Mechanics of Fluids

152 The momentum equation
for the frictionless fluid assumed, shear forces are absent. Consequently the
only net force F on the fluid in the axial direction is that produced by the
actuator disc. Therefore, for steady flow,
F = ϱQ (u4 − u1) (4.10)
This is equal in magnitude to the net force on the disc. Since there is no
change of velocity across the disc this force is given by (p3 − p2)A. Equating
this to eqn 4.10 and putting Q = Au2, where A represents the cross-sectional
area of the disc, we obtain
p3 − p2 = ϱu2 (u4 − u1) (4.11)
Applying Bernoulli’s equation between sections 1 and 2 gives
p1 + 1
2 ϱu2 1 = p2 + 1
2 ϱu2 2
(4.12)
the axis being assumed horizontal for simplicity. Similarly, between
sections 3 and 4:
p3 + 1
2 ϱu2 3 = p4 + 1
2 ϱu2 4
(4.13)
Now u2 = u3 and also p1 = p4 = pressure of undisturbed fluid. Therefore,
adding eqns 4.12 and 4.13 and rearranging gives
p3 − p2 = 1
2ϱ �
u 2 4 − u2 �
1
Eliminating p3 − p2 from eqns 4.11 and 4.14 we obtain
u2 = u1 + u4
2
(4.14)
(4.15)
The velocity through the disc is the arithmetic mean of the upstream and
downstream velocities; in other words, half the change of velocity occurs
before the disc and half after it (as shown in Fig. 4.9). This result, known
as Froude’s theorem after William Froude (1810–79), is one of the principal
assumptions in propeller design.
If the undisturbed fluid be considered stationary, the propeller advances
through it at velocity u1. The rate at which useful work is done by the
propeller is given by the product of the thrust and the velocity:
Power output = Fu1 = ϱQ (u4 − u1) u1
(4.16)
In addition to the useful work, kinetic energy is given to the slipstream which
is wasted. Consequently the power input is
ϱQ (u4 − u1) u1 + 1
2 ϱQ (u4 − u1) 2
(4.17)