Environmental Engineering Reference
In-Depth Information
Figure 1.13.
Lobes of a debris-flow deposit near the Rif Paulin stream (Hautes-Alpes, France)
approximation, the long-wave approximation, etc. Here, by fast motion, we refer
to situations where inertia, rheological effects and pressure all play a role in flow
dynamics. However, the flow velocity must not be too high; otherwise instabilities
occur at the free surface [BAL 04, COU 97, LIU 90b, TRO 87].
The Saint-Venant approach involves integrating the momentum and mass balance
equations over the depth. A considerable body of work has been published on this
method for Newtonian and non-Newtonian fluids, including viscoplastic [COU 97,
HUA 98, SIV 05] and granular materials [BOU 03, CHU 03, GRA 98, IVE 01,
KER 05, POU 02, PUD 03, SAV 89]. Here, we shall briefly recall the principle and
then directly provide the resulting governing equations. Let us start with the local
mass balance equation [1.2]. 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
[1.16]
At the free surface and the bottom, the
y
-component of velocity
v
satisfies the
boundary conditions (equation [1.4]). We then easily deduce
∂h
∂t
+
∂hu
∂x
=0
,
[1.17]
