Mechanics of Fluids

where the hs represent vertical depths of liquid in the channel, and the zs
the heights of the channel bed above datum level. To take account of nonuniformity
of velocity over the cross-section, we may write
h1 + z1 + α1u 2 1 /2g − hf = h2 + z2 + α2u 2 2 /2g (10.3)
The rate at which mechanical energy is lost to friction may be expressed
by h f/l, where l represents the length of channel over which the head loss
h f takes place. This quantity h f/l may be termed the energy gradient since it
corresponds to the slope of a graph of the total mechanical energy divided
by weight plotted against distance along the channel. In the special case
of uniform flow, u1 = u2, α1 = α2 and h1 = h2 in eqn 10.3. Therefore
h f = z1 − z2. The energy gradient is thus the same as the actual, geometrical,
gradient of the channel bed and of the liquid surface. This, it must be
emphasized, is true only for uniform flow in open channels. In discussing
non-uniform flow it is important to take note of the differences between the
energy gradient, the slope of the free surface and the slope of the bed.
10.4 STEADY UNIFORM FLOW – THE CHÉZY EQUATION
Steady uniform flow is the simplest type of open channel flow to analyse,
although in practice it is not of such frequent occurrence as might at first be
supposed. Uniform conditions over a length of the channel are achieved only
if there are no influences to cause a change of depth, there is no alteration of
the cross-section of the stream, and there is no variation in the roughness of
the solid boundaries. Indeed, strictly uniform flow is scarcely ever achieved in
practice, and even approximately uniform conditions are more the exception
than the rule. Nevertheless, when uniform flow is obtained the free surface
is parallel to the bed of the channel (sometimes termed the invert) and the
depth from the surface to the bed is then termed the normal depth.
The basic formula describing uniform flow is due to the French engineer
Antoine de Chézy (1718–98). He deduced the equation from the results of
experiments conducted on canals and on the River Seine in 1769. Here,
however, we shall derive the expression analytically.
In steady uniform (or normal) flow there is no change of momentum, and
thus the net force on the liquid is zero. Figure 10.5 represents a stretch of
a channel in which these conditions are found. The slope of the channel
is constant, the length of channel between the planes 1 and 2 is l and the
(constant) cross-sectional area is A. It is assumed that the stretch of the
channel considered is sufficiently far from the inlet (or from a change of
slope or of other conditions) for the flow pattern to be fully developed.
Now the control volume of liquid between sections 1 and 2 is acted on by
hydrostatic forces F1 and F2 at the ends. However, since the cross-sections
at 1 and 2 are identical, F1 and F2 are equal in magnitude and have the
same line of action; they thus balance and have no effect on the motion of
the liquid. Hydrostatic forces acting on the sides and bottom of the control
volume are perpendicular to the motion, and so they too have no effect.
The only forces we need consider are those due to gravity and the resistance
Steady uniform flow – the Chézy equation 419