Mechanics of Fluids

564 Unsteady flow
celerity is sometimes used rather than velocity. It represents the rate at which
a message can be telegraphed through the fluid. As another example of a
message being sent through the fluid we may consider a pump delivering
fluid through a long pipe-line into a reservoir. If the power to the pump
suddenly fails, fluid continues to be discharged from the outlet end of the
pipe until the pressure wave produced by the stopping of the pump reaches
there: the pressure wave – this time a negative one – provides the only means
by which the fluid at the outlet end ‘knows’ that the input has failed.
The magnitude of the rise in pressure caused by reducing the velocity from
u to zero is given by eqn 12.5 as ϱcu. In a more general case the velocity would
be changed form u1 to u2. The analysis, however, would be modified only
in so far as the zero of the velocity scale would be altered; the corresponding
rise in pressure would be given by ϱc(u1 − u2) This may be expressed as a
change of head by dividing by ϱg (the change of ϱ, we recall, is small):
h2 − h1 = c
g (u1 − u2) i.e. �h =− c
�u (12.9)
g
Substituting in eqn 12.9 values appropriate for water shows that a reduction
of velocity of 3 m · s −1 corresponds to an increase of head of about 440 m
(about 4.3 MPa). Such an increase is too large to be neglected in the design
of a pipe system. The equation shows too that, just as a sudden reduction of
velocity gives rise to an increase of pressure, so a sudden increase of velocity
causes a reduction of pressure. It will be noticed also that the change of head
is independent of the length of the pipe – in distinction to eqn 12.1 for the
inertia head in an incompressible fluid.
The kinetic energy lost by the liquid when its velocity is reduced is converted
entirely into strain energy by the compression of the liquid itself and the
stretching of the pipe (if there is no friction loss). No mechanical energy has
been lost: it is merely stored in the compressed material and may be released
when the strain is removed.
Not all pressure waves result from sudden changes of velocity or pressure.
A gradual change, however, may be regarded as the sum of a series of
small instantaneous changes. Thus the transmission of such a gradual pressure
wave along a pipe is similar to that of a sudden wave. Consequently,
for a pipe-line of unlimited length, any particular pattern of change at one
point will be reproduced at another point; there is, however, a time delay
corresponding to the time that the pressure wave takes to travel the distance
between the two points.
12.3.2 The reflection of waves
The problem is somewhat complicated if the pipe-line is not of unlimited
length. This is because a pressure wave, on reaching the end of the pipe,
is reflected; in other words, another wave is produced which returns along
the pipe to the starting point of the first wave. This reflected wave may
be larger or smaller than the first, according to the conditions at the end
where the reflection takes place. The simplest case is that for the horizontal
pipe illustrated in Fig. 12.4. Let us assume that flow towards the valve is
stopped by the instantaneous closing of the valve. Frictional effects are for