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National Advisory Committee for Aeronautics, Technical Notes - Analysis of an Axial Compressor Stage with Infinitesimal and Finite Blade Spacing

A method of designing circular blade systems of finite spacing is
developed.

First, the theory of a flow through a system of infinitesimally
spaced surfaces is formulated by means of a continuous axially symmetric
force field which is uniform in the circumferential direction. This
force field replaces the effect of the blade system, with its hub and
shroud boundary surfaces.

Second, the force field in the space between_the blades, the hub,
and the shroud is replaced in the equations_of finite spacing by those
inertia and pressure terms which were omitted in the equations of.
infinitesimal spacing. These terms_will change the values of the flow
variables of infinitesimal spacing.

In order to determine these changes, a series development is used
for the velocity components and pressure functions in powers of a param-
eter function t. The series development in powers of I leads to
equations of first, second, and higher orders for the determination of
increments which correct the velocity and pressure functions. In this
report only the equations of first and second order are derived explic-
itly. The solution of first order shows the important result that, at
the entrance and the exit of the compressor stage, the force vector used
for the infinitesimal spacing or, what is equivalent, the increments of
odd order must be zero at entrance and exit. This is necessary in order
to prevent discontinuity of pressure and velocity at inflow and outflow.
This conclusion is taken into account in a detailed determination of the
flow variables of infinitesimal spacing. In this way it is insured that
in finite spacing no discontinuity appears, neither for the pressures
nor for the velocities, except across the blade surfaces.

The fellowing steps are covered in the present paper in the develop-
ment of a.method of designing circular blade systems of finite spacing.
First, the theory of a flow through a system of infinitesimally spaced
surfaces is formulated by means of a continuous axially symmetric force
field which is uniform in the circumferential_direction. It replaces
the effect of the blade, hub, and shroud boundary surfaces. Second, the
force field in the space between the blades, the hub, and the shroud is
replaced in the equations of finite spacing by those inertia and pressure
terms which were omitted in the equations of infinitesimal spacing.