Page 80 - Fluid Mechanics and Thermodynamics of Turbomachinery

Two-dimensional Cascades 61
Using eqn. (3.14) together with eqn. (3.7),

sp 0 cos ˛ m s 3
C D D D cos ˛ m . (3.17)
1 2 l
m
2 c l
With eqn. (3.15)
2
sc .tan ˛ 1 tan ˛ 2 / sec ˛ m sp 0 sin ˛ m
x
C L D
1 2
m
2 c l
s
D 2 cos ˛ m .tan ˛ 1 tan ˛ 2 / C D tan ˛ m . .3.18/
l
Alternatively, employing eqns. (3.9) and (3.17),

s sin 2˛ m
C L D cos ˛ m C f . (3.19)
l 2
Within the normal range of operation in a cascade, values of C D are very much less
than C L .As ˛ m is unlikely to exceed 60 deg, the quantity C D tan ˛ m in eqn. (3.18)
can be dropped, resulting in the approximation,
2
L C L 2 sec ˛ m C f 2
D + .tan ˛ 1 tan ˛ 2 / D sec ˛ m . (3.20)
D C D

Circulation and lift

The lift of a single isolated aerofoil for the ideal case when D D 0 is given by
the Kutta Joukowski theorem

L D c, (3.21)

where c is the relative velocity between the aerofoil and the fluid at infinity and  is
the circulation about the aerofoil. This theorem is of fundamental importance in the
development of the theory of aerofoils (for further information see Glauert (1959).
In the absence of total pressure losses, the lift force per unit span of a blade in
cascade, using eqn. (3.15), is

2
L D sc .tan ˛ 1 tan ˛ 2 / sec ˛ m
x
c y2 /. .3.22/
D sc m .c y1
Now the circulation is the contour integral of velocity around a closed curve. For
the cascade blade the circulation is
c y2 /. (3.23)
 D s.c y1
Combining eqns. (3.22) and (3.23),

L D c m . (3.24)
As the spacing between the cascade blades is increased without limit (i.e.
s !1), the inlet and outlet velocities to the cascade, c 1 and c 2 , becomes equal in