Environmental Engineering Reference
In-Depth Information
For the dam-break problem, the initial and boundary conditions are
−∞ <x<∞,u
(
x,
0) = 0
,
x<
0
,h
(
x,
0) =
h
i
,
x>
0
,h
(
x,
0) = 0
.
[1.23]
The analytical solutions to equations [1.21-1.22] are the well-known Ritter solutions.
We are looking for a similarity solution in the form [GRA 94]
u
∗
=
t
∗β/α
U
(
ζ
∗
) and
h
∗
=
t
∗γ/α
H
(
ζ
∗
)
,
with
ζ
∗
=
x
∗
/t
∗α
the similarity variable and
H
and
U
two unknown functions.
Substituting
u
∗
and
h
∗
with their similarity forms into equations [1.21-1.22], we find:
β
+
α
=1and
γ
+2
α
=2. For this solution to satisfy the initial and boundary
conditions, we must pose
β
=
γ
=0, hence
α
=1. We then infer
U
H
=0
,
HU− ζ
∗
U − ζ
∗
·
kg
cos
θ
where the prime denotes the
ζ
∗
-derivative. For this system to admit a non-constant
solution, its determinant must vanish, which leads to
kgH
cos
θ
=(
U − ζ
∗
)
2
.
On substituting this relation into the system above, we deduce
U
=2
ζ
∗
/
3, thus
U
=2(
ζ
∗
+
c
)
/
3, where
c
is a constant of integration,
H
=4(
c−
2
ζ
∗
)
2
/
(9
kg
cos
θ
).
The constant
c
is found using the boundary condition
s and by
assuming that the
undisturbed flow slides at constant velocity
δt
:
c
=
√
kgh
i
cos
θ
. Returning to the
original variables, we find
x
t
u
(
x, t
)=
u
∗
+
δt
=
2
3
+
δt
+
c
,
[1.24]
2
1
9
kg
cos
θ
−
x
t
+
2
t
+2
c
h
(
x, t
)=
.
[1.25]
The boundary conditions also imply that the solution is valid over the
ζ
-range [
−c −
δt,
2
c
+
δt/
2]; the lower bound corresponds to the upstream condition
u
=0, while the
upper bound is given by the downstream condition
h
=0. It is worth noting that the
front velocity
u
f
=2
c
+
δt/
2 is constantly increasing or decreasing depending on the
sign of
δ
. When
δ<
0 (friction in excess of slope angle), the front velocity vanishes
at
t
=4
c/|δ|
. Figure 1.14 shows that the shape of the tip region is parabolic at short
times (
δt c
), in agreement with experimental data [BAL 05, SIA 05]. Solutions
corresponding to finite released volumes were also obtained by Ancey
et al.
[ANC 08],
Hogg [HOG 06], and Savage and Nohguchi [SAV 88], Savage and Hutter [SAV 89].
