Environmental Engineering Reference
In-Depth Information
2.2.4.
Intermediate models (depth-averaged models)
Simple models can provide approximate predictions concerning runout distance,
the impact pressure or deposit thickness. However they are limited for many reasons.
For instance, they are restricted to one-dimensional path profiles (the spreading of
the avalanche cannot be calculated) and the parameters used are fit to past events
and cannot be measured in the field or in the laboratory (rheometry), apart from
airborne models if the analogy with turbidity currents is used. More refined models use
depth-averaged mass and momentum equations to calculate the flow characteristics.
With such models, the limitations of simple models are alleviated. For instance, it
is possible to calculate the spreading of avalanches in their runout zone or relate
mechanical parameters used in the models to the rheological properties of snow. As far
as we know, the early depth-averaged models were developed in the 1970s by Russian
scientists [BAR 98, EGL 83, EGL 98] and French researchers [BRU 81, VIL 86] for
flowing avalanches. For airborne avalanches, the first stage was probably the model
developed by Parker
et al.
[PAR 86], which, though devoted to submarine turbidity
currents, contains almost all the ingredients used in subsequent models of airborne
avalanches. Considerable progress in the development of numerical depth-averaged
models has been made possible, thanks to the increase in computer power and
breakthrough in the numerical treatment of hyperbolic partial differential equation
systems [LEV 02].
2.2.4.1.
Depth-averaged motion equations
Here, we shall address the issue of slightly transient flows. We focus exclusively
on gradually varied flows, namely, flows that are not far from a steady uniform state
for the time interval under consideration. Moreover, we first consider flows without
entrainment of the surrounding fluid and variation in density:
≈
¯
. Accordingly
the bulk density may be merely replaced by its mean value. In this context, the motion
equations may be inferred in a way similar to the usual procedure used in hydraulics to
derive the shallow water equations (or Saint-Venant equations): it involves integrating
the momentum and mass balance equations over the depth. As such a method has been
extensively used in hydraulics for water flow [CHO 59] as well for non-Newtonian
fluids [BOU 03, SAV 91], we shall briefly recall the principle and then directly provide
the resulting motion equations. Let us consider the local mass balance:
∂/∂t
+
∇·
(
u
)=0. Integrating this equation over the flow depth leads to:
h
(
x,t
)
∂u
∂x
d
y
=
∂
∂x
h
+
∂v
∂y
u
(
x,y,t
)d
y − u
(
h
)
∂h
∂x
− v
(
x,h,t
)
− v
(
x,
0
,t
)
,
0
0
[2.21]
where
u
and
v
denote the
x
- and
y
-component of the local velocity. At the free
surface and the bottom, the
y
-component of velocity satisfies the following boundary
