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National Advisory Committee for Aeronautics, Technical Notes - A Method of Solving the Direct and Inverse Problem of Supersonic Flow Along Arbitrary Stream Filaments of Revolution in Turbomachines

Analysis of the supersonic flow in two two-dimensional high—
solidity cascades and in a partly supersonic symmetrical nozzle shows
that there is, in general, significant deviation of the mean streamline
shape-from that of the mean blade line and that the effect of blade
thickness and blade curvature on the specific mass flow along the mean
streamline is to increase the specific mass flow along the mean stream-
line about 9 percent above that given by a one-dimensional estimate.

In order to determine these effects more accurately for turbomachines
of arbitrary hub and casing shapes to be used for the three—dimensional
through-flow calculation, a method is developed for the determination of
the supersonic flow along stream surfaces of revolution in turbomachines.
In this method, the shapes of the stream surfaces are arbitrary, and the
method also takes into account the distance between adjacent stream
surfaces, which varies along the flow path. Thus, the method can be
applied to turbomachines with arbitrary hub and casing shapes.

In addition to their use for direct problems, these equations can
be used to design blade elements in supersonic flow along an arbitrary
stream filament of revolution in turbomachines.

Recent investigations of the applicability of supersonic flow in
compressors have shown the desirability of using such flow to increase
the pressure ratio per stage (references 1 and 2). For axial-flow
compressors having blades with short radial length, the over-all
performance analysis is often based on a one-dimensional approximation,
in which only average values in the channel are considered (for example,
references 5 and 4). For radially long blades and variable root or tip
radii, methods are proposed for analyzing the flow by a three-
dimensional "through—flow" calculation, which considers the axial and
radial variations of the flow but only a mean value in the circumferen-
tial direction (references 5 and 6). The flow on the relative mean
stream surface, which divides circumferentially the mass flow passing
through the_channel between two blades into two equal parts, is taken
to represent the mean flow through the blading. It is suggested in
reference 5 that the shape of this mean stream surface and the correction
factor b for a finite number of thick blades be obtained from the
analysis of a number of two-dimensional flows along general surfaces of
revolution starting at different inlet radii. A method was therefore
developed at the NACA Lewis laboratory to determine the flow variation on
these flow surfaces in a supersonic turbomachine. When the flow surfaces
may be approximated by cylindrical surfaces, the flow equations reduce to
the usual plane flow where the hodograph characteristics are applicable
for the flow analysis if the entrance shock is weak.