Page 104 - Fluid Mechanics and Thermodynamics of Turbomachinery

Two-dimensional Cascades 85
effort of calculating the wall boundary layer development. In initial calculations of
performance it is probably sufficient to use the earlier result of Dunham and Came,
eqn. (3.49), to achieve a reasonably accurate result.
(iii) The tip clearance loss coefficient Y k depends upon the blade loading Z and
the size and nature of the clearance gap k. Dunham and Came presented an amended
version of Ainley’s original result for Y k :
0.78
l k
Y k D B Z (3.51)
H l
where B D 0.5 for a plain tip clearance, 0.25 for shrouded tips.

Reynolds number correction

Ainley and Mathieson (1951) obtained their data for a mean Reynolds number of
5
2 ð 10 based on the mean chord and exit flow conditions from the turbine state.
4
They recommended for lower Reynolds numbers, down to 5 ð 10 , that a correction
be made to stage efficiency according to the rough rule:
.1 tt / / Re 1/5 .
Dunham and Came (1970) gave an optional correction which is applied directly to
the sum of the profile and secondary loss coefficients for a blade row using the
Reynolds number appropriate to that row. The rule is:

Y p C Y s / Re 1/5 .

Flow outlet angle from a turbine cascade
It was pointed out by Ainley (1948) that the method of defining deviation angle
as adopted in several well-known compressor cascade correlations had proved to
be impracticable for turbine blade cascade. In order to predict fluid outlet angle ˛ 2 ,
steam turbine designers had made much use of the simple empirical rule that
˛ 2 D cos 1 /s (3.52a)

where  is the opening at the throat, depicted in Figure 3.26, and s is the pitch. This
widely used rule gives a very good approximation to measured pitchwise averaged
flow angles when the outlet Mach number is at or close to unity. However, at low
Mach numbers substantial variations have been found between the rule and observed
flow angles. Ainley and Mathieson (1951) recommended that for low outlet Mach
numbers 0 <M 2 5 0.5, the following rule be used:

1
˛ 2 D f.cos /s/ C 4s/e (deg) (3.52b)
2
where f.cos 1 /s/ D11.15 C 1.154 cos 1 /s and e D j /.8z/ is the mean
radius of curvature of the blade suction surface between the throat and the trailing
edge. At a gas outlet Mach number of unity Ainley and Mathieson assumed, for a
turbine blade row, that

˛ 2 D cos 1 A t /A n2 (3.52c)