Environmental Engineering Reference
In-Depth Information
Finally a combination of the two equations results in a general equation
for non-steady flow:
General equation of
non-steady flow in a
rectangular channel
(2.54)
c
)
∂
(
v
±
2
c
)
∂
(
v
±
2
c
)
(
v
±
+
−
g
(
S
o
−
S
f
)
=
0
∂x
∂t
Remember that the general mathematical expression for a change of
a function
f
in the
x
-
t
-diagram when going from a given point 1 to a
neighbouring point 2 can be given by the partial derivatives of the function
in the
t
-direction and the
x
-direction.
∂f
∂t
d
t
∂f
∂x
d
x
d
f
d
t
=
∂f
∂x
d
x
d
t
+
∂f
∂t
d
f
=
+
or
In this equation
f
is a variable dependent on the two independent variables
x
and
t
, and the equations give the rate of change of
f
if
x
and
t
are
simultaneously varied in a prescribed manner, given by d
x
/d
t
. Assume
that the function
f
is, for example, the water depth
y
(
f
=
y
(
x
,
t
)).
d
y
d
t
=
∂y
∂x
d
x
d
t
+
∂y
∂t
Assuming that the function
y
is the water depth then the situation can be
considered in the following way: to an observer walking with a speed
d
x
/d
t
along a dike of an open canal, the depth
y
will appear to vary with
time at the rate given by the equation for d
y
/d
t
. A similar result would of
course be true for any other parameter such as
v
,
q
,or
c
.
The total differential d
u
/d
t
for the function
u
=
v
+
2
c
can be found in
the same way as d
y
/d
t
and will give:
d
u
d
t
=
∂u
∂x
d
x
d
t
+
∂u
∂t
d
x
d
t
=
with
v
=
f
(
x
,
t
) and
v
+
c
Assume that
S
o
is constant and that
S
o
and
S
f
are both relatively small.
Then from this assumption follows that
g
(
S
o
−
S
f
) is also very small and
a new set of equations for the unsteady flow can be written as:
c
)
∂
(
v
±
2
c
)
∂
(
v
±
2
c
)
(
v
±
+
=
g
(
S
o
−
S
f
)
(2.55)
∂x
∂t
Equation of non-steady
flow in a rectangular
channel
c
)
∂
(
v
±
2
c
)
∂
(
v
±
2
c
)
(
v
±
+
=
0
(2.56)
∂x
∂t
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