Mechanics of Fluids

The steady-flow energy equation for open channels 417
the mechanical energy divided by weight is represented by the sum of three
terms:
p u2
+ + z
ϱg 2g
Now if the streamlines are sensibly straight and parallel – and even in gradually
varied flow the curvature of streamlines is usually very slight – there is a
hydrostatic variation of pressure over the cross-section. In other words, the
pressure at any point in the stream is governed only by its depth below the
free surface. (Where there is appreciable curvature of the streamlines – as
in rapidly varied flow – there are accelerations perpendicular to them, and
consequently differences of pressure additional to the hydrostatic variation.
Also, if the slope of the channel is exceptionally large, say greater than 1 in
10, there is a modification of the hydrostatic pressure variation even when
the streamlines are straight and parallel. This is because lines perpendicular
to the streamlines – along which the piezometric pressure is constant
(Section 3.6) – cannot then be considered vertical.)
When the pressure variation is hydrostatic, a point at which the (gauge)
pressure is p is at a depth p/ϱg below the surface, and so the sum (p/ϱg) + z
(see Fig. 10.2) represents the height of the surface above datum level. The
expression for the mechanical energy divided by weight is thus simplified to
Height of surface above datum + u 2 /2g (10.1)
We see that the height of the individual streamline above datum has no place
in the expression. If it be further assumed that at the section considered the
velocity is the same along all streamlines, then the expression 10.1 has the
same value for the entire stream.
In practice, however, a uniform distribution of velocity over a section is
never achieved. The actual velocity distribution in an open channel is influenced
both by the solid boundaries (as in closed conduits such as pipes) and
by the free surface. Bends in the channel and irregularities in the boundaries
also have an effect. The irregularities in the boundaries of open channels
are usually so large, and occur in such a random manner, that each channel
has its own peculiar pattern of velocity distribution. Nevertheless, it may in
general be said that the maximum velocity usually occurs at a point slightly
below the free surface (at from 0.05 to 0.25 times the full depth) and that
the average velocity, which is usually of the order of 85% of the velocity at
Fig. 10.2