Page 209 - Fluid Mechanics and Thermodynamics of Turbomachinery

190 Fluid Mechanics, Thermodynamics of Turbomachinery
denoted by  0 and at position .x, r A / by . The important result of actuator disc
theory is that velocity perturbations decay exponentially away from the disc. This
is also true for the upstream flow field .x 5 0). The result obtained for the decay
rate is
/ 0 D 1 exp[Ý x/.r t r h /], (6.42)

where the minus and plus signs above apply to the flow regions x = 0 and
x 5 0 respectively. Equation (6.42) is often called the settling-rate rule. Since
1
c x1 D c x01 C , c x2 D c x02  and noting that  0 D .c x11 c x12 /, eqns. (6.41)
2
and (6.42) combine to give,
1
c x1 D c x11 .c x11 c x12 / exp[ x/.r t r h /], .6.43a/
2
1
c x2 D c x12 C .c x11 c x12 / exp[ x/.r t r h /]. .6.43b/
2
At the disc, x D 0, eqns. (6.43) reduce to eqn. (6.41). It is of particular interest to
note, in Figures 6.9 and 6.10, how closely isolated actuator disc theory compares
with experimentally derived results.

Blade row interaction effects

The spacing between consecutive blade rows in axial turbomachines is usually
sufficiently small for mutual flow interactions to occur between the rows. This
interference may be calculated by an extension of the results obtained from isolated
actuator disc theory. As an illustration, the simplest case of two actuator discs
situated a distance υ apart from one another is considered. The extension to the case
of a large number of discs is given in Hawthorne and Harlock (1962).
Consider each disc in turn as though it were in isolation. Referring to Figure 6.13,
disc A, located at x D 0, changes the far upstream velocity c x11 to c x12 far down-
stream. Let us suppose for simplicity that the effect of disc B, located at x D υ,
exactly cancels the effect of disc A (i.e. the velocity far upstream of disc B is c x12
which changes to c x11 far downstream). Thus, for disc A in isolation,

1 jxj
c x D c x11 .c x11 c x12 / exp , x 5 0, .6.44/
2 H

1 jxj
c x D c x12 C .c x11 c x12 / exp , x = 0, .6.45/
2 H
where jxj denotes modulus of x and H D r t r h .
For disc B in isolation,

1 jx υj
c x D c x12 .c x12 c x11 / exp , x 5 υ, .6.46/
2 H

1 jx υj
c x D c x11 C .c x12 c x11 / exp , x = υ. .6.47/
2 H
Now the combined effect of the two discs is most easily obtained by extracting
from the above four equations the velocity perturbations appropriate to a given