Geoscience Reference
In-Depth Information
0.5
x
+
=5
0.4
2
u
+
1
0.3
x
+
=20
x
+
=10
0.2
3
0.1
0
0
2
4
6
8
10
12
t
+
Fig. 5.10 Comparison between the unit response function
u
+
=
/
(
S
0
V
0
) for the upstream inflow
problem, obtained with the complete shallow water equations (1) and obtained with the
diffusion approximation (2 and 3). The curves indicated by 2 are obtained with the diffusivity
D
0
=
bq
0
/
h
0
u
S
0
. Each unit response
curve is shown as a function of time
t
+
at different distances
x
+
downstream from the release
point of the unit impulse. The constant is
a
=
2
/
3 and the Froude number is taken as
Fr
0
=
0
.
5. Some of the curves indicated by 1 and 3 are also shown in Figures 5.8 and 5.9.
S
0
1
a
2
Fr
0
and those indicated by 3 with
D
0
=
bq
0
/
−
As can be seen in Figure 5.10, for Fr
0
=
0
.
5, the diffusivity (5.97) leads to an improved
agreement with the solution of the full shallow water equations. However, for smaller values
of Fr
0
, or for large values of
x
+
, the effect of this difference in diffusivity between Equations
(5.89) and (5.97) can be expected to be small.
5.4.3
The quasi-steady-uniform flow approach: a second approximation
It was indicated earlier (see (5.81), for example) that typically the first three terms
in the momentum equation (5.22) (or (5.25)), tend to be some two to three orders of
magnitude smaller than those representing the effects of gravity and of friction, namely
S
0
and
S
f
. In the previous section the dynamic terms were omitted, but the term
x
was kept in the formulation, and this was shown to lead to the diffusion analogy. Often,
however, under conditions that turn out to be quite common in nature, it is possible to
neglect that term as well, and to keep only
S
0
and
S
f
. In other words, it is assumed that
water flows downhill but it is prevented from accelerating or decelerating very much,
because the frictional resistance of the bed is overwhelming all other factors. Thus
under such conditions the momentum equation (5.22) (or (5.25)) can be simplified to the
following
∂
h
/∂
S
f
=
S
0
(5.100)
Again, as in the previous section, the continuity equation (5.13) (or (5.24)) is left intact.
This is the basis of the so-called quasi-steady-uniform flow or kinematic wave approxi-
mation.
