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National Advisory Committee for Aeronautics, Technical Notes - The Aerodynamic Design of High Mach Number Nozzles Utilizing Axisymmetric Flow with Application to a Nozzle of Square Test Section

A method is given for the design of three—dimensional nozzles uti-
lizing axisymmetric flow. The nozzle can be designed to produce uniform
flow in a test chamber of arbitrary cross section. The method is applied
to obtain the final coordinates of a Mach number 10 nozzle for which a
square test section is specified to reduce the possibility of axisym—
metric imperfections at the wall and to provide for the installation of
schlieren windows.

Radial flow is used in a portion of the flow field to reduce
the computation time. The remainder of the flow field is computed
by the method of characteristics, but a simplified method is used
near the axis. Tables which facilitate computation of the radial
flow and the flow near the axis are included. Transition streamlines
determined from the analytic expressions of Kuno Foelsch are compared
with the streamlines obtained from the characteristics net of the Mach
number 10 nozzle. The Foelsch streamlines deviate from the flow—net
streamlines by as much as 12 percent. Similar analytic expressions are
derived from the geometric properties'of the flow. These new expressions
result in transition streamlines with a maximum error of about h percent.

Recent research has indicated that three-dimensional supersonic
nozzles may become more desirable for high Mach number tunnels than
conventional two-dimensional nozzles. Fbr example, in twofdimensional
nozzles designed for test Mach numbers much greater than 5, the flow is
very sensitive to any change of the extremely small dimensions at the
minimum section. The high temperatures required to avoid liquefaction
of the air at these high Mach numbers make the problem of obtaining
dimensional stability of the small slit-like minimum extremely difficult.
(See reference 1.) In addition, the excess growth of boundary layer
along the center of the nozzle side plates may also interfere with the
design flow. A two—stage nozzle discussed in reference 2 avoids the
first difficulty but fails to operate satisfactorily primarily because
of the excess boundary layer. Consideration of three—dimensional nozzles
thus becomes imperative not only for tests of stationary models at the
higher Mach numbers but also for ballistic tests where the projectile
must pass through the minimum of a supersonic nozzle.