Chapter X: Flow in Rivers and Canals (2)

_Motive Work of the Weight and Pressures._--Consider a small filament
a0a1 which comes in [theta] seconds to c0c1. The work done by gravity
during that movement is the same as if the portion a0c0 were carried
to a1c1. Let dQ[theta] be the volume of a0c0 or a1c1, and y0, y1 the
depths of a0, a1 from the surface of the stream. Then the volume
dQ[theta] or GdQ[theta] pounds falls through a vertical height z + y1
- y0, and the work done by gravity is

G dQ[theta](z + y1 - y0).

Putting p_a for atmospheric pressure, the whole pressure per unit of
area at a0 is Gy0 + p_a, and that at a1 is - (Gy1 + p_a). The work of
these pressures is

G(y0 + p_a/G - y1 - p_a/G) dQ[theta] = G(y0 - y1) dQ[theta].

Adding this to the work of gravity, the whole work is GzdQ[theta]; or,
for the whole cross section,

GzQ[theta]. (2)

_Work expended in Overcoming the Friction of the Stream Bed._--Let
A´B´, A´´B´´ be two cross sections at distances s and s + ds from
A0B0. Between these sections the velocity may be treated as uniform,
because by hypothesis the changes of velocity from section to section
are gradual. Hence, to this short length of stream the equation for
uniform motion is applicable. But in that case the work in overcoming
the friction of the stream bed between A´B´ and A´´B´´ is

GQ[theta][zeta](u²/2g)([chi]/[Omega]) ds,

where u, [chi], [Omega] are the mean velocity, wetted perimeter, and
section at A´B´. Hence the whole work lost in friction from A0B0 to
A1B1 will be
_
/ l
GQ[theta] | [zeta](u²/2g)([chi]/[Omega]) ds. (3)
_/ 0

Equating the work given in (2) and (3) to the change of kinetic energy
given in (1),

[alpha](GQ[theta]/2g)(u1² - u0²)
_
/ l
= GQz[theta] - GQ[theta] | [zeta](u²/2g)([chi]/[Omega]) ds;
_/ 0
_
/ l
.: z = [alpha](u1² - u0²)/2g + | [zeta](u²/2g)([chi]/[Omega]) ds.
_/ 0

§ 116. _Fundamental Differential Equation of Steady Varied
Motion._--Suppose the equation just found to be applied to an
indefinitely short length ds of the stream, limited by the end
sections ab, a1b1, taken for simplicity normal to the stream bed (fig.
120). For that short length of stream the fall of surface level, or
difference of level of a and a1, may be written dz. Also, if we write
u for u0, and u + du for u1, the term (u0² - u1²)/2g becomes udu/g.
Hence the equation applicable to an indefinitely short length of the
stream is

dz = udu/g + ([chi]/[Omega])[zeta](u²/2g) ds. (1)

From this equation some general conclusions may be arrived at as to
the form of the longitudinal section of the stream, but, as the
investigation is somewhat complicated, it is convenient to simplify it
by restricting the conditions of the problem.

_Modification of the Formula for the Restricted Case of a Stream
flowing in a Prismatic Stream Bed of Constant Slope._--Let i be the
constant slope of the bed. Draw ad parallel to the bed, and ac
horizontal. Then dz is sensibly equal to a´c. The depths of the
stream, h and h + dh, are sensibly equal to ab and a´b´, and therefore
dh = a´d. Also cd is the fall of the bed in the distance ds, and is
equal to ids. Hence

dz = a´c = cd - a´d = i ds - dh. (2)

Since the motion is steady--

Q = [Omega]u = constant.

Differentiating,

[Omega] du + u d[Omega] = 0;

.:du = -u d[Omega]/[Omega].

Let x be the width of the stream, then d[Omega] = xdh very nearly.
Inserting this value,

du = -(ux/[Omega]) dh. (3)

Putting the values of du and dz found in (2) and (3) in equation (1),

i ds - dh = -(u²x/g[Omega]) dh + ([chi]/[Omega])[zeta](u²/2g) ds.

dh/ds = {i - ([chi]/[Omega]) [zeta] (u²/2g)}/{1 - (u²/g)(x/[Omega])}. (4)

_Further Restriction to the Case of a Stream of Rectangular Section
and of Indefinite Width._--The equation might be discussed in the form
just given, but it becomes a little simpler if restricted in the way
just stated. For, if the stream is rectangular, [chi]h = [Omega], and
if [chi] is large compared with h, [Omega]/[chi] = xh/x = h nearly.
Then equation (4) becomes

dh/ds = i(1 - [zeta]u²/2gih)/(1 - u²/gh). (5)

§ 117. _General Indications as to the Form of Water Surface furnished
by Equation_ (5).--Let A0A1 (fig. 121) be the water surface, B0B1 the
bed in a longitudinal section of the stream, and ab any section at a
distance s from B0, the depth ab being h. Suppose B0B1, B0A0 taken as
rectangular coordinate axes, then dh/ds is the trigonometric tangent
of the angle which the surface of the stream at a makes with the axis
B0B1. This tangent dh/ds will be positive, if the stream is increasing
in depth in the direction B0B1; negative, if the stream is diminishing
in depth from B0 towards B1. If dh/ds = 0, the surface of the stream
is parallel to the bed, as in cases of uniform motion. But from
equation (4)

dh/ds = 0, if i - ([chi]/[Omega])[zeta](u²/2g) = 0;

.: [zeta](u²/2g) = ([Omega]/[chi])i = mi,

which is the well-known general equation for uniform motion, based on
the same assumptions as the equation for varied steady motion now
being considered. The case of uniform motion is therefore a limiting
case between two different kinds of varied motion.

Consider the possible changes of value of the fraction

(1 - [zeta]u²/2gih)/(1 - u²/gh).

As h tends towards the limit 0, and consequently u is large, the
numerator tends to the limit -[oo]. On the other hand if h = [oo], in
which case u is small, the numerator becomes equal to 1. For a value H
of h given by the equation

1 - [zeta]u²/2giH = 0,

H = [zeta]u²/2gi,

we fall upon the case of uniform motion. The results just stated may
be tabulated thus:--

For h = 0, H, > H, [oo],

the numerator has the value -[oo], 0, > 0, 1.

Next consider the denominator. If h becomes very small, in which case
u must be very large, the denominator tends to the limit -[oo]. As h
becomes very large and u consequently very small, the denominator
tends to the limit 1. For h = u²/g, or u = [root](gh), the denominator
becomes zero. Hence, tabulating these results as before:--

For h = 0, u²/g, > u²/g, [oo],

the denominator becomes

-[oo], 0, > 0, 1.

§ 118. _Case_ 1.--Suppose h > u²/g, and also h > H, or the depth
greater than that corresponding to uniform motion. In this case dh/ds
is positive, and the stream increases in depth in the direction of
flow. In fig. 122 let B0B1 be the bed, C0C1 a line parallel to the bed
and at a height above it equal to H. By hypothesis, the surface A0A1
of the stream is above C0C1, and it has just been shown that the depth
of the stream increases from B0 towards B1. But going up stream h
approaches more and more nearly the value H, and therefore dh/ds
approaches the limit 0, or the surface of the stream is asymptotic to
C0C1. Going down stream h increases and u diminishes, the numerator
and denominator of the fraction (1 - [zeta]u²/2gih)/(1 -u²/gh) both
tend towards the limit 1, and dh/ds to the limit i. That is, the
surface of the stream tends to become asymptotic to a horizontal line
D0D1.

The form of water surface here discussed is produced when the flow of
a stream originally uniform is altered by the construction of a weir.
The raising of the water surface above the level C0C1 is termed the
backwater due to the weir.

§ 119. _Case_ 2.--Suppose h > u²/g, and also h < H. Then dh/ds is
negative, and the stream is diminishing in depth in the direction of
flow. In fig. 123 let B0B1 be the stream bed as before; C0C1 a line
drawn parallel to B0B1 at a height above it equal to H. By hypothesis
the surface A0A1 of the stream is below C0C1, and the depth has just
been shown to diminish from B0 towards B1. Going up stream h
approaches the limit H, and dh/ds tends to the limit zero. That is, up
stream A0A1 is asymptotic to C0C1. Going down stream h diminishes and
u increases; the inequality h>u²/g diminishes; the denominator of the
fraction (1 - [zeta]u²/2gih)/(1 - u²/gh) tends to the limit zero, and
consequently dh/ds tends to [infinity]. That is, down stream A0A1
tends to a direction perpendicular to the bed. Before, however, this
limit was reached the assumptions on which the general equation is
based would cease to be even approximately true, and the equation
would cease to be applicable. The filaments would have a relative
motion, which would make the influence of internal friction in the
fluid too important to be neglected. A stream surface of this form may
be produced if there is an abrupt fall in the bed of the stream (fig.
124).

On the Ganges canal, as originally constructed, there were abrupt
falls precisely of this kind, and it appears that the lowering of the
water surface and increase of velocity which such falls occasion, for
a distance of some miles up stream, was not foreseen. The result was
that, the velocity above the falls being greater than was intended,
the bed was scoured and considerable damage was done to the works.
"When the canal was first opened the water was allowed to pass freely
over the crests of the overfalls, which were laid on the level of the
bed of the earthen channel; erosion of bed and sides for some miles up
rapidly followed, and it soon became apparent that means must be
adopted for raising the surface of the stream at those points (that
is, the crests of the falls). Planks were accordingly fixed in the
grooves above the bridge arches, or temporary weirs were formed over
which the water was allowed to fall; in some cases the surface of the
water was thus raised above its normal height, causing a backwater in
the channel above" (Crofton's _Report on the Ganges Canal_, p. 14).
Fig. 125 represents in an exaggerated form what probably occurred, the
diagram being intended to represent some miles' length of the canal
bed above the fall. AA parallel to the canal bed is the level
corresponding to uniform motion with the intended velocity of the
canal. In consequence of the presence of the ogee fall, however, the
water surface would take some such form as BB, corresponding to Case 2
above, and the velocity would be greater than the intended velocity,
nearly in the inverse ratio of the actual to the intended depth. By
constructing a weir on the crest of the fall, as shown by dotted
lines, a new water surface CC corresponding to Case 1 would be
produced, and by suitably choosing the height of the weir this might
be made to agree approximately with the intended level AA.

§ 120. _Case_ 3.--Suppose a stream flowing uniformly with a depth
h<u²/g. For a stream in uniform motion [zeta]u²/2g = mi, or if the
stream is of indefinitely great width, so that m = H, then
[zeta]u²/2g = iH, and H = [zeta]u²/2gi. Consequently the condition
stated above involves that

[zeta]u²/2gi < u²/g, or that i > [zeta]/2.

If such a stream is interfered with by the construction of a weir
which raises its level, so that its depth at the weir becomes h1 >
u²/g, then for a portion of the stream the depth h will satisfy the
conditions h < u²/g and h > H, which are not the same as those assumed in the two
previous cases. At some point of the stream above the weir the depth h
becomes equal to u²/g, and at that point dh/ds becomes infinite, or
the surface of the stream is normal to the bed. It is obvious that at
that point the influence of internal friction will be too great to be
neglected, and the general equation will cease to represent the true
conditions of the motion of the water. It is known that, in cases such
as this, there occurs an abrupt rise of the free surface of the
stream, or a standing wave is formed, the conditions of motion in
which will be examined presently.

It appears that the condition necessary to give rise to a standing
wave is that i > [zeta]/2. Now [zeta] depends for different channels
on the roughness of the channel and its hydraulic mean depth. Bazin
calculated the values of [zeta] for channels of different degrees of
roughness and different depths given in the following table, and the
corresponding minimum values of i for which the exceptional case of
the production of a standing wave may occur.

+-----------------------------+----------------+-------------------------+
| | Slope below | Standing Wave Formed. |
| |which a Standing| |
| Nature of Bed of Stream. | Wave is +-------------+-----------+
| | impossible in |Slope in feet|Least Depth|
| | feet peer foot.| per foot. | in feet. |
+-----------------------------+----------------+-------------+-----------+
| | | / 0.002 | 0.262 |
| Very smooth cemented surface| 0.00147 | < 0.003 | .098 |
| | | \ 0.004 | .065 |
| | | | |
| | | / 0.003 | .394 |
| Ashlar or brickwork | 0.00186 | < 0.004 | .197 |
| | | \ 0.006 | .098 |
| | | | |
| | | / 0.004 | 1.181 |
| Rubble masonry | 0.00235 | < 0.006 | .525 |
| | | \ 0.010 | .262 |
| | | | |
| | | / 0.006 | 3.478 |
| Earth | 0.00275 | < 0.010 | 1.542 |
| | | \ 0.015 | .919 |
+-----------------------------+----------------+-------------+-----------+

STANDING WAVES

§ 121. The formation of a standing wave was first observed by Bidone.
Into a small rectangular masonry channel, having a slope of 0.023 ft.
per foot, he admitted water till it flowed uniformly with a depth of
0.2 ft. He then placed a plank across the stream which raised the
level just above the obstruction to 0.95 ft. He found that the stream
above the obstruction was sensibly unaffected up to a point 15 ft.
from it. At that point the depth suddenly increased from 0.2 ft. to
0.56 ft. The velocity of the stream in the part unaffected by the
obstruction was 5.54 ft. per second. Above the point where the abrupt
change of depth occurred u² = 5.54² = 30.7, and gh = 32.2 × 0.2 =
6.44; hence u² was > gh. Just below the abrupt change of depth u =
5.54 × 0.2/0.56 = 1.97; u² = 3.88; and gh = 32.2 × 0.56 = 18.03; hence
at this point u² < gh. Between these two points, therefore, u² = gh;
and the condition for the production of a standing wave occurred.

The change of level at a standing wave may be found thus. Let fig. 126
represent the longitudinal section of a stream and ab, cd cross
sections normal to the bed, which for the short distance considered
may be assumed horizontal. Suppose the mass of water abcd to come to
a´b´c´d´ in a short time t; and let u0, u1 be the velocities at ab and
cd, [Omega]0, [Omega]1 the areas of the cross sections. The force
causing change of momentum in the mass abcd estimated horizontally is
simply the difference of the pressures on ab and cd. Putting h0, h1
for the depths of the centres of gravity of ab and cd measured down
from the free water surface, the force is G(h0[Omega]0 - h1[Omega]1)
pounds, and the impulse in t seconds is G (h0[Omega]0 - h1[Omega]1) t
second pounds. The horizontal change of momentum is the difference of
the momenta of cdc´d´ and aba´b´; that is,

(G/g)([Omega]1u1² - [Omega]0u0²)t.

Hence, equating impulse and change of momentum,

G(h0[Omega]0 - h1[Omega]1)t = (G/g)([Omega]1u1² - [Omega]0u0²)t;

.: h0[Omega]0 - h1[Omega]1 = ([Omega]1u1² - [Omega]0u0²)/g. (1)

For simplicity let the section be rectangular, of breadth B and depths
H0 and H1, at the two cross sections considered; then h0 = ½H0, and h1
= ½H1. Hence

H0² - H1² = (2/g)(H1u1² - H0u0²).

But, since [Omega]0u0 = [Omega]1u1, we have

u1² = u0²H0²/H1²,

H0² - H1² = (2u0²/g)(H0²/H1 - H0). (2)

This equation is satisfied if H0 = H1, which corresponds to the case
of uniform motion. Dividing by H0 - H1, the equation becomes

(H1/H0)(H0 + H1) = 2u0²/g; (3)

.: H1 = [root](2u0²H0/g + ¼H0²) - ½H0. (4)

In Bidone's experiment u0 = 5.54, and H0 = 0.2. Hence H1 = 0.52, which
agrees very well with the observed height.

§ 122. A standing wave is frequently produced at the foot of a weir.
Thus in the ogee falls originally constructed on the Ganges canal a
standing wave was observed as shown in fig. 127. The water falling
over the weir crest A acquired a very high velocity on the steep slope
AB, and the section of the stream at B became very small. It easily
happened, therefore, that at B the depth h < u²/g. In flowing along
the rough apron of the weir the velocity u diminished and the depth h
increased. At a point C, where h became equal to u²/g, the conditions
for producing the standing wave occurred. Beyond C the free surface
abruptly rose to the level corresponding to uniform motion with the
assigned slope of the lower reach of the canal.

A standing wave is sometimes formed on the down stream side of bridges
the piers of which obstruct the flow of the water. Some interesting
cases of this kind are described in a paper on the "Floods in the
Nerbudda Valley" in the _Proc. Inst. Civ. Eng._ vol. xxvii. p. 222, by
A. C. Howden. Fig. 128 is compiled from the data given in that paper.
It represents the section of the stream at pier 8 of the Towah
Viaduct, during the flood of 1865. The ground level is not exactly
given by Howden, but has been inferred from data given on another
drawing. The velocity of the stream was not observed, but the author
states it was probably the same as at the Gunjal river during a
similar flood, that is 16.58 ft. per second. Now, taking the depth on
the down stream face of the pier at 26 ft., the velocity necessary for
the production of a standing wave would be u = [root](gh) =
[root](32.2 × 26) = 29 ft. per second nearly. But the velocity at this
point was probably from Howden's statements 16.58 × {40/26} = 25.5 ft.
per second, an agreement as close as the approximate character of the
data would lead us to expect.