Mechanics of Fluids

friction, the steady-flow momentum equation is:
ϱgh1
2 h1 − F − ϱgh2
2 h2 = ϱq (u2 − u1) (10.28)
for unit width of a uniform rectangular channel.
This equation is true whatever the values of h1 and h2 in relation to the
critical depth: both h1 and h2 may be greater than hc, both may be less than
hc or one may be greater and one less than hc. It may, however, be shown
that if the initial flow is tranquil the effect of the applied force is to reduce
the depth of the stream (so that h2 < h1), although a limit is set at the critical
depth hc because the specific energy is then a minimum. A further increase
in the obstructing force F beyond the value giving h2 = hc merely raises the
upstream level. If, on the other hand, the initial flow is rapid, the depth is
increased by application of the force (i.e. h2 > h1). Indeed, if the force is
large enough – because, for example, the obstacle is large compared with the
cross-sectional area of the channel – the depth may be increased beyond the
critical value via a hydraulic jump. In all these cases eqn 10.28 holds and
the force F may thus be calculated.
10.11 THE OCCURRENCE OF CRITICAL CONDITIONS
In analysing problems of flow in open channels it is important to know at
the outset whether critical flow occurs anywhere and, if so, at which section
it is to be found, especially because these conditions impose a limitation on
the discharge (as indicated by Fig. 10.19).
Critical conditions are of course to be expected at a section where tranquil
flow changes to rapid flow. Such a situation is illustrated in Fig. 10.26. A long
channel of mild slope (i.e. s < sc) is connected to a long channel of steep
slope (i.e. s > sc). (The slopes are greatly exaggerated in the diagram.) At a
sufficiently large distance from the junction the depth in each channel is the
normal depth corresponding to the particular slope and rate of flow; that
is, in the channel of mild slope there is uniform tranquil flow, and in the
other channel there is uniform rapid flow. Between these two stretches of
uniform flow the flow is non-uniform, and at one position the depth must
pass through the critical value as defined, for example by eqn 10.19. This
The occurrence of critical conditions 443
Fig. 10.26