fluid_mechanics

3.6 Examples of Use of the Bernoulli Equation 121
V 1
z 1
a
Sluice gates
(1)
Sluice gate
width = b
b
z 2
V 2 (2)
Q
a
(3)
(4)
(a)
F I G U R E 3.19
(b)
Sluice gate geometry. (Photograph courtesy of Plasti-Fab, Inc.)
give
Thus, we apply the Bernoulli equation between points on the free surfaces at 112 and 122 to
p 1 1 2rV 2 1 gz 1 p 2 1 2rV 2 2 gz 2
Also, if the gate is the same width as the channel so that A 1 bz 1 and A 2 bz 2 , the continuity
equation gives
The flowrate under
a sluice gate depends
on the water
depths on either
side of the gate.
Q A 1 V 1 bV 1 z 1 A 2 V 2 bV 2 z 2
With the fact that p 1 p 2 0, these equations can be combined and rearranged to give the flowrate
as
2g1z 1 z 2 2
Q z 2 b B 1 1z 2z 1 2 2
In the limit of z 1 z 2 this result simply becomes
(3.21)
Q z 2 b12gz 1
This limiting result represents the fact that if the depth ratio, z 1z 2 , is large, the kinetic energy of
the fluid upstream of the gate is negligible and the fluid velocity after it has fallen a distance
1z 1 z 2 2 z 1 is approximately V 2 12gz 1 .
The results of Eq. 3.21 could also be obtained by using the Bernoulli equation between points
132 and 142 and the fact that p 3 gz 1 and p 4 gz 2 since the streamlines at these sections are straight.
In this formulation, rather than the potential energies at 112 and 122, we have the pressure contributions
at 132 and 142.
The downstream depth, z 2 , not the gate opening, a, was used to obtain the result of Eq. 3.21.
As was discussed relative to flow from an orifice 1Fig. 3.142, the fluid cannot turn a sharp 90° corner.
A vena contracta results with a contraction coefficient, C c z 2a, less than 1. Typically C c is
approximately 0.61 over the depth ratio range of 0 6 az 1 6 0.2. For larger values of az 1 the
value of C c increases rapidly.
E XAMPLE 3.12
Sluice Gate
GIVEN Water flows under the sluice gate shown in Fig. E3.12a. FIND Determine the approximate flowrate per unit width of
the channel.