Environmental Engineering Reference
In-Depth Information
are used extensively in engineering throughout the world. Despite their simplicity
and approximate character, they can provide valuable results, the more so as their
parameters and the computation procedures combining expert rules and scientific
basis have benefited from many improvements over the last few decades [ANC 03,
ANC 04b, ANC 04c, ANC 05].
2.2.3.1.
Simple models for flowing avalanches
The early models date back to the beginning of the 20th century [MOU 22]. For
the Olympic Games at Chamonix in 1924, the Swiss professor Lagotala [LAG 27]
calculated the velocity of avalanches in the Favrand path. His method was then
extended by Voellmy [VOE 55], who popularized it. Many models have elaborated on
Voellmy's work. The Voellmy-Salm-Gubler (VSG) model [SAL 90] and the Perla-
Cheng-McClung model [PER 80] are probably the best-known avalanche-dynamics
models used throughout the world.
Here we outline the VSG model. The avalanche is assumed to behave as a rigid
body, which moves along an inclined plane. The position of the center of mass is given
by its abscissa
x
in the downward direction. The momentum equation is
d
d
t
=
g
sin
θ −
F
m
,
[2.2]
where
m
is the avalanche mass,
u
the velocity, and
θ
the mean slope of the path. In
this model, a flowing avalanche is considered as a sliding block subject to a frictional
force:
F
=
mg
u
2
ξh
+
μmg
cos
θ,
[2.3]
where
h
is the mean flow depth of the avalanche,
μ
a friction coefficient related to
the snow fluidity, and
ξ
a coefficient of dynamic friction related to path roughness.
If these last two parameters cannot be measured directly, they can be adjusted from
several series of past events. It is generally accepted that the friction coefficient
μ
only depends on the avalanche size and ranges from 0.4 (small avalanches) to
0.155 (very large avalanches) [SAL 90]; in practice, lower values can be observed
for large-volume avalanches [ANC 04b]. Likewise, the dynamic parameter
ξ
reflects
the influence of the path on avalanche motion. When an avalanche runs down a wide
open rough slope,
ξ
is close to 1000 or more. Conversely, for avalanches moving down
confined straight gullies,
ξ
can be taken as being equal to 400. In a steady state, the
velocity is directly inferred from the momentum balance equation:
u
=
ξh
cos
θ
(tan
θ − μ
)
.
[2.4]
According to this equation two flow regimes can occur depending on path inclination.
For tan
θ>μ
, equation [2.4] has a real solution and a steady regime can occur. For
