Mechanics of Fluids

568 Unsteady flow
Fig. 12.6 Effect of friction.
Variation of pressure next
to the valve with time after
complete valve closure.
Theoretical (no friction)
line shown dotted.
of this motion may then produce a positive pressure greater than that caused
by the initial valve closure. It is therefore important to avoid, if possible,
the vaporization of the liquid (or the release of substantial amounts of air or
other gases from solution).
The effects of friction losses are indicated in Fig. 12.6, which shows the
perhaps surprising fact that a greater rise of pressure may occur with friction
than without it. When the velocity of the fluid is reduced, so is the head
lost to friction; the head available at the downstream end of the pipe consequently
rises somewhat as layer after layer of the fluid is slowed down. This
secondary effect is transmitted back from each layer in turn with celerity c,
and so the full effect is not felt at the valve until a time 2l/c after its closure.
In Fig. 12.6 this effect is indicated by the upward slope of the line ab. During
the second time interval of 2l/c velocities and pressure amplitudes have
reversed signs, and thus the line de slopes slightly downwards. However,
energy is also dissipated by viscous forces during the small movements of
individual particles as the fluid is compressed and expanded. This dissipation
of energy, known as damping, always tends to reduce the amplitude of
the pressure waves and so bring ab and de nearer to the horizontal. Indeed,
the effect of the damping may sometimes exceed that of the pipe friction, and
so the lines ab and de may even converge towards the equilibrium pressure
line (which, for complete closure of the valve, corresponds to the static pressure
line). Friction forces also oppose the flow of the fluid back towards the
reservoir, so the velocity of this flow is somewhat less in magnitude than the
original. Consequently the amplitude of a wave is reduced at each reflection
until the final equilibrium pressure is reached. Except in pipes of very small
diameter, friction does not appreciably affect the celerity with which waves
are propagated.
It is difficult in calculations to account accurately for friction. The neglect
of friction is often justified because the friction head is small compared
with the head produced by the water hammer. In all cases, however, values
‘on the safe side’ are obtained by assuming that the initial head at the valve
is the same as the head in the reservoir, and neglecting subsequent frictional
effects.