Environmental Engineering Reference
In-Depth Information
where we have introduced depth-averaged variables defined as
h
(
x,t
)
1
h
(
x,t
)
¯
f
(
x,t
)=
f
(
x,y, t
)d
y.
0
The same procedure is applied to the momentum balance equation [1.3]. Without any
difficulty, we can deduce the averaged momentum equation from the
x
-component of
the momentum equation
+
∂hu
2
∂x
∂hu
∂t
=
ρgh
sin
θ −
∂hp
∂x
+
∂hσ
xx
∂x
ρ
− τ
b
,
[1.18]
where we have introduced the bottom shear stress:
τ
b
=
σ
xy
(
x,
0
,t
). In the present
form, the system of equations [1.17-1.18] is not closed since the number of variables
exceeds the number of equations. A common approximation involves introducing a
parameter (sometimes called the Boussinesq momentum coefficient), which links the
mean velocity to the mean square velocity
h
u
2
=
1
h
u
2
(
y
)d
y
=
αu
2
.
[1.19]
0
Most of the time, the coefficient
α
is set to 1.
Another helpful (and common) approximation, not mentioned in the above
system, concerns the computation of stress. Within the framework of long-wave
approximation, we assume that longitudinal motion outweighs vertical motion: for
any quantity
m
related to motion, we have
∂m/∂y ∂m/∂x
. This allows us to
consider that every vertical slice of flow can be treated as if it was locally uniform.
In such conditions, it is possible to infer the bottom shear stress by extrapolating its
steady-state value and expressing it as a function of
u
and
h
. Using this approximation,
Coussot [COU 94, COU 97] obtained the following bottom shear stress
1+2
n
1+
n
n
u
n
τ
b
=
μ
((2 +
n
−
1
)
h − h
0
)
n
,
h
n
+1
0
for Herschel-Bulkley fluids. Using the first-order approximation of the
y
-component
of the momentum balance equation [1.3], he found that the pressure was hydrostatic,
which leads to a flow-depth averaged pressure
p
=
1
2
ρgh
cos
θ.
The effects of normal stresses can be neglected to first order. Note that this derivation
is not the only way of deriving the Saint-Venant equations for a Bingham fluid;
