Page 188 - Fluid Mechanics and Thermodynamics of Turbomachinery

CHAPTER 6

Three-dimensional Flows in

Axial Turbomachines

It cost much labour and many days before all these things were brought to
perfection. (DEFOE, Robinson Crusoe.)

Introduction
IN CHAPTERS 4 and 5 the fluid motion through the blade rows of axial turboma-
chines was assumed to be two-dimensional in the sense that radial (i.e. spanwise)
velocities did not exist. This is a not unreasonable assumption for axial turboma-
chines of high hub tip ratio. However, with hub tip ratios less than about 4/5, radial
velocities through a blade row may become appreciable, the consequent redistribu-
tion of mass flow (with respect to radius) seriously affecting the outlet velocity
profile (and flow angle distribution). It is the temporary imbalance between the
strong centrifugal forces exerted on the fluid and radial pressures restoring equi-
librium which is responsible for these radial flows. Thus, to an observer travelling
with a fluid particle, radial motion will continue until sufficient fluid is transported
(radially) to change the pressure distribution to that necessary for equilibrium. The
flow in an annular passage in which there is no radial component of velocity,
whose streamlines lie in circular, cylindrical surfaces and which is axisymmetric, is
commonly known as radial equilibrium flow.
An analysis called the radial equilibrium method, widely used for three-
dimensional design calculations in axial compressors and turbines, is based upon
the assumption that any radial flow which may occur, is completed within a blade
row, the flow outside the row then being in radial equilibrium. Figure 6.1 illustrates
the nature of this assumption. The other assumption that the flow is axisymmetric
implies that the effect of the discrete blades is not transmitted to the flow.

Theory of radial equilibrium

Consider a small element of fluid of mass dm, shown in Figure 6.2, of unit depth
and subtending an angle d at the axis, rotating about the axis with tangential
velocity, c at radius r. The element is in radial equilibrium so that the pressure
forces balance the centrifugal forces;

2
1
.p C dp/.r C dr/d prd .p C dp/drd D dmc /r.
2
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