determining the
2-dimensional potential flow of a nonviscous incom-
pressible fluid around bodies of arbitrary shape can be
made to depend on a theorem in conformal represen-
tation stated by Riemann almost a century ago,
known as the fundamental theorem of conformal rep-
resentation. This theorem is equivalent to the state—
ment that it is possible to transform the region
bounded by a simple curve into the region bounded by
a circle in such a way that all equipotential lines and
stream lines of the first region transform respectively
into those of the circle. The theorem will be stated
more precisely in the body of this report and its sig—
nificance for wing section theory shown—suffice it at
present to state that if the analytic transformation by
which the one region is transformed conformally into
the region bounded by the circle is known, the poten—
tial field of this region is readily obtained in terms of
the potential field of the circle.