Environmental Engineering Reference
In-Depth Information
Hence, the 2D convection-diffusion equation reduces to:
ε
z
∂c
∂z
∂c
∂t
+
u
∂c
w
s
∂c
∂
∂z
∂x
=
∂z
+
(5.122)
•
the concentration at the upstream boundary at each time step is
known;
•
the concentration
c
x,z,t
(Figure 5.30) is presented in a depth-averaged
concentration
c
x,t
(Figure 5.31);
•
the bed-load concentration
c
(bed)
is a function of the flow and sediment
parameters.
Wang and Ribberink (1986) studied the validity of the Galappatti
model and they concluded that the use of the model is not suitable for
large deviations of the concentration profile from the equilibrium pro-
file. They recommended some specific requirements to be applied in the
Galappatti model for the computation of suspended sediment transport.
These requirements are:
- the Galappatti model is only valid for fine sediment. The factor
w
s
/
u
*
should be much smaller than unity; recommended values of
w
s
/
u
* are
between 0.3 and 0.4;
- the time scale of the flow variations should be much larger than
h
/
u
*;
- the length scale of the flow variations should be larger than
Vh
/
u
*.
where:
u
∗
=
local shear velocity (m/s)
w
s
=
fall velocity (m/s)
h
=
water depth (m)
V
=
mean velocity (m/s)
The boundary condition for the bed is not applied at the bed (
z
=
z
a
),
but at a small distance
z
from the bed
z
z
a
+
z
. Galappatti uses
in his analysis one type of bed boundary condition, i.e. the value of the
concentration near the bed
c
(bed)
at
z
=
=
z
a
+
z
. He has assumed that
c
(bed)
at
z
z
is known in terms of local flow and sediment parameters.
In other words,
c
(bed)
is known in advance. Equation (5.122), if integrated
vertically with the pre-set boundary conditions, gives the depth-averaged
equation.
=
z
a
+
∂
(
hc
)
∂t
∂
(
h
uc
)
∂x
·
+
=
E
(5.123)
Z
A
+
Z
+
h
h
·
=
uc
uc
d
z
(5.124)
Z
A
+
Z
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