Environmental Engineering Reference
In-Depth Information
1.4
1.2
1
0.8
0.6
0.4
0.2
−
1.5
−
1
−
0.5
0
0.5
1
1.5
2
z
Figure 1.14.
Flow-depth profile generated just after the wall retaining a granular material is
removed. Computations made with c
=1
m/s. The similarity variable ζ is ζ
=
x/t
In the second application, we use the method of characteristics to find a solution to
the governing equations for Bingham flows that are stretched thin when they are nearly
steady uniform. In section 1.4.3, we found that for mild slopes, when the aspect ratio
is very low, the inertial and pressure contributions can be neglected. This means that
the flow-depth averaged velocity is very close to the mean velocity reached for steady
uniform flows
1
−
h
0
3
h
u
s
=
u
p
,
where
u
p
is the plug velocity
u
p
=
ρgh
0
sin
θ
2
μ
,
where
h
is the flow depth and
h
0
=
h − τ
c
/
(
ρg
sin
θ
) the yield-surface elevation;
h
0
must be positive or no steady flow occurs. We then use the kinematic-wave
approximation introduced by Lighthill and Whitham [LIG 55] to study floods on
long rivers; this approximation was then extensively used in hydraulic applications
[ARA 94, HUN 94, HUA 97, HUA 98, WEI 83]. It involves substituting the mean
velocity into the mass balance equation [1.17] by its steady-state value
=0
.
∂h
∂t
+
∂
h −
h
3
∂x
u
p
[1.26]
Introducing the plug thickness
h
p
=
h − h
0
=
τ
c
/
(
ρg
sin
θ
), we obtain an expression
that is a function of
h
and its time and space derivative
+
K
h
2
− hh
p
∂h
∂x
∂h
∂t
=0
,
