Environmental Engineering Reference
In-Depth Information
alternative procedures have been proposed [HUA 97, HUA 98, PAS 04]. For instance,
Huang and Garcìa further considered two partial differential equations to supplement
the governing equations [1.17-1.18] [HUA 97, HUA 98]: one equation governing the
elevation h 0 of the yield surface and another providing the bottom shear stress.
For Coulomb materials, the same procedure can be repeated. The only
modification concerns the momentum balance equation [1.18], which takes the form
[IVE 01, SAV 89]
∂hu
∂t
= ρgh
sin θ − k cos θ ∂h
∂x
+ ∂hu 2
∂x
ρ
− τ b ,
[1.20]
with k a proportionality coefficient between the normal stresses σ xx and σ yy , which
is calculated by assuming limiting Coulomb equilibrium in compression ( x ¯ u< 0)or
extension ( x ¯ u> 0); the coefficient is called the active/passive pressure coefficient.
In equation [1.20], the bottom shear stress can be calculated by using the Coulomb
law τ b =( σ yy | y =0 − p b )tan ϕ , with σ yy | y =0 = ρgh cos θ and p b the pore pressure at
the bed level.
Analytical solutions can be obtained for the Saint-Venant equations. Most of them
were derived by seeking self-similarity solutions (see [CHU 03, SAV 88, SAV 89]
for the Coulomb model and [HOG 04] for viscoplastic and hydraulic models). Some
solutions can also be obtained using the method of characteristics. We are going to see
two applications based on these methods.
In the first application, we use the fact that the Saint-Venant equations for
Coulomb materials are structurally similar to those used in hydraulics when the
bottom drag can be neglected. The only difference lies in the non-hydrostatic pressure
term and the source term (bottom shear stress). However, using a change in variable
makes it possible to retrieve the usual shallow-water equations and seek similarity
solutions to derive the Ritter solutions [KAR 00, KER 05, MAN 00, SHE 63]. The
Ritter solutions are the solutions to the so-called dam-break problem, where an infinite
volume of material at rest is suddenly released and spreads over a dry bed (i.e. no
material laying along the bed). Much attention has been paid to this problem, notably
in geophysics because it is used as a paradigm for studying rapid surge motion. We
pose
x = x − 2 t 2 , t = t , u = u − δt , and h = h,
where we introduced the parameter δ = g cos θ (tan θ − μ ). We deduce
∂h
∂t
+ ∂h u
∂x
=0 ,
[1.21]
∂u
∂t
+ u ∂u
∂x
+ gk cos θ ∂h
∂x
=0 .
[1.22]
 
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