Environmental Engineering Reference
In-Depth Information
If the flow is kept at a constant value, the question arises, which water
levels result from the formula. In order to answer the question, the area has to
be expressed in terms of water level. For a rectangular cross-section one
obtains:
q
2
2
gh
2
þ h
cos
ðÞ¼H
e
where
q
denotes the flow rate per unit width. The formula yields an energy
height as function of water depth
h
. The situation can be visualized using
MATLAB
®
.
The corresponding M-file is included in the accompanying software
under the name
“OpenChannel.m”
Figure
11.5
illustrates that there is a minimum energy height
H
e
,
min
. The
water level, corresponding with
H
e
,
min
, is commonly referred to as critical
height
h
crit
. For all possible levels
H
e
> H
e,min
, there are two possible water
table positions; one above
h
crit
and one below
h
crit
. Both values for the height
of the water column are denoted as
conjugated heights
. In the former situation
the velocity is lower than in the latter situation. That's why the first case is
called
subcritical
, while the second case is called
supercritical
. Height and
velocity at the critical state are given by:
p
q
2
p
gh
crit
cos
h
crit
¼
3
=g
cos
ðbÞ
v
crit
¼
ðbÞ
which can be derived from the condition
@H
e
=@h ¼
0. For
b ¼
0 the critical
state can be related to the
Froude
9
number
v
gh
p
Fr
¼
<
>
1. In open
channels and regulated rivers changes from subcritical to supercritical or vice
versa can be observed at locations where the channel characteristics change, as
channel boundary roughness, channel bottom slope, lower boundary elevation,
etc. (Rouse
1978
). Most natural rivers are in the sub-critical regime in most
parts (Olsen
2002
), except in the vicinity of waterfalls, weirs or other structures.
Flow is subcritical flow for Fr
1 and supercritical for Fr
(
continued
)
9
William Froude (1810-1879) English engineer.
