Environmental Engineering Reference
In-Depth Information
From this general, continuity equation follows the equation for three-
dimensional, steady and incompressible flow (
ρ
=
constant):
δu
δx
+
δv
δy
+
δw
δz
=
0
(2.7)
In the most common case, namely a two-dimensional flow, the equation
becomes:
δu
δx
+
δv
δy
=
0
(2.8)
When
δv/δy
=
0, the flow is one-dimensional and the equation reads:
δu
δx
=
0
(2.9)
This equation results in the fact that the velocity
u
is constant and it
represents uniform flow; the direction and magnitude of the velocity in all
points are the same. For a steady flow, the discharge
Q
is constant along
the canal and the continuity principle reads:
A
Q
=
v
d
A
=¯
vA
=
constant
Q
=¯
v
1
A
1
=¯
v
2
A
2
(2.10)
where:
A
area of the cross section (m
2
)
=
mean velocity perpendicular to the cross-section (m/s)
When the mean velocity in an open channel is constant (
v
1
=
v
¯
=
v
2
) then the
area
A
of the cross sections is the same; the channel has a prismatic cross
section with
A
1
=
A
2
. When the flow is uniform, the water depth will be
the same in all sections.
Conservation of momentum
The second law of Newton states that the sum of all external forces
P
equals the rate of change of momentum. The change of momentum per
unit of time is equal to the resultant of all external forces (hydrostatic,
friction, weight) acting on the body of flowing water:
P
(
m
v
)
=
lim
t
→
0
(2.11)
t
P
d(
m
v
)
out
−
d(
m
v
)
in
(
m
1
v
1
)
t
+
t
−
(
m
1
v
1
)
t
=
+
(2.12)
d
t
d
t
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