Mechanics of Fluids

460 Flow with a free surface
✷
result is complicated. The solution is therefore normally obtained by either
a numerical or a graphical integration.
The problem under investigation is more often that of determining the
position in the channel at which a particular depth is reached, that is, l is
required for a particular value of h, so eqn 10.42 may be inverted. Then
l =
� h2
h1
1 − u2 /gh
dh (10.43)
s − i
The following example illustrates the general method of solution.
Example 10.4 A dam is built across a channel of rectangular crosssection
which carries water at the rate of 8.75 m 3 · s −1 . As a result the
depth just upstream of the dam is increased to 2.5 m. The channel is
5 m wide and the slope of the bed is 1 in 5000. The channel is lined
with concrete (Manning’s n = 0.015). How far upstream is the depth
within 100 mm of the normal depth?
Solution
To determine the normal depth h0 Manning’s formula is applied (with
metre-second units):
8.75 m3 · s−1 (5 × h0) m2 = u = m2/3i1/2 =
n
1
� �2/3 � �1/2 5h0 1
0.015 5 + 2h0 5000
Solution by trial gives h0 = 1.800 m. Since the channel is rectangular
in section the critical depth hc = � q2 /g �1/3 � �
= 1.752 1/3
/9.81 m =
0.678 m. This value is considerably less than the actual depth and the
flow is therefore tranquil.
We require a solution of eqn 10.43 between the limits 2.5 m and
(1.800+0.100) m = 1.9 m. To illustrate a simple technique of numerical
integration it will be sufficient to divide the range of depth, 2.5–
1.9 m, into three equal parts. Then, starting at the dam (the position
where the depth is known) and proceeding upstream, the calculations
may be set out in tabular form (Table 10.2; Fig. 10.41).
This method is well suited to computer programming and it may be applied
to channels of variable cross-section, roughness and slope by adopting suitable
average values of these parameters over each increment of length �l.
Moreover, although the given example concerns a backwater curve in tranquil
flow, the same technique may be used for any type of gradually varied
flow.
For example, it may be used in calculating the position of a hydraulic
jump. As we saw in Section 10.10 the depth h2 downstream of the jump is
determined by the conditions there (e.g. the normal depth appropriate to the