Environmental Engineering Reference
In-Depth Information
conditions:
v
(
x,h,t
)=
d
d
t
=
∂h
∂t
+
u
(
x,h, t
)
∂h
∂x
,
[2.22]
v
(
x,
0
,t
)=0
.
[2.23]
We easily deduce:
∂h
∂t
+
∂hu
∂x
=0
,
[2.24]
where we have introduced depth-averaged values defined as:
h
(
x,t
)
1
h
(
x,t
)
¯
f
(
x,t
)=
f
(
x,y,t
)d
y.
[2.25]
0
The same procedure is applied to the momentum balance equation:
d
u/
d
t
=
ρg
+
∇·
σ
, where
σ
denotes the stress tensor. Without difficulty, we can deduce the averaged
momentum equation from the
x
-component of the momentum equation:
∂hu
∂t
=¯
gh
sin
θ
+
∂hσ
xx
∂x
+
∂hu
2
∂x
¯
− τ
p
,
[2.26]
where we have introduced the bottom shear stress:
τ
p
=
σ
xy
(
x,
0
,t
). In the present
form, the motion equation system [2.24]-[2.26] 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
.
[2.27]
0
Another helpful (and common) approximation, not mentioned in the above system,
concerns the computation of stress [CHO 59]. 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
. A point often neglected is
that this method and its results are only valid for flow regimes that are not too far away
from a steady-state uniform regime. In flow parts where there are significant variations
in the flow depth (e.g. at the leading edge and when the flow widens or narrows
substantially), corrections should be made to the first-order approximation of stress.
Recent studies however showed that errors made with the shallow-flow approximation
for the leading edge are not significant [ANC 07b, ANC 09a, ANC 09b].
