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National Advisory Committee for Aeronautics, Report - Equations for the Design of Two Dimensional Supersonic Nozzles

Equations are presented for obtaining the wall coordinates
of two-dimensional supersonic nozzles. The equations are
based on the application of the method of characteristics to
irrotational flow of perfect gases in channels. Curves and
tables are included for obtaining the parameters required by
the equations for the wall coordinates.

A bn'ef discussion of characteristics as applied to nozzle
design is given to assist in understanding and using the nozzle—
design method of this report. A sample design is shown.

In a properly designed nozzle, there are no compression or
expansion waves in the flow downstream of the straightening
portion. A streamline crossing such waves wouldbe altered
in direction and Mach number, which is generally undesir-
able.

The method of characteristics provides, a means for ob-
taining the properties of a fluid moving at» supersonic speed
past solid surfaces. A particular application of the method
of characteristics permits the solution of the inverse problem
of obtaining the profile of the solid boundary that would
create a desired supersonic flow.

Graphical methods for designing two-dimensional nozzles
by the method of characteristics, for example, are reviewed
in reference 1. Graphical methods employing characteristics
for obtaining nozzles free from waves in the final flow, how-
ever, are tedious and subject to the error inherent in con—
struction involving the plotting of many consecutive lines.

The application of the method of characteristics to the
analytical design of two-dimensional supersonic nozzles was
completed at the N ACA Cleveland laboratory in February
1947. Analytical expressions are obtained for the wall con—
tours of the supersonic part of the two-dimensional nozzle.

A working knowledge of the method of characteristics is
desirable in order to understand and use the nozzle-design
method. For this reason, the form of the method of charac-
teristics most convenient for discussing the method of nozzle
design considered is given in an appendix. A summary of
the design equations and a sample nozzle design are included.