Page 103 - Fluid Mechanics and Thermodynamics of Turbomachinery

84 Fluid Mechanics, Thermodynamics of Turbomachinery
t/l either 0.25 or 0.15. By substituting ˛ 1 D ˛ 2 and t/l D 0.2 in eqn. (3.46), the
zero incidence loss coefficient for the impulse blades Y p.˛1D˛2/ given in Figure 3.25
is recovered. Similarly, with ˛ 1 D 0at t/l D 0.2 in eqn. (3.46) gives Y p.˛1D0/ of
Figure 3.25.
A feature of the losses given in Figure 3.25 is that, compared with the impulse
blades, the nozzle blades have a much lower loss coefficient. This trend confirms
the results shown in Figure 3.14, that flow in which the mean pressure is falling
has a lower loss coefficient than a flow in which the mean pressure is constant
or increasing.
(ii) The secondary losses arise from complex three-dimensional flows set up as
a result of the end wall boundary layers passing through the cascade. There is
substantial evidence that the end wall boundary layers are convected inwards along
the suction-surface of the blades as the main flow passes through the blade row,
resulting in a serious mal-distribution of the flow, with losses in stagnation pressure
often a significant fraction of the total loss. Ainley found that secondary losses could
be represented by

2
C Ds D C L /.s/l/ (3.47)
where is parameter which is a function of the flow acceleration through the blade
row. From eqn. (3.17), together with the definition of Y, eqn. (3.45) for incompress-
2
3
ible flow, C D D Y.s/l/ cos ˛ m / cos ˛ 2 , hence
2 2 2
C Ds cos ˛ 2 C L cos ˛ 2
Y s D D D Z (3.48)
3 3
.s/l/ cos ˛ m s/l cos ˛ m
where Z is the blade aerodynamic loading coefficient. Dunham (1970) subsequently
found that this equation was not correct for blades of low aspect ratio, as in small
turbines. He modified Ainley’s result to include a better correlation with aspect ratio
and at the same time simplified the flow acceleration parameter. The correlation,
given by Dunham and Came (1970), is

l cos ˛ 2
Y s D 0.0334 Z (3.49)
H cos ˛ 1 0
and this represents a significant improvement in the prediction of secondary losses
using Ainley’s method.
Recently, more advanced methods of predicting losses in turbine blade rows have
been suggested which take into account the thickness of the entering boundary layers
on the annulus walls. Came (1973) measured the secondary flow losses on one end
wall of several turbine cascades for various thicknesses of inlet boundary layer. He
correlated his own results, and those of several other investigators, and obtained a
modified form of Dunham’s earlier result, viz.,
2

cos ˛ 1 l cos ˛ 2
Y s D 0.25Y 1 C 0.009 Z Y 1 (3.50)
2 0
cos ˛ 2 H cos ˛ 1
which is the net secondary loss coefficient for one end wall only and where Y 1 is
a mass-averaged inlet boundary layer total pressure loss coefficient. It is evident
that the increased accuracy obtained by use of eqn. (3.50) requires the additional