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National Advisory Committee for Aeronautics, Technical Notes - An Accurate and Rapid Method for the Design of Supersonic Nozzles

A procedure is given for designing two-dimensional nozzles in which
the streamline coordinates are computed directly from tabulated flow
parameters and appropriate equations. The method of characteristics is
used to obtain the first part of the flow, which consists of a continuous
expansion from a uniform sonic flow to a radial flow. The Foelsch equa-
tions are then used for the transition from_radial flow to the final uni—
form flow. Information is presented which enables the designer to select
and compute rapidly the wall contour for any nozzle or series of nozzles
for a wide range of length-to-height ratio, mach number, and wall angle
at the inflection point. In general, a nozzle is determinedfby specifying
any two of these three parameters.

Recent experience obtained at the Langley Gas Dynamics Branch of the
National Advisory Committee for Aeronautics with the aerodynamic design
and flow calibration of two—dimensional supersonic nozzles for wind tun-
nels has indicated that a need exists for a more accurate and rapid design
method than the graphical and computational methods in common use. The
analytic equations derived independently by Foelsch and Atkin (refs. 1
and 2) partially fulfill this need since they are an exact solution to the
problem of generating uniform parallel flow from a supersonic divergent
radial flow. These equations give the required streamline coordinates
directly in terms of the assumed radial flow. The Foelsch—Atkin method,
however, has not been widely used because of the difficulties associated
with generating a divergent radial flow from a parallel sonic flow.

Atkin (ref. 2) reduced these difficulties to some extent by deriving
analogous expressions for expanding a uniform supersonic stream to a
divergent radial flow at a larger Mach number. Pinkel (ref. 5) has sug—
gested several convenient procedures for obtaining the transition from a
parallel sonic flow to a uniform flow at a slightly higher Mach number,
which is required in the Atkin solution, either by adaptations of the
Prandtl-Meyer solution or by the use of a minimum-length "subnozzle" to
be computed by graphical methods. Although the procedures of reference 5
are in theory satisfactory, the peculiar shape of the resulting streams
lines will ordinarily introduce practical construction difficulties or
other prdblems connected with boundary—layer development.