Environmental Engineering Reference
In-Depth Information
Thus:
∂y
∂x
−
v
g
∂v
∂x
−
1
g
∂v
∂t
S
f
=
S
o
−
(
de Saint Venant equation
)
(2.38)
v
g
∂v
∂x
−
1
g
∂v
∂t
=
v
2
C
2
R
∂y
∂x
−
S
f
=
S
o
−
(2.39)
This is the de Saint Venant equation, where:
y
=
water depth (m)
t
=
time (s)
v
=
average velocity (m/s)
x
=
longitudinal distance (m)
S
o
=
channel slope (m/m)
S
f
=
friction slope
=
slope of the energy line (m/m)
By definition
Q
=
vA
and
v
=
Q/A
∂y
∂x
−
v
gA
∂Q
∂x
−
1
gA
∂Q
∂t
=
S
o
−
S
f
−
0
(2.40)
Dynamic equation for gradually varied unsteady flow
This general dynamic equation for gradually varied unsteady flow is only
true when the pressure is hydrostatic, meaning that the vertical component
of the acceleration is negligible and that the flow lines are straight and
parallel.
The first two terms on the right hand side (
S
o
−
S
f
=
0) signifies
steady uniform flow. The first four terms together [
S
o
−
S
f
−
∂y/∂x
−
0] express a steady gradually varied flow.
The dynamic equation may also be written in another form:
v/gA
(
∂Q/∂x
)
=
∂y
∂x
+
v
g
∂v
∂x
+
1
g
∂v
∂t
+
S
f
−
S
o
=
0
(2.41)
g
∂y
v
∂v
∂v
∂t
+
∂x
+
∂x
+
g
(
S
f
−
S
o
)
=
0
(2.42)
These equations clearly show the four variables in unsteady, open channel
flow, namely
x
,
t
,
v
and
y
.
Continuity equation
The law of continuity for an unsteady, one-dimensional flow can be found
by considering the conservation of mass of a small canal reach (d
x
)
between two cross-sections. The difference in outflow and inflow in the
reach during a time step d
t
is equal to the change of storage over that
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