Environmental Engineering Reference
In-Depth Information
r
v
¼
0
(11.6)
is fulfilled at a time instant, this property remains valid further on. It can be shown
that under condition (
11.6
) a potential
'
exists, which is characterized by the
property
v
¼r'
(11.7)
and that it fulfills the Bernoulli theorem (for example: Gallavotti
2002
):
r
@'
@t
þ
2
v
2
þ rf þ p ¼ C
(11.8)
where
).
C
is a constant in
a simply connected domain. If condition (
11.6
) is skipped,
C
is a constant for each
streamline but not in the entire flow domain. For steady states the first term in (
11.8
)
can be omitted. In fact, the Bernoulli theorem yields a relation between pressure and
velocity. In following chapters methods will be explained how the potential can be
determined, from which the velocity field is derived.
f
denotes the potential of the force vector f (i.e. f
¼rf
Sidebar 11.2 Open Channel Flow
Open channel flow is defined as flow in any situation in which a liquid has
a free surface, such as in channels, rivers, streams, ditches, uncovered
conduits and discharge from tailings ponds. There is open channel flow in
closed channels, such as pipes, tunnels or adits, if the liquid does not fill the
entire cross-section. Open channel flow is not under pressure, with gravity as
the driving force.
Some characteristics of open channel flow can be derived from the
Bernoulli theorem. For steady conditions according to (
11.8
) and (
11.10
),
one can state that the left hand side of the Bernoulli (
11.8
)
v
2
2
g
þ h
cos
ðÞ¼H
e
is a constant, which represents total energy in the dimension of height and is
therefore denoted as
H
e
, the energy height (see also textbooks on fluid mechan-
ics, for example: Schr
oder
1995
).
h
denotes the height of the water column, i.e.
the water level with reference to the zero level at the bottom of the flowing
water body. Using the formula between mean flow velocity
v
,cross-sectional
area
A
and flux
Q ¼ Av
, one may write the equation in the form:
€
Q
2
2
gAðhÞ
2
þ h
cos
ðÞ¼H
e
(
continued
)
