Chapter X: Flow in Rivers and Canals (1)

§ 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.

§ 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.

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)

This may be put in the form

v = [root](2g/[zeta]) [root](mi) = c [root](mi); (2a)

where c is a coefficient depending on the roughness and form of the
channel.

The coefficient of friction [zeta] varies greatly with the degree of
roughness of the channel sides, and somewhat also with the velocity.
It must also be made to depend on the absolute dimensions of the
section, to eliminate the error of neglecting the variations of
velocity in the cross section. A common mean value assumed for [zeta]
is 0.00757. The range of values will be discussed presently.

It is often convenient to estimate the fall of the stream in feet per
mile, instead of in feet per foot. If f is the fall in feet per mile,

f = 5280 i.

Putting this and the above value of [zeta] in (2a), we get the very
simple and long-known approximate formula for the mean velocity of a
stream--

v = ¼ ½ [root](2mf). (3)

The flow down the stream per second, or discharge of the stream, is

Q = [Omega]v = [Omega]c [root](mi). (4)

§ 97. _Coefficient of Friction for Open Channels._--Various
expressions have been proposed for the coefficient of friction for
channels as for pipes. Weisbach, giving attention chiefly to the
variation of the coefficient of friction with the velocity, proposed
an expression of the form

[zeta] = [alpha](1 + [beta]/v), (5)

and from 255 experiments obtained for the constants the values

[alpha] = 0.007409; [beta] = 0.1920.

This gives the following values at different velocities:--

+----------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| v = | 0.3 | 0.5 | 0.7 | 1 | 1½ | 2 | 3 | 5 | 7 | 10 | 15 |
| | | | | | | | | | | | |
| [zeta] = |0.01215|0.01025|0.00944|0.00883|0.00836|0.00812|0.90788|0.00769|0.00761|0.00755|0.00750|
+----------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+

In using this value of [zeta] when v is not known, it is best to
proceed by approximation.

§ 98. _Darcy and Bazin's Expression for the Coefficient of
Friction._--Darcy and Bazin's researches have shown that [zeta] varies
very greatly for different degrees of roughness of the channel bed,
and that it also varies with the dimensions of the channel. They give
for [zeta] an empirical expression (similar to that for pipes) of the
form

[zeta] = a(1 + [beta]/m); (6)

where m is the hydraulic mean depth. For different kinds of channels
they give the following values of the coefficient of friction:--

+-------------------------------------------------+--------+------+
| Kind of Channel. | [alpha]|[beta]|
+-------------------------------------------------+--------+------+
| I. Very smooth channels, sides of smooth | | |
| cement or planed timber | .00294 | 0.10 |
| II. Smooth channels, sides of ashlar, brickwork,| | |
| planks | .00373 | 0.23 |
|III. Rough channels, sides of rubble masonry or | | |
| pitched with stone | .00471 | 0.82 |
| IV. Very rough canals in earth | .00549 | 4.10 |
| V. Torrential streams encumbered with detritus | .00785 | 5.74 |
+-------------------------------------------------+--------+------+

The last values (Class V.) are not Darcy and Bazin's, but are taken
from experiments by Ganguillet and Kutter on Swiss streams.

The following table very much facilitates the calculation of the mean
velocity and discharge of channels, when Darcy and Bazin's value of
the coefficient of friction is used. Taking the general formula for
the mean velocity already given in equation (2a) above,

v = c [root](mi),

where c = [root](2g/[zeta]), the following table gives values of c for
channels of different degrees of roughness, and for such values of the
hydraulic mean depths as are likely to occur in practical
calculations:--

Values of c in v = c[root](mi), deduced from Darcy and Bazin's Values.

+----------+-----------+----------+---------+----------+--------------+
| |Very Smooth| Smooth | Rough |Very Rough| Excessively |
| Mean | Channels, | Channels,|Channels,| Channels,|Rough Channels|
|Depth = m.| Cement. |Ashlar or | Rubble |Canals in | encumbered |
| | |Brickwork.| Masonry.| Earth. |with Detritus.|
+----------+-----------+----------+---------+----------+--------------+
| .25 | 125 | 95 | 57 | 26 | 18.5 |
| .5 | 135 | 110 | 72 | 36 | 25.6 |
| .75 | 139 | 116 | 81 | 42 | 30.8 |
| 1.0 | 141 | 119 | 87 | 48 | 34.9 |
| 1.5 | 143 | 122 | 94 | 56 | 41.2 |
| 2.0 | 144 | 124 | 98 | 62 | 46.0 |
| 2.5 | 145 | 126 | 101 | 67 | .. |
| 3.0 | 145 | 126 | 104 | 70 | 53 |
| 3.5 | 146 | 127 | 105 | 73 | .. |
| 4.0 | 146 | 128 | 106 | 76 | 58 |
| 4.5 | 146 | 128 | 107 | 78 | .. |
| 5.0 | 146 | 128 | 108 | 80 | 62 |
| 5.5 | 146 | 129 | 109 | 82 | .. |
| 6.0 | 147 | 129 | 110 | 84 | 65 |
| 6.5 | 147 | 129 | 110 | 85 | .. |
| 7.0 | 147 | 129 | 110 | 86 | 67 |
| 7.5 | 147 | 129 | 111 | 87 | .. |
| 8.0 | 147 | 130 | 111 | 88 | 69 |
| 8.5 | 147 | 130 | 112 | 89 | .. |
| 9.0 | 147 | 130 | 112 | 90 | 71 |
| 9.5 | 147 | 130 | 112 | 90 | .. |
| 10.0 | 147 | 130 | 112 | 91 | 72 |
| 11 | 147 | 130 | 113 | 92 | .. |
| 12 | 147 | 130 | 113 | 93 | 74 |
| 13 | 147 | 130 | 113 | 94 | .. |
| 14 | 147 | 130 | 113 | 95 | .. |
| 15 | 147 | 130 | 114 | 96 | 77 |
| 16 | 147 | 130 | 114 | 97 | .. |
| 17 | 147 | 130 | 114 | 97 | .. |
| 18 | 147 | 130 | 114 | 98 | .. |
| 20 | 147 | 131 | 114 | 98 | 80 |
| 25 | 148 | 131 | 115 | 100 | .. |
| 30 | 148 | 131 | 115 | 102 | 83 |
| 40 | 148 | 131 | 116 | 103 | 85 |
| 50 | 148 | 131 | 116 | 104 | 86 |
| [oo] | 148 | 131 | 117 | 108 | 91 |
+----------+-----------+----------+---------+----------+--------------+

§ 99. _Ganguillet and Kutter's Modified Darcy Formula._--Starting from
the general expression v = c[root]mi, Ganguillet and Kutter examined
the variations of c for a wider variety of cases than those discussed
by Darcy and Bazin. Darcy and Bazin's experiments were confined to
channels of moderate section, and to a limited variation of slope.
Ganguillet and Kutter brought into the discussion two very distinct
and important additional series of results. The gaugings of the
Mississippi by A. A. Humphreys and H. L. Abbot afford data of
discharge for the case of a stream of exceptionally large section and
or very low slope. On the other hand, their own measurements of the
flow in the regulated channels of some Swiss torrents gave data for
cases in which the inclination and roughness of the channels were
exceptionally great. Darcy and Bazin's experiments alone were
conclusive as to the dependence of the coefficient c on the dimensions
of the channel and on its roughness of surface. Plotting values of c
for channels of different inclination appeared to indicate that it
also depended on the slope of the stream. Taking the Mississippi data
only, they found

c = 256 for an inclination of 0.0034 per thousand,
= 154 " " 0.02 "

so that for very low inclinations no constant value of c independent
of the slope would furnish good values of the discharge. In small
rivers, on the other hand, the values of c vary little with the slope.
As regards the influence of roughness of the sides of the channel a
different law holds. For very small channels differences of roughness
have a great influence on the discharge, but for very large channels
different degrees of roughness have but little influence, and for
indefinitely large channels the influence of different degrees of
roughness must be assumed to vanish. The coefficients given by Darcy
and Bazin are different for each of the classes of channels of
different roughness, even when the dimensions of the channel are
infinite. But, as it is much more probable that the influence of the
nature of the sides diminishes indefinitely as the channel is larger,
this must be regarded as a defect in their formula.

Comparing their own measurements in torrential streams in Switzerland
with those of Darcy and Bazin, Ganguillet and Kutter found that the
four classes of coefficients proposed by Darcy and Bazin were
insufficient to cover all cases. Some of the Swiss streams gave
results which showed that the roughness of the bed was markedly
greater than in any of the channels tried by the French engineers. It
was necessary therefore in adopting the plan of arranging the
different channels in classes of approximately similar roughness to
increase the number of classes. Especially an additional class was
required for channels obstructed by detritus.

To obtain a new expression for the coefficient in the formula

v = [root](2g/[zeta]) [root](mi) = c [root](mi),

Ganguillet and Kutter proceeded in a purely empirical way. They found
that an expression of the form

c = [alpha]/(1 + [beta]/[root]m)

could be made to fit the experiments somewhat better than Darcy's
expression. Inverting this, we get

1/c = 1/[alpha] + [beta]/[alpha] [root]m,

an equation to a straight line having 1/[root]m for abscissa, 1/c for
ordinate, and inclined to the axis of abscissae at an angle the
tangent of which is [beta]/[alpha].

Plotting the experimental values of 1/c and 1/[root]m, the points so
found indicated a curved rather than a straight line, so that [beta]
must depend on [alpha]. After much comparison the following form was
arrived at--

c = (A + l/n)/(1 + An/[root]m),

where n is a coefficient depending only on the roughness of the sides
of the channel, and A and l are new coefficients, the value of which
remains to be determined. From what has been already stated, the
coefficient c depends on the inclination of the stream, decreasing as
the slope i increases.

Let

A = a + p/i.

Then

c = (a + l/n + p/i)/{1 + (a + p/i)n/[root]m},

the form of the expression for c ultimately adopted by Ganguillet and
Kutter.

For the constants a, l, p Ganguillet and Kutter obtain the values 23,
1 and 0.00155 for metrical measures, or 41.6, 1.811 and 0.00281 for
English feet. The coefficient of roughness n is found to vary from
0.008 to 0.050 for either metrical or English measures.

The most practically useful values of the coefficient of roughness n
are given in the following table:--

Nature of Sides of Channel. Coefficient of
Roughness n.
Well-planed timber 0.009
Cement plaster 0.010
Plaster of cement with one-third sand 0.011
Unplaned planks 0.012
Ashlar and brickwork 0.013
Canvas on frames 0.015
Rubble masonry 0.017
Canals in very firm gravel 0.020
Rivers and canals in perfect order, free from stones \
or weeds / 0.025
Rivers and canals in moderately good order, not \
quite free from stones and weeds / 0.030
Rivers and canals in bad order, with weeds and \
detritus / 0.035
Torrential streams encumbered with detritus 0.050

Ganguillet and Kutter's formula is so cumbrous that it is difficult to
use without the aid of tables.

Lowis D'A. Jackson published complete and extensive tables for
facilitating the use of the Ganguillet and Kutter formula (_Canal and
Culvert Tables_, London, 1878). To lessen calculation he puts the
formula in this form:--

M = n(41.6 + 0.00281/i);

v = ([root]m/n) {(M + 1.811)/(M + [root]m)} [root](mi).

The following table gives a selection of values of M, taken from
Jackson's tables:--

+--------+--------------------------------------------------------------+
| | Values of M for n = |
| i = +--------+--------+--------+--------+--------+--------+--------+
| | 0.010 | 0.012 | 0.015 | 0.017 | 0.020 | 0.025 | 0.030 |
+--------+--------+--------+--------+--------+--------+--------+--------+
| .00001 | 3.2260 | 3.8712 | 4.8390 | 5.4842 | 6.4520 | 8.0650 | 9.6780 |
| .00002 | 1.8210 | 2.1852 | 2.7315 | 3.0957 | 3.6420 | 4.5525 | 5.4630 |
| .00004 | 1.1185 | 1.3422 | 1.6777 | 1.9014 | 2.2370 | 2.7962 | 3.3555 |
| .00006 | 0.8843 | 1.0612 | 1.3264 | 1.5033 | 1.7686 | 2.2107 | 2.6529 |
| .00008 | 0.7672 | 0.9206 | 1.1508 | 1.3042 | 1.5344 | 1.9180 | 2.3016 |
| .00010 | 0.6970 | 0.8364 | 1.0455 | 1.1849 | 1.3940 | 1.7425 | 2.0910 |
| .00025 | 0.5284 | 0.6341 | 0.7926 | 0.8983 | 1.0568 | 1.3210 | 1.5852 |
| .00050 | 0.4722 | 0.5666 | 0.7083 | 0.8027 | 0.9444 | 1.1805 | 1.4166 |
| .00075 | 0.4535 | 0.5442 | 0.6802 | 0.7709 | 0.9070 | 1.1337 | 1.3605 |
| .00100 | 0.4441 | 0.5329 | 0.6661 | 0.7550 | 0.8882 | 1.1102 | 1.3323 |
| .00200 | 0.4300 | 0.5160 | 0.6450 | 0.7310 | 0.8600 | 1.0750 | 1.2900 |
| .00300 | 0.4254 | 0.5105 | 0.6381 | 0.7232 | 0.8508 | 1.0635 | 1.2762 |
+--------+--------+--------+--------+--------+--------+--------+--------+

A difficulty in the use of this formula is the selection of the
coefficient of roughness. The difficulty is one which no theory will
overcome, because no absolute measure of the roughness of stream beds
is possible. For channels lined with timber or masonry the difficulty
is not so great. The constants in that case are few and sufficiently
defined. But in the case of ordinary canals and rivers the case is
different, the coefficients having a much greater range. For
artificial canals in rammed earth or gravel n varies from 0.0163 to
0.0301. For natural channels or rivers n varies from 0.020 to 0.035.

In Jackson's opinion even Kutter's numerous classes of channels seem
inadequately graduated, and he proposes for artificial canals the
following classification:--

I. Canals in very firm gravel, in perfect order n = 0.02
II. Canals in earth, above the average in order n = 0.0225
III. Canals in earth, in fair order n = 0.025
IV. Canals in earth, below the average in order n = 0.0275
V. Canals in earth, in rather bad order, partially\
overgrown with weeds and obstructed by > n = 0.03
detritus. /

Ganguillet and Kutter's formula has been considerably used partly from
its adoption in calculating tables for irrigation work in India. But
it is an empirical formula of an unsatisfactory form. Some engineers
apparently have assumed that because it is complicated it must be more
accurate than simpler formulae. Comparison with the results of
gaugings shows that this is not the case. The term involving the slope
was introduced to secure agreement with some early experiments on the
Mississippi, and there is strong reason for doubting the accuracy of
these results.

§ 100. _Bazin's New Formula._--Bazin subsequently re-examined all the
trustworthy gaugings of flow in channels and proposed a modification
of the original Darcy formula which appears to be more satisfactory
than any hitherto suggested (_Étude d'une nouvelle formule_, Paris,
1898). He points out that Darcy's original formula, which is of the
form mi/v² = [alpha] + [beta]/m, does not agree with experiments on
channels as well as with experiments on pipes. It is an objection to
it that if m increases indefinitely the limit towards which mi/v²
tends is different for different values of the roughness. It would
seem that if the dimensions of a canal are indefinitely increased the
variation of resistance due to differing roughness should vanish. This
objection is met if it is assumed that [root](mi/v²) = [alpha] +
[beta]/[root]m, so that if a is a constant mi/v² tends to the limit a
when m increases. A very careful discussion of the results of gaugings
shows that they can be expressed more satisfactorily by this new
formula than by Ganguillet and Kutter's. Putting the equation in the
form [zeta]v²/2g = mi, [zeta] = 0.002594(1 + [gamma]/[root]m), where
[gamma] has the following values:--

I. Very smooth sides, cement, planed plank, [gamma] = 0.109
II. Smooth sides, planks, brickwork 0.290
III. Rubble masonry sides 0.833
IV. Sides of very smooth earth, or pitching 1.539
V. Canals in earth in ordinary condition 2.353
VI. Canals in earth exceptionally rough 3.168

§ 101. _The Vertical Velocity Curve._--If at each point along a
vertical representing the depth of a stream, the velocity at that
point is plotted horizontally, the curve obtained is the vertical
velocity curve and it has been shown by many observations that it
approximates to a parabola with horizontal axis. The vertex of the
parabola is at the level of the greatest velocity. Thus in fig. 104 OA
is the vertical at which velocities are observed; v0 is the surface;
v_z the maximum and v_d the bottom velocity. B C D is the vertical
velocity curve which corresponds with a parabola having its vertex at
C. The mean velocity at the vertical is

v_m = (1/3)[2v_z + v_d + (d_z/d)(v0 - v_d)].

_The Horizontal Velocity Curve._--Similarly if at each point along a
horizontal representing the width of the stream the velocities are
plotted, a curve is obtained called the horizontal velocity curve. In
streams of symmetrical section this is a curve symmetrical about the
centre line of the stream. The velocity varies little near the centre
of the stream, but very rapidly near the banks. In unsymmetrical
sections the greatest velocity is at the point where the stream is
deepest, and the general form of the horizontal velocity curve is
roughly similar to the section of the stream.

§ 102. _Curves or Contours of Equal Velocity._--If velocities are
observed at a number of points at different widths and depths in a
stream, it is possible to draw curves on the cross section through
points at which the velocity is the same. These represent contours of
a solid, the volume of which is the discharge of the stream per
second. Fig. 105 shows the vertical and horizontal velocity curves and
the contours of equal velocity in a rectangular channel, from one of
Bazin's gaugings.

§ 103. _Experimental Observations on the Vertical Velocity Curve._--A
preliminary difficulty arises in observing the velocity at a given
point in a stream because the velocity rapidly varies, the motion not
being strictly steady. If an average of several velocities at the same
point is taken, or the average velocity for a sensible period of time,
this average is found to be constant. It may be inferred that though
the velocity at a point fluctuates about a mean value, the
fluctuations being due to eddying motions superposed on the general
motion of the stream, yet these fluctuations produce effects which
disappear in the mean of a series of observations and, in calculating
the volume of flow, may be disregarded.

In the next place it is found that in most of the best observations on
the velocity in streams, the greatest velocity at any vertical is
found not at the surface but at some distance below it. In various
river gaugings the depth d_z at the centre of the stream has been
found to vary from 0 to 0.3d.

§ 104. _Influence of the Wind._--In the experiments on the Mississippi
the vertical velocity curve in calm weather was found to agree fairly
with a parabola, the greatest velocity being at (3/10)ths of the depth
of the stream from the surface. With a wind blowing down stream the
surface velocity is increased, and the axis of the parabola approaches
the surface. On the contrary, with a wind blowing up stream the
surface velocity is diminished, and the axis of the parabola is
lowered, sometimes to half the depth of the stream. The American
observers drew from their observations the conclusion that there was
an energetic retarding action at the surface of a stream like that due
to the bottom and sides. If there were such a retarding action the
position of the filament of maximum velocity below the surface would
be explained.

It is not difficult to understand that a wind acting on surface
ripples or waves should accelerate or retard the surface motion of the
stream, and the Mississippi results may be accepted so far as showing
that the surface velocity of a stream is variable when the mean
velocity of the stream is constant. Hence observations of surface
velocity by floats or otherwise should only be made in very calm
weather. But it is very difficult to suppose that, in still air, there
is a resistance at the free surface of the stream at all analogous to
that at the sides and bottom. Further, in very careful experiments, P.
P. Boileau found the maximum velocity, though raised a little above
its position for calm weather, still at a considerable distance below
the surface, even when the wind was blowing down stream with a
velocity greater than that of the stream, and when the action of the
air must have been an accelerating and not a retarding action. A much
more probable explanation of the diminution of the velocity at and
near the free surface is that portions of water, with a diminished
velocity from retardation by the sides or bottom, are thrown off in
eddying masses and mingle with the rest of the stream. These eddying
masses modify the velocity in all parts of the stream, but have their
greatest influence at the free surface. Reaching the free surface they
spread out and remain there, mingling with the water at that level and
diminishing the velocity which would otherwise be found there.

_Influence of the Wind on the Depth at which the Maximum Velocity is
found._--In the gaugings of the Mississippi the vertical velocity
curve was found to agree well with a parabola having a horizontal axis
at some distance below the water surface, the ordinate of the parabola
at the axis being the maximum velocity of the section. During the
gaugings the force of the wind was registered on a scale ranging from
0 for a calm to 10 for a hurricane. Arranging the velocity curves in
three sets--(1) with the wind blowing up stream, (2) with the wind
blowing down stream, (3) calm or wind blowing across stream--it was
found that an upstream wind lowered, and a down-stream wind raised,
the axis of the parabolic velocity curve. In calm weather the axis was
at (3/10)ths of the total depth from the surface for all conditions of
the stream.

Let h´ be the depth of the axis of the parabola, m the hydraulic mean
depth, f the number expressing the force of the wind, which may range
from +10 to -10, positive if the wind is up stream, negative if it is
down stream. Then Humphreys and Abbot find their results agree with
the expression

h´/m = 0.317 ± 0.06f.

Fig. 106 shows the parabolic velocity curves according to the American
observers for calm weather, and for an up- or down-stream wind of a
force represented by 4.

It is impossible at present to give a theoretical rule for the
vertical velocity curve, but in very many gaugings it has been found
that a parabola with horizontal axis fits the observed results fairly
well. The mean velocity on any vertical in a stream varies from 0.85
to 0.92 of the surface velocity at that vertical, and on the average
if v0 is the surface and v_m the mean velocity at a vertical v_m =
6/7 v0, a result useful in float gauging. On any vertical there is a
point at which the velocity is equal to the mean velocity, and if this
point were known it would be useful in gauging. Humphreys and Abbot in
the Mississippi found the mean velocity at 0.66 of the depth; G. H. L.
Hagen and H. Heinemann at 0.56 to 0.58 of the depth. The mean of
observations by various observers gave the mean velocity at from 0.587
to 0.62 of the depth, the average of all being almost exactly 0.6 of
the depth. The mid-depth velocity is therefore nearly equal to, but a
little greater than, the mean velocity on a vertical. If v_(md) is the
mid-depth velocity, then on the average v_m = 0.98v_(md).

§ 105. _Mean Velocity on a Vertical from Two Velocity
Observations._--A. J. C. Cunningham, in gaugings on the Ganges canal,
found the following useful results. Let v0 be the surface, v_m the
mean, and v_(xd) the velocity at the depth xd; then

v_m = ¼[v0 + 3v_(2/3d)]
= ½[v_(.211)^d + v_(.789)^d].

§ 106. _Ratio of Mean to Greatest Surface Velocity, for the whole
Cross Section in Trapezoidal Channels._--It is often very important to
be able to deduce the mean velocity, and thence the discharge, from
observation of the greatest surface velocity. The simplest method of
gauging small streams and channels is to observe the greatest surface
velocity by floats, and thence to deduce the mean velocity. In general
in streams of fairly regular section the mean velocity for the whole
section varies from 0.7 to 0.85 of the greatest surface velocity. For
channels not widely differing from those experimented on by Bazin, the
expression obtained by him for the ratio of surface to mean velocity
may be relied on as at least a good approximation to the truth. Let v0
be the greatest surface velocity, v_m the mean velocity of the stream.
Then, according to Bazin,

v_m = v0 - 25.4 [root](mi).

But

v_m = c [root](mi),

where c is a coefficient, the values of which have been already given
in the table in § 98. Hence

v_m = cv0/(c + 25.4).

_Values of Coefficient c/(c + 25.4) in the Formula v_m = cv0/(c +
25.4)._

+----------+---------+----------+---------+----------+----------+
|Hydraulic | Very | Smooth | Rough |Very Rough| Channels |
|Mean Depth| Smooth |Channels. |Channels.| Channels.|encumbered|
| = m. |Channels.|Ashlar or | Rubble | Canals in| with |
| | Cement. |Brickwork.| Masonry.| Earth. | Detritus.|
+----------+---------+----------+---------+----------+----------+
| | | | | | |
| 0.25 | .83 | .79 | .69 | .51 | .42 |
| 0.5 | .84 | .81 | .74 | .58 | .50 |
| 0.75 | .84 | .82 | .76 | .63 | .55 |
| 1.0 | .85 | .. | .77 | .65 | .58 |
| 2.0 | .. | .83 | .79 | .71 | .64 |
| 3.0 | .. | .. | .80 | .73 | .67 |
| 4.0 | .. | .. | .81 | .75 | .70 |
| 5.0 | .. | .. | .. | .76 | .71 |
| 6.0 | .. | .84 | .. | .77 | .72 |
| 7.0 | .. | .. | .. | .78 | .73 |
| 8.0 | .. | .. | .. | .. | .. |
| 9.0 | .. | .. | .82 | .. | .74 |
| 10.0 | .. | .. | .. | .. | .. |
| 15.0 | .. | .. | .. | .79 | .75 |
| 20.0 | .. | .. | .. | .80 | .76 |
| 30.0 | .. | .. | .82 | .. | .77 |
| 40.0 | .. | .. | .. | .. | .. |
| 50.0 | .. | .. | .. | .. | .. |
| [oo] | .. | .. | .. | .. | .79 |
+----------+---------+----------+---------+----------+----------+

§ 107. _River Bends._--In rivers flowing in alluvial plains, the
windings which already exist tend to increase in curvature by the
scouring away of material from the outer bank and the deposition of
detritus along the inner bank. The sinuosities sometimes increase till
a loop is formed with only a narrow strip of land between the two
encroaching branches of the river. Finally a "cut off" may occur, a
waterway being opened through the strip of land and the loop left
separated from the stream, forming a horseshoe shaped lagoon or marsh.
Professor James Thomson pointed out (_Proc. Roy. Soc._, 1877, p. 356;
_Proc. Inst. of Mech. Eng._, 1879, p. 456) that the usual supposition
is that the water tending to go forwards in a straight line rushes
against the outer bank and scours it, at the same time creating
deposits at the inner bank. That view is very far from a complete
account of the matter, and Professor Thomson gave a much more
ingenious account of the action at the bend, which he completely
confirmed by experiment.

When water moves round a circular curve under the action of gravity
only, it takes a motion like that in a free vortex. Its velocity is
greater parallel to the axis of the stream at the inner than at the
outer side of the bend. Hence the scouring at the outer side and the
deposit at the inner side of the bend are not due to mere difference
of velocity of flow in the general direction of the stream; but, in
virtue of the centrifugal force, the water passing round the bend
presses outwards, and the free surface in a radial cross section has a
slope from the inner side upwards to the outer side (fig. 108). For
the greater part of the water flowing in curved paths, this difference
of pressure produces no tendency to transverse motion. But the water
immediately in contact with the rough bottom and sides of the channel
is retarded, and its centrifugal force is insufficient to balance the
pressure due to the greater depth at the outside of the bend. It
therefore flows inwards towards the inner side of the bend, carrying
with it detritus which is deposited at the inner bank. Conjointly with
this flow inwards along the bottom and sides, the general mass of
water must flow outwards to take its place. Fig. 107 shows the
directions of flow as observed in a small artificial stream, by means
of light seeds and specks of aniline dye. The lines CC show the
directions of flow immediately in contact with the sides and bottom.
The dotted line AB shows the direction of motion of floating particles
on the surface of the stream.

§ 108. _Discharge of a River when flowing at different Depths._--When
frequent observations must be made on the flow of a river or canal,
the depth of which varies at different times, it is very convenient to
have to observe the depth only. A formula can be established giving
the flow in terms of the depth. Let Q be the discharge in cubic feet
per second; H the depth of the river in some straight and uniform
part. Then Q = aH + bH², where the constants a and b must be found by
preliminary gaugings in different conditions of the river. M. C.
Moquerey found for part of the upper Saône, Q = 64.7H + 8.2H² in
metric measures, or Q = 696H + 26.8H² in English measures.

§ 109. _Forms of Section of Channels._--The simplest form of section
for channels is the semicircular or nearly semicircular channel (fig.
109), a form now often adopted from the facility with which it can be
executed in concrete. It has the advantage that the rubbing surface is
less in proportion to the area than in any other form.

Wooden channels or flumes, of which there are examples on a large
scale in America, are rectangular in section, and the same form is
adopted for wrought and cast-iron aqueducts. Channels built with
brickwork or masonry may be also rectangular, but they are often
trapezoidal, and are always so if the sides are pitched with masonry
laid dry. In a trapezoidal channel, let b (fig. 110) be the bottom
breadth, b0 the top breadth, d the depth, and let the slope of the
sides be n horizontal to 1 vertical. Then the area of section is
[Omega] = (b + nd)d = (b0 - nd)d, and the wetted perimeter [chi] = b +
2d[root](n² + 1).

When a channel is simply excavated in earth it is always originally
trapezoidal, though it becomes more or less rounded in course of time.
The slope of the sides then depends on the stability of the earth, a
slope of 2 to 1 being the one most commonly adopted.

Figs. 111, 112 show the form of canals excavated in earth, the former
being the section of a navigation canal and the latter the section of
an irrigation canal.

§ 110. _Channels of Circular Section._--The following short table
facilitates calculations of the discharge with different depths of
water in the channel. Let r be the radius of the channel section; then
for a depth of water = [kappa]r, the hydraulic mean radius is [mu]r
and the area of section of the waterway [nu]r², where [kappa], [mu],
and [nu] have the following values:--

+---------------------------------+------+-----+-----+-----+-----+-----+-----+-----+----+----+----+----+----+----+----+-----+-----+-----+-----+-----+-----+
| Depth of water in \ [kappa] = |.01 |.05 |.10 |.15 |.20 |.25 |.30 |.35 |.40 |.45 |.50 |.55 |.60 |.65 |.70 |.75 |.80 |.85 |.90 |.95 |1.0 |
| terms of radius / | | | | | | | | | | | | | | | | | | | | | |
| Hydraulic mean depth\ [mu] = |.00668|.0321|.0523|.0963|.1278|.1574|.1852|.2142|.242|.269|.293|.320|.343|.365|.387|.408 |.429 |.449 |.466 |.484 |.500 |
| in terms of radius/ | | | | | | | | | | | | | | | | | | | | | |
| Waterway in terms of\ [nu] = |.00189|.0211|.0598|.1067|.1651|.228 |.294 |.370 |.450|.532|.614|.709|.795|.885|.979|1.075|1.175|1.276|1.371|1.470|1.571|
| square of radius / | | | | | | | | | | | | | | | | | | | | | |
+---------------------------------+------+-----+-----+-----+-----+-----+-----+-----+----+----+----+----+----+----+----+-----+-----+-----+-----+-----+-----+

§ 111. _Egg-Shaped Channels or Sewers._--In sewers for discharging
storm water and house drainage the volume of flow is extremely
variable; and there is a great liability for deposits to be left when
the flow is small, which are not removed during the short periods when
the flow is large. The sewer in consequence becomes choked. To obtain
uniform scouring action, the velocity of flow should be constant or
nearly so; a complete uniformity of velocity cannot be obtained with
any form of section suitable for sewers, but an approximation to
uniform velocity is obtained by making the sewers of oval section.
Various forms of oval have been suggested, the simplest being one in
which the radius of the crown is double the radius of the invert, and
the greatest width is two-thirds the height. The section of such a
sewer is shown in fig. 113, the numbers marked on the figure being
proportional numbers.

§ 112. _Problems on Channels in which the Flow is Steady and at
Uniform Velocity._--The general equations given in §§ 96, 98 are

[zeta] = [alpha](1 + [beta]/m); (1)

[zeta]v²/2g = mi; (2)

Q = [Omega]v. (3)

_Problem I._--Given the transverse section of stream and discharge, to
find the slope. From the dimensions of the section find [Omega] and m;
from (1) find [zeta], from (3) find v, and lastly from (2) find i.

_Problem II._--Given the transverse section and slope, to find the
discharge. Find v from (2), then Q from (3).

_Problem III._--Given the discharge and slope, and either the breadth,
depth, or general form of the section of the channel, to determine its
remaining dimensions. This must generally be solved by approximations.
A breadth or depth or both are chosen, and the discharge calculated.
If this is greater than the given discharge, the dimensions are
reduced and the discharge recalculated.

Since m lies generally between the limits m = d and m = ½d, where d is
the depth of the stream, and since, moreover, the velocity varies as
[root](m) so that an error in the value of m leads only to a much less
error in the value of the velocity calculated from it, we may proceed
thus. Assume a value for m, and calculate v from it. Let v1 be this
first approximation to v. Then Q/v1 is a first approximation to
[Omega], say [Omega]1. With this value of [Omega] design the section
of the channel; calculate a second value for m; calculate from it a
second value of v, and from that a second value for [Omega]. Repeat
the process till the successive values of m approximately coincide.

§ 113. _Problem IV. Most Economical Form of Channel for given Side
Slopes._--Suppose the channel is to be trapezoidal in section (fig.
114), and that the sides are to have a given slope. Let the
longitudinal slope of the stream be given, and also the mean velocity.
An infinite number of channels could be found satisfying the
foregoing conditions. To render the problem determinate, let it be
remembered that, since for a given discharge [Omega][oo] [cube
root][chi], other things being the same, the amount of excavation will
be least for that channel which has the least wetted perimeter. Let d
be the depth and b the bottom width of the channel, and let the sides
slope n horizontal to 1 vertical (fig. 114), then

[Omega] = (b + nd)d;

[chi] = b + 2d [root](n² + 1).

Both [Omega] and [chi] are to be minima. Differentiating, and equating
to zero.

(db/dd + n)d + b + nd = 0,

db/dd + 2[root](n² + 1) = 0;

eliminating db/dd,

{n - 2[root](n² + 1)}d + b + nd = 0;

b = 2 {[root](n² + 1) - n}d.

But

[Omega]/[chi] = (b + nd)d/{b + 2d [root](n² + 1)}.

Inserting the value of b,

m = [Omega]/[chi] = {2d[root](n² + 1) - nd}/
{4d [root](n² + 1) - 2nd} = ½d.

That is, with given side slopes, the section is least for a given
discharge when the hydraulic mean depth is half the actual depth.

A simple construction gives the form of the channel which fulfils this
condition, for it can be shown that when m = ½d the sides of the
channel are tangential to a semicircle drawn on the water line.

Since

[Omega]/[chi] = ½d,

therefore

[Omega] = ½[chi]d. (1)

Let ABCD be the channel (fig. 115); from E the centre of AD drop
perpendiculars EF, EG, EH on the sides.

Let

AB = CD = a; BC = b; EF = EH = c; and EG = d.

[Omega] = area AEB + BEC + CED,
= ac + ½bd.

[chi] = 2a + b.

Putting these values in (1),

ac + ½bd = (a + ½b)d; and hence c = d.

That is, EF, EG, EH are all equal, hence a semicircle struck from E
with radius equal to the depth of the stream will pass through F and H
and be tangential to the sides of the channel.

To draw the channel, describe a semicircle on a horizontal line with
radius = depth of channel. The bottom will be a horizontal tangent of
that semicircle, and the sides tangents drawn at the required side
slopes.

The above result may be obtained thus (fig. 116):--

[chi] = b + 2d/sin [beta]. (1)

[Omega] = d(b + d cot [beta]);

[Omega]/d = b + d cot [beta]; (2)

[Omega]/d² = b/d + cot [beta]. (3)

From (1) and (2),

[chi] = [Omega]/d - d cot [beta] + 2d/sin [beta].

This will be a minimum for

d[chi]/dd = [Omega]/d² + cot[beta] - 2/sin [beta] = 0,

or

[Omega]/d² = 2 cosec. [beta] - cot [beta]. (4)

or

d = [root]{[Omega] sin [beta]/(2 - cos [beta])}.

From (3) and (4),

b/d = 2(1 - cos [beta])/sin [beta] = 2 tan ½[beta].

_Proportions of Channels of Maximum Discharge for given Area and Side
Slopes. Depth of channel = d; Hydraulic mean depth = ½d; Area of
section =_ [Omega].

+-------------+-----------+--------+----------+---------+------------+
| |Inclination|Ratio of| Area of | |Top width = |
| |of Sides to| Side | Section | Bottom |twice length|
| | Horizon. | Slopes.| [Omega]. | Width. |of each Side|
| | | | | | Slope. |
+-------------+-----------+--------+----------+---------+------------+
| Semicircle | .. | .. | 1.571 d² | 0 | 2 d |
| Semi-hexagon| 60° 0´ | 3 : 5 | 1.732 d² | 1.155 d | 2.310 d |
| Semi-square | 90° 0´ | 0 : 1 | 2 d² | 2 d | 2 d |
| | 75° 58´ | 1 : 4 | 1.812 d² | 1.562 d | 2.062 d |
| | 63° 26´ | 1 : 2 | 1.736 d² | 1.236 d | 2.236 d |
| | 53° 8´ | 3 : 4 | 1.750 d² | d | 2.500 d |
| | 45° 0´ | 1 : 1 | 1.828 d² | 0.828 d | 2.828 d |
| | 38° 40´ | 1¼ : 1 | 1.952 d² | 0.702 d | 3.202 d |
| | 33° 42´ | 1½ : 1 | 2.106 d² | 0.606 d | 3.606 d |
| | 29° 44´ | 1¾ : 1 | 2.282 d² | 0.532 d | 4.032 d |
| | 26° 34´ | 2 : 1 | 2.472 d² | 0.472 d | 4.472 d |
| | 23° 58´ | 2¼ : 1 | 2.674 d² | 0.424 d | 4.924 d |
| | 21° 48´ | 2½ : 1 | 2.885 d² | 0.385 d | 5.385 d |
| | 19° 58´ | 2¾ : 1 | 3.104 d² | 0.354 d | 5.854 d |
| | 18° 26´ | 3 : 1 | 3.325 d² | 0.325 d | 6.325 d |
+-------------+-----------+--------+----------+---------+------------+

Half the top width is the length of each side slope. The wetted
perimeter is the sum of the top and bottom widths.

§ 114. _Form of Cross Section of Channel in which the Mean Velocity is
Constant with Varying Discharge._--In designing waste channels from
canals, and in some other cases, it is desirable that the mean
velocity should be restricted within narrow limits with very different
volumes of discharge. In channels of trapezoidal form the velocity
increases and diminishes with the discharge. Hence when the discharge
is large there is danger of erosion, and when it is small of silting
or obstruction by weeds. A theoretical form of section for which the
mean velocity would be constant can be found, and, although this is
not very suitable for practical purposes, it can be more or less
approximated to in actual channels.

Let fig. 117 represent the cross section of the channel. From the
symmetry of the section, only half the channel need be considered. Let
obac be any section suitable for the minimum flow, and let it be
required to find the curve beg for the upper part of the channel so
that the mean velocity shall be constant. Take o as origin of
coordinates, and let de, fg be two levels of the water above ob.

Let

ob = b/2; de = y, fg = y + dy, od = x, of = x + dx; eg = ds.

The condition to be satisfied is that

v = c [root](mi)

should be constant, whether the water-level is at ob, de, or fg.
Consequently

m = constant = k

for all three sections, and can be found from the section obac. Hence
also

Increment of section y dx
---------------------- = ---- = k
Increment of perimeter ds

y²dx² = k²ds² = k²(dx² + dy²) and dx = k dy/[root](y² - k²).

Integrating,

x = k log_[epsilon] {y + [root](y² - k²)} + constant;

and, since y = b/2 when x = 0,

x = k log_[epsilon] [{y + [root](y² - k²)}/{½b + [root](¼b² - k²)}].

Assuming values for y, the values of x can be found and the curve
drawn.

The figure has been drawn for a channel the minimum section of which
is a half hexagon of 4 ft. depth. Hence k = 2; b = 9.2; the rapid
flattening of the side slopes is remarkable.

STEADY MOTION OF WATER IN OPEN CHANNELS OF VARYING CROSS SECTION AND
SLOPE

§ 115. In every stream the discharge of which is constant, or may be
regarded as constant for the time considered, the velocity at
different places depends on the slope of the bed. Except at certain
exceptional points the velocity will be greater as the slope of the
bed is greater, and, as the velocity and cross section of the stream
vary inversely, the section of the stream will be least where the
velocity and slope are greatest. If in a stream of tolerably uniform
slope an obstruction such as a weir is built, that will cause an
alteration of flow similar to that of an alteration of the slope of
the bed for a greater or less distance above the weir, and the
originally uniform cross section of the stream will become a varied
one. In such cases it is often of much practical importance to
determine the longitudinal section of the stream.

The cases now considered will be those in which the changes of
velocity and cross section are gradual and not abrupt, and in which
the only internal work which needs to be taken into account is that
due to the friction of the stream bed, as in cases of uniform motion.
Further, the motion will be supposed to be steady, the mean velocity
at each given cross section remaining constant, though it varies from
section to section along the course of the stream.

Let fig. 118 represent a longitudinal section of the stream, A0A1
being the water surface, B0B1 the stream bed. Let A0B0, A1B1 be cross
sections normal to the direction of flow. Suppose the mass of water
A0B0A1B1 comes in a short time [theta] to C0D0C1D1, and let the work
done on the mass be equated to its change of kinetic energy during
that period. Let l be the length A0A1 of the portion of the stream
considered, and z the fall, of surface level in that distance. Let Q
be the discharge of the stream per second.

_Change of Kinetic Energy._--At the end of the time [theta] there are
as many particles possessing the same velocities in the space C0D0A1B1
as at the beginning. The change of kinetic energy is therefore the
difference of the kinetic energies of A0B0C0D0 and A1B1C1D1.

Let fig. 119 represent the cross section A0B0, and let [omega] be a
small element of its area at a point where the velocity is v. Let
[Omega]0 be the whole area of the cross section and u0 the mean
velocity for the whole cross section. From the definition of mean
velocity we have

u0 = [Sigma][omega]v/[Omega]0.

Let v = u0 + w, where w is the difference between the velocity at the
small element [omega] and the mean velocity. For the whole cross
section, [Sigma][omega]w = 0.

The mass of fluid passing through the element of section [omega], in
[theta] seconds, is (G/g)[omega]v[theta], and its kinetic energy is
(G/2g)[omega]v³[theta]. For the whole section, the kinetic energy of
the mass A0B0C0D0 passing in [theta] seconds is

(G[theta]/2g)[Sigma][omega]v³
= (G[theta]/2g)[Sigma][omega](u0³ + 3u0²w + 3u0² + w³),
= (G[theta]/2g){u0³[Omega] + [Sigma][omega]w²(3u0 + w)}.

The factor 3u0 + w is equal to 2u0 + v, a quantity necessarily
positive. Consequently [Sigma][omega]v³ > [Omega]0u0³, and
consequently the kinetic energy of A0B0C0D0 is greater than

(G[theta]/2g)[Omega]0u0³ or (G[theta])/2g)Qu0²,

which would be its value if all the particles passing the section had
the same velocity u0. Let the kinetic energy be taken at

[alpha](G[theta]/2g)[Omega]0u0³ = [alpha](G[theta]/2g)Qu0²,

where [alpha] is a corrective factor, the value of which was estimated
by J. B. C. J. Bélanger at 1.1.[6] Its precise value is not of great
importance.

In a similar way we should obtain for the kinetic energy of A1B1C1D1
the expression

[alpha](G[theta]/2g)[Omega]1u1³ = [alpha](G[theta]/2g)Qu1²,

where [Omega]1, u1 are the section and mean velocity at A1B1, and
where a may be taken to have the same value as before without any
important error.

Hence the change of kinetic energy in the whole mass A0B0A1B1 in
[theta] seconds is

[alpha](G[theta]/2g) Q (u1² - u0²). (1)