§ 95. _Flow of Water in Open Canals and Rivers._--When water flows in
a pipe the section at any point is determined by the form of the
boundary. When it flows in an open channel with free upper surface,
the section depends on the velocity due to the dynamical conditions.
Suppose water admitted to an unfilled canal. The channel will
gradually fill, the section and velocity at each point gradually
changing. But if the inflow to the canal at its head is constant, the
increase of cross section and diminution of velocity at each point
attain after a time a limit. Thenceforward the section and velocity at
each point are constant, and the motion is steady, or permanent regime
is established.
If when the motion is steady the sections of the stream are all equal,
the motion is uniform. By hypothesis, the inflow [Omega]v is constant
for all sections, and [Omega] is constant; therefore v must be
constant also from section to section. The case is then one of uniform
steady motion. In most artificial channels the form of section is
constant, and the bed has a uniform slope. In that case the motion is
uniform, the depth is constant, and the stream surface is parallel to
the bed. If when steady motion is established the sections are
unequal, the motion is steady motion with varying velocity from
section to section. Ordinary rivers are in this condition, especially
where the flow is modified by weirs or obstructions. Short
unobstructed lengths of a river may be treated as of uniform section
without great error, the mean section in the length being put for the
actual sections.
In all actual streams the different fluid filaments have different
velocities, those near the surface and centre moving faster than those
near the bottom and sides. The ordinary formulae for the flow of
streams rest on a hypothesis that this variation of velocity may be
neglected, and that all the filaments may be treated as having a
common velocity equal to the mean velocity of the stream. On this
hypothesis, a plane layer abab (fig. 102) between sections normal to
the direction of motion is treated as sliding down the channel to
a´a´b´b´ without deformation. The component of the weight parallel to
the channel bed balances the friction against the channel, and in
estimating the friction the velocity of rubbing is taken to be the
mean velocity of the stream. In actual streams, however, the velocity
of rubbing on which the friction depends is not the mean velocity of
the stream, and is not in any simple relation with it, for channels of
different forms. The theory is therefore obviously based on an
imperfect hypothesis. However, by taking variable values for the
coefficient of friction, the errors of the ordinary formulae are to a
great extent neutralized, and they may be used without leading to
practical errors. Formulae have been obtained based on less restricted
hypotheses, but at present they are not practically so reliable, and
are more complicated than the formulae obtained in the manner
described above.
[Illustration: FIG. 102.]
§ 96. _Steady Flow of Water with Uniform Velocity in Channels of
Constant Section._--Let aa´, bb´ (fig. 103) be two cross sections
normal to the direction of motion at a distance dl. Since the mass
aa´bb´ moves uniformly, the external forces acting on it are in
equilibrium. Let [Omega] be the area of the cross sections, [chi] the
wetted perimeter, pq + qr + rs, of a section. Then the quantity m =
[Omega]/[chi] is termed the hydraulic mean depth of the section. Let v
be the mean velocity of the stream, which is taken as the common
velocity of all the particles, i, the slope or fall of the stream in
feet, per foot, being the ratio bc/ab.
[Illustration: FIG. 103.]
The external forces acting on aa´bb´ parallel to the direction of
motion are three:--(a) The pressures on aa´ and bb´, which are equal
and opposite since the sections are equal and similar, and the mean
pressures on each are the same. (b) The component of the weight W of
the mass in the direction of motion, acting at its centre of gravity
g. The weight of the mass aa´bb´ is G[Omega]dl, and the component of
the weight in the direction of motion is G[Omega]dl × the cosine of
the angle between Wg and ab, that is, G[Omega]dl cos abc = G[Omega]dl
bc/ab = G[Omega]idl. (c) There is the friction of the stream on the
sides and bottom of the channel. This is proportional to the area
[chi]dl of rubbing surface and to a function of the velocity which may
be written f(v); f(v) being the friction per sq. ft. at a velocity v.
Hence the friction is -[chi]dl f(v). Equating the sum of the forces to
zero,
G[Omega]i dl - [chi]dl f(v) = 0,
f(v)/G = [Omega]i/[chi] = mi. (1)
But it has been already shown (§ 66) that f(v) = [zeta]Gv²/2g,
.: [zeta]v²/2g = mi. (2)