fluid_mechanics

564 Chapter 10 ■ Open-Channel Flow
(a)
F I G U R E 10.22
nappe, (b) submerged nappe.
(b)
Flow conditions over a weir without a free nappe: (a) plunging
Flowrate over a
weir depends on
whether the nappe
is free or submerged.
The above results for sharp-crested weirs are valid provided the area under the nappe is
ventilated to atmospheric pressure. Although this is not a problem for triangular weirs, for
rectangular weirs it is sometimes necessary to provide ventilation tubes to ensure atmospheric
pressure in this region. In addition, depending on downstream conditions, it is possible to obtain
submerged weir operation, as is indicated in Fig. 10.22. Clearly the flowrate will be different for
these situations than that given by Eqs. 10.30 and 10.32.
H
H/L w = 0.08
H/L w = 0.50
10.6.3 Broad-Crested Weirs
A broad-crested weir is a structure in an open channel that has a horizontal crest above which the
fluid pressure may be considered hydrostatic. A typical configuration is shown in Fig. 10.23.
Generally, to ensure proper operation, these weirs are restricted to the range 0.08 6 HL w 6 0.50.
These conditions are drawn to scale in the figure in the margin. For long weir blocks 1HL w less
than 0.082, head losses across the weir cannot be neglected. On the other hand, for short weir blocks
1HL w greater than 0.502 the streamlines of the flow over the weir block are not horizontal. Although
broad-crested weirs can be used in channels of any cross-sectional shape, we restrict our attention
to rectangular channels.
The operation of a broad-crested weir is based on the fact that nearly uniform critical flow
is achieved in the short reach above the weir block. 1If HL w 6 0.08, viscous effects are important,
and the flow is subcritical over the weir.2 If the kinetic energy of the upstream flow is negligible,
then V 2 12g y and the upstream specific energy is E 1 V 2 1 12g y 1 y 1 . Observations show
that as the flow passes over the weir block, it accelerates and reaches critical conditions, y 2 y c
and Fr 2 1 1i.e., V 2 c 2 2, corresponding to the nose of the specific energy curve 1see Fig. 10.72.
The flow does not accelerate to supercritical conditions 1Fr 2 7 12. To do so would require the
ability of the downstream fluid to communicate with the upstream fluid to let it know that there is
an end of the weir block. Since waves cannot propagate upstream against a critical flow, this
information cannot be transmitted. The flow remains critical, not supercritical, across the weir
block.
The Bernoulli equation can be applied between point 112 upstream of the weir and point 122
over the weir where the flow is critical to obtain
L w
F I G U R E 10.23 Broad-crested
or, if the upstream velocity head is negligible
H P w V 2 1
2g y c P w V 2 c
2g
H y c 1V 2 c V 2 12
V 2 c
2g 2g
(1)
(2)
y 1
V 1
P w
H
y2 = y c
L w
Weir block
V 2 = V c
weir geometry.