Geoscience Reference
In-Depth Information
The diffusion equation of free surface flow
Consider again the case of a very wide channel. Omission of the acceleration terms
reduces the momentum equation (5.22), in the absence of lateral inflow, to
∂
h
x
+
S
f
−
S
0
=
0
(5.82)
∂
If Equation (5.39) can be assumed to be valid, the friction slope can be written concisely
as
S
f
=
α
r
q
1
/
b
(5.83)
(
C
r
h
a
+
1
)
−
1
/
b
, and where as before
q
where
α
r
=
=
(
Vh
) is the flow rate per unit width.
Thus (5.82) can be written as
S
0
+
∂
h
α
r
q
1
/
b
−
x
=
0
(5.84)
∂
Proceeding in the same way as for Equation (5.67), that is applying
∂/∂
t
to the momentum
equation (5.84) and
∂/∂
x
to the continuity equation (5.13), and subtracting one from the
other, one obtains
2
q
α
r
q
−
1
+
1
/
b
∂
q
∂
t
−
∂
q
1
/
b
∂α
r
∂
b
−
1
x
2
+
t
=
0
(5.85)
∂
Because
α
r
depends only on the geometry of the cross section, which in turn is related
to the water depth
h
, the derivative in the third term becomes
∂α
r
∂
d
dh
∂
α
r
h
∂
t
=
t
Making use of the continuity equation (5.13) to replace this partial time derivative of
h
,
i.e. (
x
), and making use of (5.83) to eliminate
q
1
/
b
, one obtains from
∂
h
/∂
t
)
=−
(
∂
q
/∂
(5.85)
bq
α
r
∂
bq
S
f
∂
∂
q
∂
d
α
r
dh
q
2
q
t
−
x
=
(5.86)
∂
∂
x
2
The same derivation carried out for a channel with wide rectangular cross section
A
c
=
(
B
c
h
) can be shown to yield a similar result, namely
∂
bQ
α
r
∂
bQ
B
c
S
f
∂
2
Q
∂
Q
∂
α
r
dA
c
d
Q
∂
t
−
x
=
(5.87)
x
2
Equations (5.86) and (5.87) are in the form of a nonlinear advective diffusion equa-
tion. Accordingly, the term
D
=
bq
/
S
f
(or
bQ
/
B
c
S
f
) can be referred to as diffusivity;
S
f
(
d
α
−
b
r
for convenient reference, the term
c
d
=−
[(
bq
/α
r
)(
d
α
r
/
dh
)]
=
/
dh
) (or
c
d
=
B
c
S
f
(
d
α
−
b
r
−
dA
c
)) will henceforth be called the
advectivity
.
As usual, the dimensions of the diffusivity are [L
2
T
−
1
], and those of the advectivity
[L T
−
1
]. In general descriptive terms, the magnitude of the advectivity reflects the speed
of propagation of a flow disturbance (in
q
or
h
), whereas the diffusivity is related to the
speed with which this disturbance will spread out streamwise or, which is equivalent,
dissipate its magnitude.
[(
bQ
/α
r
)(
d
α
r
/
dA
c
)]
=
/
