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.
[Illustration: FIG. 104.]
§ 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.
[Illustration: FIG. 105.]
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.
[Illustration: FIG. 106.]
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).