Mechanics of Fluids

where 0 ≤ θ ≤ 90 ◦ and sin θ = b2/b1. Ifb1 is large compared with b2, r
becomes equal to 1.5 and eqn 10.35 may then be written
Q = b2g1/2h 3/2
1
(1.5) 3/2
The occurrence of critical conditions 453
(10.36)
This equation may be derived alternatively by substituting for h2 from
eqn 10.33 into eqn 10.32:
Q = b2g 1/2
� � �
3/2
2
3
h1 + u2 1
2g
� 3/2
(10.37)
when b1 is very large, u1 is small and u2 1 /2g may be neglected. Even for
b2/b1 = 1/3, r = 1.474 and so the error involved in neglecting the velocity
of approach and using eqn 10.36 would be only 1.7% in r and 2.54% in Q.
Corrections for velocity of approach may be applied in the same manner
as for the broad-crested weir (Section 10.11.1). If u1 is at first neglected
in eqn 10.37 an approximate value of Q may be calculated from which
u1 = Q/b1h1 can be estimated for use in a second approximation. For a
flume that has already been calibrated, however, this arithmetical process is
not necessary as the cubic equation for r will have been solved once for all.
A coefficient of discharge Cd is required to account for the small amount
of friction between inlet and throat and for the effects of curvature of the
streamlines which the given theory ignores. In practice, values of Cd are
typically between 0.95 and 0.99.
All the foregoing applies to a flume under conditions of free discharge,
that is, where the liquid surface downstream is not maintained at too high a
level. The level in the outlet from the flume continues to fall and thus rapid
flow exists where the width of the passage again increases. If conditions
downstream of the flume are such that the velocity is greater than the critical,
the surface of the liquid issuing from the flume gradually merges into the
normal depth of flow in the downstream channel, and any excess energy
possessed by the liquid emerging from the flume is dissipated by friction.
On the other hand, if conditions downstream are such as to demand a
velocity less than the critical, the flow has to change from rapid to tranquil.
This change normally takes place through a hydraulic jump (Fig. 10.38). The
jump occurs at a point where the depth of the rapid flow is such as to give
the correct depth of subsequent tranquil flow (according to eqn 10.25 if the
section is rectangular). A limiting position is that in which the jump – then
Fig. 10.38 Flow through venturi flume. The critical depth at the throat is greater
because the narrowing of the channel increases the discharge divided by width q and
for rectangular section hc = (q 2 /g) 1/3 .