Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

62 Fluid Mechanics, Thermodynamics of Turbomachinery
With eqn. (3.15)
Alternatively, employing eqns. (3.9) and (3.17),
(3.17)
(3.18)
(3.19)
Within the normal range of operation in a cascade, values of CD are very much less
than CL. As am is unlikely to exceed 60deg, the quantity CD tan am in eqn. (3.18) can
be dropped, resulting in the approximation
Circulation and lift
(3.20)
The lift of a single isolated aerofoil for the ideal case when D = 0 is given by the
Kutta–Joukowski theorem
(3.21)
where c is the relative velocity between the aerofoil and the fluid at infinity and G 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
(3.22)
Now the circulation is the contour integral of velocity around a closed curve. For the
cascade blade the circulation is
Combining eqns. (3.22) and (3.23),
(3.23)
(3.24)
As the spacing between the cascade blades is increased without limit (i.e. s Æ•),
the inlet and outlet velocities to the cascade, c1 and c2, becomes equal in magnitude and
direction. Thus c1 = c2 = c and eqn. (3.24) become identical with the Kutta–Joukowski
theorem obtained for an isolated aerofoil.