fluid_mechanics

556 Chapter 10 ■ Open-Channel Flow
Upstream velocity
profile
Bridge pier
V10.10 Big Sioux
River bridge
collapse
Horseshoe vortex
Scouring of
channel bottom
F I G U R E 10.14
The complex three-dimensional flow structure around a bridge pier.
A hydraulic jump is
a steplike increase
in fluid depth in an
open channel.
p
c
y 1
/2
y 1
10.6.1 The Hydraulic Jump
Observations of flows in open channels show that under certain conditions it is possible that the
fluid depth will change very rapidly over a short length of the channel without any change in the
channel configuration. Such changes in depth can be approximated as a discontinuity in the freesurface
elevation 1dydx 2. For reasons discussed below, this step change in depth is always
from a shallow to a deeper depth—always a step up, never a step down.
Physically, this near discontinuity, called a hydraulic jump, may result when there is a conflict
between the upstream and downstream influences that control a particular section 1or reach2 of a channel.
For example, a sluice gate may require that the conditions at the upstream portion of the channel
1downstream of the gate2 be supercritical flow, while obstructions in the channel on the downstream
end of the reach may require that the flow be subcritical. The hydraulic jump provides the mechanism
1a nearly discontinuous one at that2 to make the transition between the two types of flow.
The simplest type of hydraulic jump occurs in a horizontal, rectangular channel as is indicated
in Fig. 10.15. Although the flow within the jump itself is extremely complex and agitated, it is
reasonable to assume that the flow at sections 112 and 122 is nearly uniform, steady, and onedimensional.
In addition, we neglect any wall shear stresses, t w , within the relatively short segment
between these two sections. Under these conditions the x component of the momentum equation
1Eq. 5.222 for the control volume indicated can be written as
F 1 F 2 rQ1V 2 V 1 2 rV 1 y 1 b1V 2 V 1 2
where, as indicated by the figure in the margin, the pressure force at either section is hydrostatic.
That is, F and F 2 p c2 A 2 gy 2 1 p c1 A 1 gy 2 1b2
2b2, where p c1 gy 12 and p c2 gy 22 are
the pressures at the centroids of the channel cross sections and b is the channel width. Thus, the
momentum equation becomes
2
y 1
2 y2 2
2 V 1y 1
g 1V 2 V 1 2
In addition to the momentum equation, we have the conservation of mass equation 1Eq. 5.122
y 1 bV 1 y 2 bV 2 Q
(10.21)
(10.22)
Control
volume
h L
(2)
V 2
Energy
line
(1) V y 2
1
F2
Q
y F 1 1
x
τ w = 0
F I G U R E 10.15 Hydraulic jump geometry.