Environmental Engineering Reference
In-Depth Information
other initial conditions are at
t
=0and
x
=0,
H
=
H
0
,
L
=
L
0
,
V
0
=
k
v
H
0
L
0
, and
ρ
=
ρ
0
. At short times, the velocity is independent of the entrainment parameters and
the initial conditions (
ρ
0
and
V
0
):
2
gx
sin
θ,
Δ
ρ
0
Δ
ρ
0
+(1+
χ
)
ρ
a
≈
U ∝
2
gx
sin
θ
[2.17]
where we used
ρ
a
Δ
ρ
0
. This implies that the cloud accelerates vigorously in the
first instants (d
U/
d
x →∞
at
x
=0), then its velocity grows more slowly. At long
times for an infinite plane, the velocity reaches a constant asymptotic velocity that
depends mainly on the entrainment conditions for flows in the air:
2
gh
n
(1 +
α
l
2
)sin
θρ
s
α
v
(1 +
χ
)
ρ
a
U
∞
∝
.
[2.18]
Because of the slow growth of the velocity, this asymptotic velocity is reached only
at very long times. Without particle entrainment, the velocity reaches a maximum at
approximately
x
2
m
=(2
ρ
0
/
3
ρ
a
)
α
−
v
V
0
/
(1 +
χ
):
g
√
V
0
sin
θ
α
v
√
1+
χ
,
ρ
0
ρ
a
4
√
3
U
m
≈
then it decreases asymptotically as:
8Δ
ρ
0
3
ρ
a
gV
0
sin
θ
x
1
α
v
(1 +
χ
)
.
U ∝
[2.19]
In this case, the front position varies with time as:
x
f
∝
(
g
0
V
0
sin
θ
)
1
/
3
t
2
/
3
.
[2.20]
These simple calculations show the substantial influence of the particle entrainment
on cloud dynamics. In the absence of particle entrainment from the bed, the fluid
entrainment has a key role since it directly affects the value of the maximum velocity
that a cloud can reach.
Here, we examine only the avalanche of 25 February 1999, for which the front
velocity was recorded [DUF 01]. In Figure 2.10, we have reported the variation in the
mean front velocity
U
f
as a function of the horizontal downstream distance
y
f
: the
dots correspond to the measured data while the curves represent the solution obtained
by integrating equations [2.2-2.4] numerically and by assuming that the growth rate
coefficient depends on the overall Richardson number (solid line). For the initial
conditions, we assume that
u
0
=0,
h
0
=2
.
1m,
l
0
=20m, and
ρ
0
=
ρ
s
= 150 kg/m
3
.
Due to the high path gradient between the origin and the elevation
z
= 1800m
