WATER FLOW 1283
following flow regimes could be defined with reference to the
specific energy diagram.
Subcritical flow denotes tranquil flow in which the
Froude number and the mean velocity are less than unity and
the celerity of the gravity wave respectively.
Critical flow represents the discharge phenomenon
where: (1) for a constant specific energy, the discharge is a
maximum; (2) the specific energy is a minimum for a con-
stant discharge; (3) the critical velocity equals the celerity of
a small gravity wave; (4) the Froude number equals unity;
(5) the critical depth is also the depth of minimum pressure-
momentum force.
Supercritical flow which is also known as shooting
or rapid flow, is that state of water flow where the Froude
number and mean velocity exceed unity and the speed of
transmission of a surface wave respectively.
Based on the equations developed earlier for critical
flow, particular formulae can be derived for a rectangular
section:Dq
c
c
g⎛
⎝⎜⎞
⎠⎟13/
(55)DEcc2
3() (56)^DV
c
c2g(57)^V
Dc
g c1. (58)Flow Transition The concept of the normal depth is an impor-
tant parameter in the study of flow transition. For a given
channel and any fixed discharge, uniform flow will occur at
one unique depth. It is the depth attained in a long channel
when the component of gravity force is just balanced by the
frictional resistance of the channel.
When the normal and critical depth are equal, the flow
is critical and the bed and energy slopes are the same.
The channel bed then has a critical slope. The bed slope
is termed mild when the normal depth exceeds the critical
depth; the bed slope is then less than the critical energy
slope and the flow regime is subcritical. When the normal
depth lies below the critical depth and, hence, the bed slope
is greater than the critical energy slope, the channel slope
is considered to be steep and the supercritical flow regime
prevails.
When water makes a transition from a channel with a
mild slope to another with a steep slope, or vice versa, the
flow passes through the critical depth close to the junction
of the two channels. The section in which the water depth is
critical defines a channel control. The weir acts as a control
when water flows over it as critical depth is attained there.
Hydraulic Jump A hydraulic jump occurs when supercritical
flow makes a transition to subcritical flow. A common occur-
rence of a hydraulic jump takes place at the base of a chutespillway. Figure 8 shows the energy momentum and depth
relations for a hydraulic jump and also defines the symbols
to be used.
In developing the equations for the hydraulic jump, the
following assumptions are used:1) the bed slope is considered small and neglected,
2) frictional resistance along the bed and sides of the
channel are omitted.Consider the control volume between Sections (1) and (2).
Applying the impulse-momentum principle:FF QVV 12 21 g
g() (59)orFQVF QV112 2gg
gg(60)in which F 1 and F 1 denote the hydrostatic forces at sections 1
and 2 respectively. The term (F g/gQV) is given the name
pressure-momentum force. Let A flow area, y– distance
of centroid of flow area from surface; then they hydrostatic
force gAy– and Eq. (60) can be written as:AyQ
AAyQ(^11) A
2
1
21
2
2
gg
. (61)
Equation (60) states the condition for the formation of a
hydraulic jump and suggests a graphical solution. A plot of
Eq. (60) for any fixed discharge is shown in Figure 8. For any
given up-stream supercritical water depth, which is usually
known such as at the toe of a spillway, the subcritical hydraulic
jump depth or sequent depth can be obtained from the graph.
For a rectangular cross-section channel and utilising the
continuity relation:QVA VA^11 2 2. (62)^DS.E. P.M.F.D D
TELEj V
2 /2g2FV 12 /2gDc F 2(1) (2)E P.M.F.FLOWFIGURE 8 Hydraulic jump relations.C023_003_r03.indd 1283C023_003_r03.indd 1283 11/18/2005 11:12:14 AM11/18/2005 11:12:14 AM