Environmental Engineering Reference
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models proposed so far rely on analogy with other physical phenomena: typical
examples include analogies with granular flows [CUI 07, SAV 89a, SAV 89b, TAI 01],
Newtonian fluids [HUN 94], power-law fluids [NOR 86], and viscoplastic flows
[ANC 01, DEN 82]. From a purely rheological point of view, these models rely on
a purely speculative foundation. Indeed, most of the time, the rheological parameters
used in these models have been estimated by matching the model predictions (such as
the leading-edge velocity and the runout distance) with field data [BUS 80, DEN 80,
SCH 74]. However, this obviously does not provide evidence that the constitutive
equation is appropriate.
Avalanches can be considered at different spatial scales (see Figure 2.6). The
larger scale, corresponding to the entire flow, leads to the simplest models. The chief
parameters include the location of the gravity center and its velocity. Mechanical
behavior is mainly reflected by the friction force F exerted by the bottom (ground
or snowpack) on the avalanche. The smallest scale, close to the size of snow particles
involved in the avalanches, leads to complicated rheological and numerical problems.
The flow characteristics (velocity, stress) are calculated at any point of the occupied
space. Intermediate models have also been developed. They benefit from being less
complex than three-dimensional numerical models and yet more accurate than simple
ones. Such intermediate models are generally obtained by integrating the motion
equations across the flow depth in a way similar to what is done in hydraulics for
shallow water equations.
2.2.3. Simple models
Simple models have been developed for almost 80 years in order to crude
estimations of avalanche features (velocity, pressure, and runout distance). They
u
F
h ( x , t )
u ( x , t )
t
p
s
( x , y , t )
u ( x , y , t )
Figure 2.6. Different spatial scales used for describing avalanches
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