Mechanics of Fluids

456 Flow with a free surface
✷
Hence
h2 =
0.45 m
1.467
= 0.307 m
(d) Q = g 1/2 b2h 3/2
2 = (9.81 m · s−2 ) 1/2 × 0.3 m
× (0.307 m) 3/2 = 0.160 m 3 · s −1 .
10.12 GRADUALLY VARIED FLOW
Uniform flow, which we studied in Sections 10.4 to 10.7, is generally to
be found only in artificial channels because the condition requires a crosssection
constant in shape and area. Consequently the liquid surface must be
parallel to the bed of the channel, and this in turn demands that the slope of
the bed be constant. With a natural stream, such as a river, the shape and
size of cross-section and also the slope of the bed usually vary appreciably,
and true uniform flow is extremely rare. Indeed, even for artificial channels
uniform flow is a condition that is approached asymptotically and so, strictly
speaking, is never attained at all. The equations for uniform flow therefore
give results that are only approximations to the truth when applied to flow
in natural channels and, even so, care should be taken that they are not
applied to long lengths of the channels over which the conditions are not
even approximately constant.
For a particular shape of channel and for a given discharge and bed slope,
there is only one depth at which uniform flow can take place. This depth is
known as the normal depth. There are, however, innumerable ways in which
the same steady rate of flow can pass along the same channel in non-uniform
flow. The liquid surface is then not parallel to the bed, and takes the form
of a curve.
There are, broadly speaking, two kinds of steady, non-uniform flow. In
one the changes of depth and velocity take place over a long distance. Such
flow is termed gradually varied flow. In the other type of non-uniform flow
the changes of depth and velocity take place in only a short distance and
may, in fact, be quite abrupt (as in a hydraulic jump). This local non-uniform
flow is termed rapidly varied flow. There is in practice no rigid dividing line
between these two types, but for the purposes of analysis gradually varied
flow is regarded as that in which the changes occur slowly enough for the
effects of the acceleration of the liquid to be negligible. It is important to
realize this limitation of the analysis: formulae based on the assumption of
gradually varied flow should not be applied to flow in which the changes
take place more rapidly.
Gradually varied flow may result from a change in the geometry of the
channel – for example, a change in the shape of the cross-section, a change
of slope, or an obstruction – or from a change in the frictional forces at the