Chapter 7 continued
Open Channel Flow
Specific Energy, Critical Flow,
Froude Numbers, Hydraulic Jump
Bernoulli
• Return to the Bernoulli equation for open
channels:
Pressure ,
kinetic energy,
and potential energy head
• H is the total head. The units are length.
Depth and height above datum
• As you know, we can separate z into two
components: the depth of water h and the
depth to some lower datum z0, maybe to sealevel.
Specific Energy
• We can define a component of the total energy that
only contains the flow depth and the velocity term.
*Note, no P/g
• This is called the specific energy. Notice we changed
h to y, and H (energy head) to E for energy
• * this is appropriate for open channels, since nearby areas have about the same pressure
Specific Energy for Flow Rate
• Now redefine specific energy in terms of
discharge Q instead of velocity V.
• Substitute V = Q/A
• The velocity in kinetic energy was squared in
the previous slide, so we will get a discharge
term squared Q2/A2 .
Flow Rate per unit width
• Let’s take the simple example of a rectangular
channel, and then define q = Q/width. The
area for a rectangle is A = base (i.e. width) x
height, so only the height part is left after we
divide by the width (the width of a unit width
is 1. )
We changed from Q to q to show that it’s discharge per unit width.
On the right hand side of the equation, q2/2gy2 is the specific kinetic energy,
and y is the specific potential energy. Notice that all terms have units of
length, for the depth.
Plot of Specific Energy vs. Depth
• Lets plot E, the specific energy, against y, for a
particular flow rate (discharge) per unit width, q.
At the red line specific energy is minimum, so the exact slope at that
point is zero, dE/dy = 0. For higher energy (blue line) there are
two possible depths for the same specific energy.
Same plot with depth vertical
• Let’s turn the graph on its side, as in the text.
We can again graph how flow depth y changes
for any change in Specific Energy E. For some
constant q:
For the energy line shown,
there are two possible
depths where it crosses the
blue plot of some flow per
unit width q. The upper one
is mostly potential energy
(the water is elevated) ,
and the velocity is small;
the lower one has greater
velocity and is not as high
Critical Depth
• There’s also one specific depth, yc, the critical depth, for which energy E in the
system is minimized. This is the lowest specific energy for a given discharge q. If the
flow is deeper (higher on the graph) than this, the velocity drops, but if the flow is
shallower than this, the velocity increases.
Solving for Critical Depth
• This critical point occurs where the derivative (slope)
dE/dy is 0. So, take the derivative of E with respect to
y. Only E and y are variables.
which is
Then
or
Setting this equal to zero
gives
At the minimum specific energy
the ratio of velocity squared to depth
times the gravitational acceleration is
one.
Froude Number
• V2/gy is the Froude Number, squared. Notice that it
is dimensionless, i.e. all the units cancel. It is the
ratio of kinetic to potential energy, and is used to
characterize open channel flow.
• And so, returning to the text, at the minimum
specific energy the dimensionless Froude Number
is:
Flow deeper than a Froude Number of Fr=1 (large depth in
denominator so Fr <1) is called subcritical flow. It is higher and
slower.
Flow shallower than Fr=1 (Fr>1) is called supercritical or
shooting flow. It is lower and faster.
Never design a channel on a slope that is near
critical (Fr = 1) because of the unpredictable water
surface.
Hydraulic Jump
• What happens if the Froude number crosses from Fr>1 (shallow,
fast) to Fr < 1 (deep, slow)? At the transition, the flow has to
suddenly change from one flow depth to the other. It forms a
jump between one and the other. The two regions are separated
by a continuously collapsing wall of water referred to as a
hydraulic jump.
The Depth Ratio for a Hydraulic Jump
The ratio of the depths is:
Momentum Balance for a Rectangular Channel
Again, Bedient
skips the
derivation. Here it
is.
Example 7-6: Calculation of a Hydraulic Jump
A sluice gate is constructed across an open channel.
Water flowing under it creates a hydraulic jump.
Determine the depth just downstream of the jump
(point b) if the depth of flow at point a is 0.0563
meters and the velocity at point a is 5.33 meters/sec.