Mechanics of Fluids

452 Flow with a free surface
liquid surface downstream is not maintained at too high a level, maximum
discharge is achieved through the narrowest section, known as the throat,
for the same reasons that the discharge over a broad-crested weir reaches a
maximum in the absence of appreciable restrictions downstream. The flow
at the throat is therefore critical, and the flume is said to be under free discharge
conditions. For a rectangular section in which the streamlines are
straight and parallel the velocity at the throat is therefore given by � (gh2),
and the discharge by
�
Q = b2h2 (gh2) (10.32)
where b2 represents the width of the throat, and h2 the corresponding depth.
Measurements of b2 and h2 would therefore be sufficient for the calculation
of the discharge. The exact position at which the critical conditions
exist, however, is not easy to determine, and measurement of h2 is thus
impracticable. If, however, there is negligible friction in the upstream, converging,
part of the flume and the slope of the bed is also negligible over this
distance, then
h1 + u2 1
2g = h2 + u2 2
2g = h2 + h2 3
=
2 2 h2
where suffix 1 refers to quantities upstream of the contraction.
Moreover
u1 = Q
b1h1
= b2h2
�
(gh2)
b1h1
and this expression substituted for u1 in eqn 10.33 gives
h1 + 1
� �2 b2 h3 2
2g b1 h2 g =
1
3
2 h2
that is,
(h1/h2) 3 + 1
2 (b2/b1) 2 = 3
2
(h1/h2) 2
(10.33)
(10.34)
Equation 10.34 shows that, since the ratio b2/b1 is fixed for a particular
flume, the ratio of depths h1/h2 is constant whatever the rate of flow,
provided that the discharge is free (i.e. the velocity at the throat is critical).
Therefore eqn 10.32 may be rewritten
Q = b2g 1/2 h 3/2
2 = b2g1/2h 3/2
1
r 3/2
(10.35)
where r = h1/h2, and the rate of flow through a given flume may be
determined by measuring the upstream depth h1.
Since b1 > b2, the discharge divided by width q1 < q2, and thus, as
Fig. 10.19 shows, h1 > h2 for tranquil flow upstream, that is, r > 1.
The value of r is, of course, given by the solution of eqn 10.34. Of the
three roots the only one meeting the requirement r > 1is
r = 0.5 + cos(2θ/3)