fluid_mechanics

542 Chapter 10 ■ Open-Channel Flow
Chapter 3 that the energy line is located a distance z 1the elevation from some datum to the channel
bottom2 plus the pressure head 1 pg2 plus the velocity head 1V 2 2g2 above the datum. Therefore,
Eq. 10.6 can be written as
y 1 y 2 1V 2
2 V 2 12
1S
2g
f S 0 2/
(10.7)
If there is no head loss, the energy line is horizontal 1S f 02, and the total energy of the
flow is free to shift between kinetic energy and potential energy in a conservative fashion. In
the specific instance of a horizontal channel bottom 1S 0 02 and negligible head loss 1S f 02,
Eq. 10.7 simply becomes
y 1 y 2 1V 2 2 V 2 12
2g
E
The specific energy
is the sum of potential
energy and
kinetic energy (per
unit weight).
q = constant
y
10.3.1 Specific Energy
The concept of the specific energy or specific head, E, defined as
E y V 2
2g
(10.8)
is often useful in open-channel flow considerations. The energy equation, Eq. 10.7, can be written
in terms of E as
E 1 E 2 1S f S 0 2/
(10.9)
If head losses are negligible, then S f 0 so that 1S f S 0 2/ S 0 / z 2 z 1 and the sum of the
specific energy and the elevation of the channel bottom remains constant 1i.e., E 1 z 1 E 2 z 2 ,
a statement of the Bernoulli equation2.
If we consider a simple channel whose cross-sectional shape is a rectangle of width b, the
specific energy can be written in terms of the flowrate per unit width, q Qb Vybb Vy, as
E y
2gy 2
(10.10)
which is illustrated by the figure in the margin.
For a given channel of constant width, the value of q remains constant along the channel, although
the depth, y, may vary. To gain insight into the flow processes involved, we consider the specific energy
diagram, a graph of E E1y2, with q fixed, as shown in Fig. 10.7. The relationship between the flow
depth, y, and the velocity head, V 2 2g, as given by Eq. 10.8 is indicated in the figure.
q2
y
V 2
2g
y = E
y sub
y
q = q" > q'
y c
E min E
y sup
q = q'
y neg
F I G U R E 10.7
diagram.
Specific energy