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National Advisory Committee for Aeronautics, Technical Notes - Graphical and Analytical Methods for the Determination of a Flow of a Compressible Fluid Around an Obstacle

Chaplygin introduced the hodograph method in the the_ory
of compressible fluid flows and dev.eloped a method for aan'_
constructing stream' functions of such flows. This method,
which has been extensively used in investigation of compress-
ible fluid flows, is limited in certain respects:- The ex—
pression for the stream function obtained in this manner can
represent onlylcertain types of flow patterns, :In general.
flow patterns obtained in this way cannot represent the whole
flow around an obstacle, but only a part of such a flow, and
therefore several expressions are needed in order'to obtain
the whole flow.- On the other hand, in many instances, it is
important to have a single expression representing the whole.
flow.

In the Present report this formula is employed to con—
struct two—damensicnal subsonic compressible fluid flows
around a body similar in shape to a given symmetric obstacle.

In a companion report (reference 1) the authpr derived a
formula'transforming an arbitrary analytic function of-a com—
plex variable, g(Z), into a function- w(x;y) which satis*
fies the differential equation-for the stream function'of at
two—dimensional potential gas flow. (It'is assumed that-the
pressure of the fluid-is a function of its density.)

This report describes the application of this formula'to
the determination of a flow nround an obstacle. First; the
actual'comnutation'of the stream function generated by-an
analytic function is described in detail. Then various meth—
ods for choosing the function which yields a flow around a
given obstacle (or at least around a shape similar to the
aiven'one) are discussed.

In the case of an incompressible fluid this function
would lead to the'circulation—free flow around a circular
cylinder. In the case of a.compressible-fluid a flow around
a cylinder of somewhat distorted cross section is obtained.

For the mathematical justification of the formulas used,
the reader is referred to reference 1.

This investigation. conducted. at -the Brown University,-
was sponsored by and conducted with the financial assistance
of the -National Advisorv Committee for Aeronautics.