Mechanics of Fluids

12 Unsteady flow
12.1 INTRODUCTION
Previous chapters of this book have been concerned almost exclusively with
steady flow – that is, flow in which the velocity, pressure, density and so on at
a particular point do not change with time. Admittedly, flow is rarely steady
in the strictest sense of the term: in turbulent flow, for example, countless
small variations of velocity are superimposed on the main velocity. But if the
values of velocity and of other quantities at any particular point are averaged
over a period of time and the resulting mean values are unchanging, then
steadiness of flow may be assumed.
In unsteady or non-steady flow, however, the mean values at a particular
point do vary with time. Such variations add considerably to the difficulties
of solving problems that involve unsteady flow. Indeed, the majority of
such problems are too complex for normal algebraic methods to yield a
solution. Nevertheless, certain problems of unsteady flow are amenable
to analytical solution, and two or three will be briefly considered in this
chapter.
Problems of unsteady flow may be put into one of three broad categories,
according to the rate at which the change occurs. In the first
group are problems in which the changes of mean velocity, although significant,
take place slowly enough for the forces causing the temporal
acceleration to be negligible compared with other forces involved. An
example of this sort of problem is the continuous filling or emptying
of a reservoir, discussed in Section 7.10. The second category embraces
problems in which the flow changes rapidly enough for the forces producing
temporal acceleration to be important: this happens in reciprocating
machinery, such as positive displacement pumps, and in hydraulic and
pneumatic servo-mechanisms. In the third group may be placed those
instances in which the flow is changed so quickly, as for example by
the sudden opening or closing of a valve, that elastic forces become
significant.
Oscillatory motions, in which certain cycles of events are repeated, are
also classified as unsteady. Examples of such motion are tidal movements,
the oscillation of liquids in U-tubes and other vessels, and the vibrations
encountered in acoustics.