Page 78 - Fluid Mechanics and Thermodynamics of Turbomachinery

Two-dimensional Cascades 59
Energy losses

A real fluid crossing the cascade experiences a loss in total pressure p 0 due to
skin friction and related effects. Thus
1
p 0 p 1 p 2 2 2
D C .c 1 c /. (3.4)
2
2
2
2
Noting that c 2 c D .c 2 C c / .c 2 2 c y2 /, substitute
x
1 2 y1 x y2 C c / D .c y1 C c y2 /.c y1
eqns. (3.2) and (3.3) into eqn. (3.4) to derive the relation,
p 0 1
D . X C Y tan ˛ m /, (3.5)
s
where
1
tan ˛ m D .tan ˛ 1 C tan ˛ 2 /. (3.6)
2
A non-dimensional form of eqn. (3.5) is often useful in presenting the results of
cascade tests. Several forms of total pressure-loss coefficient can be defined of
which the most popular are,
2
1
D p 0 /. c / (3.7a)
2 x
and
1
2
ω D p 0 /. c /. (3.7b)
2 1
Using again the same reference parameter, a pressure rise coefficient C p and a
tangential force coefficient C f may be defined
X
p 2 p 1
C p D D , .3.8/
1 c 2 1 sc 2
2 x 2 x
Y
C f D D 2.tan ˛ 1 tan ˛ 2 /, .3.9/
1 sc 2
2 x
using eqns. (3.2) and (3.3a).
Substituting these coefficients into eqn. (3.5) to give, after some rearrangement,

. (3.10)
C p D C f tan ˛ m
Lift and drag

A mean velocity c m is defined as
c m D c x / cos ˛ m , (3.11)

where ˛ m is itself defined by eqn. (3.6). Considering unit depth of a cascade blade,
a lift force L acts in a direction perpendicular to c m and a drag force D in a direction
parallel to c m . Figure 3.4 shows L and D as the reaction forces exerted by the blade
upon the fluid.