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National Advisory Committee for Aeronautics, Technical Notes - An Investigation by the Hodograph Method of Flow Through a Symmetrical Nozzle with Locally Supersonic Regions

The flow of a compressible fluid through a channel having locally
supersonic regions is studied by using the Tricomi equation in the hodo-
graph variables as an approximation in the sonic region to the equatiOn
of flow of an irrotational, inviscid gas. It is shown that this is
equivalent to studying the flow of a gas having a pressure-density rela-
tion matching the isentropic relation to the third derivative at the
sonic point. A one—parameter family of solutions of the Tricomi equa-
tion is used which provides symmetrical accelerated-decelerated flows.
The variation of this parameter alters the Mach number at the center of
the throat, the velocity distribution and gradient along the center.
streamline, as well as the shape of the channel, that is, the curvature
of the bounding streamline.

As specific examples, flows are computed having Mach number equal
to unity and to 0.86 at the center of the throat section. Constant-
velocity lines are plotted and it is found that the velocity gradient
becomes zero at three places along each streamline outside of'a limiting
streamline for'values of the parameter greater than zero (M < l at
center of throat section). For the phrameter equal to zero (M = l at
center of throat section), the velocity gradient alOng the streamlines
and the curvature is discontinuous at all points of the two character—
istics which meet the center streamline.

Other solutions to the Tricomi equation are discussed which may:be
used to formulate channel flows. The exact nature of these flows has
not yet been investigated.

In the investigations of the effect of free~stream Mach number on
the lift and drag of an airfoil in a uniform stream, it was found that
the lift did not begin to decrease or the drag increase until the free-
stream velocity exceeded the critical free-stream velocity, that is,
the velocity for which the local velocity of sound is just attained at
some point of the body. Thus it appeared that, for free-stream velocities
in between the critical free-stream velocity and the velocity for which
the lift characteristics changed rapidLy with Mach number, there might
exist about the body a locally supersonic region which contained no shocks,
although such shock-free flows have never been Observed experimentally.
A similar phenomenon occurs in the study of flows through a gradual
constriction. When the exit pressure of a nozzle is decreased, the
mean velocity at the narrowest cross section and also the total mass
flow are increased.