Download Chapter 7b Specific head_Critical Depth_Hydraulic Jump b

Specific head, Critical depth, Hydraulic Jumps
Return to the Bernoulli equation for open channels:
H is called the total head. The units are meters
We can separate z into two components: the depth of water h and the depth to
some lower datum z0, maybe to sea-level.
water
sand, gravel
We can define a component of the total head that only contains the flow depth
and the velocity head.
This is called the specific head. Notice we changed h to y
Now redefine specific head in terms of discharge V = Q/A instead of velocity V.
The velocity is squared, so we will get a discharge term squared Q2/A2 .
Let’s take the simple example of a rectangular channel, and then
define q = Q/width. The area for a rectangle is A = base x height, so only the
height part is left after we divide by the width
On the right hand side of the equation, q2/2gy2 is the specific kinetic energy head,
and y is the potential energy head. Notice that all terms have units meters, depth.
We can graph how flow depth h changes for any change in Specific Energy E.
For some constant q:
This graph says that above some critical depth, there are two permissible flow
depths that will yield an identical discharge q.
The flow depth is on the vertical axis as in nature. There’s one flow depth where
most of the specific head is held as potential energy (y), and just a little is
held as kinetic energy (V2/2g), AND there’s another one where most of the
energy is kinetic, and little is potential.
There’s also one specific depth, the critical depth, and some velocity for which
energy E in the system is minimized. This is the lowest specific head for a given
discharge q.It says that if the flow is deeper than this, the velocity drops, but if the
flow is shallower than this the velocity increases.
This critical point occurs where the derivative of E is 0. So,
take the derivative of E with respect to y. Starting with the q form will be easiest
which equals
because 1/y2 equals y-2
and setting
gives
Near here Bedient takes a big jump without comment. What follows is a
demonstration of the details:
but at the minimum
so
(1)
and
And so, returning to the text, at the minimum Energy the dimensionless
Froude Number is:
Flow deeper than a Froude Number of F=1 is called subcritical or streaming flow.
Flow shallower than F=1 is called supercritical or shooting flow.
What happens if the flow crosses from one region to another? 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 either as a hydraulic jump, a standing wave,
or a bore.