Geoscience Reference
In-Depth Information
10
9
8
7
6
5
4
3
2
1
0
0
5
10
15
20
25
[m]
30
35
40
45
50
Fig. 2 Example 1 - Parameters:
h
L
0.
Solid line
is the height profile (m),
dashed line
the velocity (m/s), and
thin solid line
is the bottom
profile;
t
¼
10.0 m,
h
R
¼
1.0 m,
Y
bl
¼
1.0 m, a
¼
0.2, and d
¼
¼
1s
The fundamental nondimensional parameters that describe the initial conditions
are Ferreira et al. (
2006
):
h
R
þ
min
ð
0
;
Y
bl
Þ
Y
bl
a
Y
bl
Þ
;
d
(24)
h
L
þ
max
ð
0
;
h
L
þ
max
ð
0
;
Y
bl
Þ
where
h
R
and
h
L
are the water heights of the initial right and left states, respectively,
and
Y
bl
is the bed elevation on the left side considering the right side the reference
horizontal plane.
Example 1 is a 1D result. The initial left state was a 10.0 m water column. The
initial right state comprised a 1.0 m water column over a 1.0 m bed step. The
solution, presented in Fig.
2
, contains a left-moving rarefaction wave, a stationary
shock at the step, and a right-moving shock wave. The presence of the step leads to
a reduction of the discharge to the right side of the domain, induced by a head loss
in the stationary shock. The position of the fan expansion and the shock are correct
(cf. Alcrudo and Benkhaldoun
2001
) and no numerical oscillations are produced.
Example 2 is a 1D simulation with dry bed. The initial left state had
h
i
¼
0m.
The solution, presented in Fig.
3
, contains a left-moving rarefaction wave and a
right-moving wetting advancing front. It is comparable to an exact Ritter solution.
The scheme being first order, the solution presents itself smoothed, without the
typical discontinuity present in the exact solution. The velocity profile shows that
the maximum speed is near 14 ms
1
, against the 19.7 ms
1
given by the Ritter
solution. This can result from the use of a coarse mesh, together with the tendency
of the scheme to present numerical diffusion and the design of the wetting-drying
algorithm. A correct behavior is observed in the zone where a sonic rarefaction was
expected, showing the effectiveness of the entropy corrections.
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