Mechanics of Fluids

566 Unsteady flow
Although the fluid is at rest it is not in equilibrium. Let us assume that
the reservoir is very large in relation to the cross-sectional area of the pipe,
so that the velocity in the reservoir is always negligible and the pressure
therefore constant. There is now a discontinuity between the pipe and the
reservoir: the fluid in the pipe is at the increased pressure, whereas that in the
reservoir is at the original pressure and its velocity is already zero. As a result
of the discontinuity of pressure, fluid begins to flow from the pipe back into
the reservoir so as to equalize the pressures at that end of the pipe. In other
words, the discontinuity that constitutes the pressure wave now moves back
towards the valve. Because of the relation between the changes of pressure
and velocity (eqn 12.5), the velocity with which the fluid now moves towards
the reservoir is of magnitude u. The result is equivalent to the superposition
of a negative or unloading wave on the original positive one so as to nullify
it. In other words, the reflection of a wave at a completely open end (i.e. an
end connected to a reservoir of infinite extent) gives a second wave, equal in
magnitude to the first but opposite in sign. (See fourth pressure diagram in
Fig. 12.4.)
The reflected wave travels the length of the pipe in a time l/c, so it reaches
the valve at a time 2l/c after the closing of the valve. The pressure has now
been reduced everywhere to its original value, the fluid density and the pipe
diameter are back to their original values – but all the fluid is moving back
from the valve. This reverse movement decompresses the fluid immediately
next to the valve, with the result that there is a fall in pressure. The magnitude
of this pressure change is – ideally – the same as the magnitude of the
previous changes because the corresponding change of velocity is again u.
Therefore a negative wave is now propagated from the valve to the reservoir.
This illustrates the nature of a reflection from a completely closed end:
the magnitude of the reflected wave equals that of the incident wave, and
the sign remains unchanged. When, after the next time interval of l/c, the
negative wave reaches the open, reservoir, end there is an unbalanced state,
with a higher pressure in the reservoir than in the pipe. Therefore fluid flows
from the reservoir into the pipe; in other words, reflection of the wave takes
place with a change of sign so that the existing negative wave is nullified
by a positive wave. This positive wave is propagated towards the valve and
reaches it at the end of the fourth time interval of l/c.
Conditions have now been reached that are identical with those existing at
the moment the valve was closed: all the fluid in the pipe is moving towards
the valve with velocity u, and the pressure is back to the original value. The
complete cycle of events (as illustrated in Fig. 12.4) is therefore repeated
and, in the absence of friction, would be repeated indefinitely, each cycle
occupying a period of time 4l/c. In practice, energy is gradually dissipated
by friction and imperfect elasticity, so the waves diminish in intensity and
die away.
The time needed for a pressure wave to make the round trip from valve to
reservoir and back again is 2l/c. Thus, for an instantaneous closing of the
valve, the excess pressure at a point immediately before the valve remains
constant for a time interval 2l/c; the pressure is then altered by the arrival of
the negative, unloading, wave. Similarly, the subsequent drop of pressure