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National Advisory Committee for Aeronautics, Technical Notes - Three Dimensional Supersonic Nozzles and Inlets of Arbitrary Exit Cross Section

Because of the great complexity of the method of characteristics
in three dimensions (see, for example, reierences l and 2), the accurate
theoretical design of general three—dimensional supersonic nozzles and
diffusers has not been feasible. The designer is thus limited to two—
dimensional and axisymmetric flows. In many cases this may be a serious
disadvantage. In the case, particularly, of a hypersonic wind tunnel,
it would be very helpful if a three-dimensional expansion could be
used to avoid the usual thin slit which would appear as the throat of
such a nozzle if it were two dimensional. Furthermore, such a nozzle
might have better secondary flow characteristics than the corresponding
two—dimensional nozzle in which transverse pressure gradients may cause
cross flow in, and local buildrup of, the boundary layer. However, the
usual three-dimensional nozzle is axially symmetric and thus precludes
the use of flat schlieren windows. In the design of supersonic inlets,
considerations of space, as well as of permissible flow turning, make
the use of shapes other than two-dimensional or axisymmetric ones
desirable.

A simple method for obtaining three-dimensional nozzles from
axisymmetrical ones has been developed at the NASA Lewis laboratory
and is presented herein. The procedure is exact within the limitations
of the method of characteristics.

In any inviscid fluid flow, the streamsheets, by definition, form
surfaces across which there is no flow and hence may be replaced by
solid boundaries. This fact will herein be applied to find unsymmetric
nozzles and inlets from axially symmetric flows. Two salient features
of an axisymmetric flow are that the streamlines lie in planes through
the streamwise aids and that the flow in any one such plane is the
same as that in any other.

In order to find the nozzle of a desired shape, the axisymmetric
nozzle having the desired length and Mach number characteristics is
computed and then, choosing the desired cross section at some station,
for example, the exit, the streamlines which pass through the periphery
of that section are found. The streamlines can be determined either
by constructing them from the known local flow directions or by the
better method of integrating the aflsymmetric stream function (appendix).
The streamsheet formed by these streamlines will constitute the walls
of the desired inlet or nozzle.