Environmental Engineering Reference
In-Depth Information
Mathematical models for the simulation of non-equilibrium, sus-
pended sediment transport in open canals might be based on the
solution of:
•
the 1-D, 2-D or 3-D convection-diffusion equation;
•
depth-integrated models.
Depth-integrated models are based on a depth-integrated approach
for the suspended sediment transport; the model describes how the mean
concentration adapts in time and space towards the local mean equilib-
rium concentration. The suspended sediment transport model can be used
together with the depth-averaged hydrodynamic equations.
Galappatti (1983) developed a depth-integrated suspended sediment
transport model for suspended sediment transport in unsteady and non-
uniform flow based on an asymptotic solution for the two-dimensional
convection-diffusion equation in the vertical plane. In this model the ver-
tical dimensions are eliminated by means of an asymptotic solution in
which the concentration
c
(
x
,
z
, and
t
) is expressed in terms of the depth-
averaged concentration
c
(
x
,
t
). The latter concentration is represented by
a series of previously determined profile functions.
Among the depth-integrated models for suspended sediment transport
this model has two advantages over others; firstly no empirical relation
has been used during the derivation of the model and secondly all pos-
sible bed boundary conditions can be used (Wang and Ribberink, 1986).
Moreover it includes the boundary condition near the bed, and hence an
empirical relation for deposition/pick-up rate near the bed is not necessary
(Ribberink, 1986). The partial differential equation that governs the trans-
port of suspended sediment by convection and turbulent diffusion under
gravity is given by (Galappatti, 1983):
ε
x
∂c
∂x
ε
y
∂c
∂y
∂c
∂t
+
u
∂c
ν
∂c
w
∂c
w
s
∂c
∂
∂x
∂
∂y
∂x
+
∂y
+
∂z
=
∂z
+
+
ε
z
∂c
∂z
∂
∂z
+
(5.119)
Neglecting the diffusion terms other than the vertical, the equation for a
two dimensional flow in the vertical plane becomes:
ε
z
∂c
∂z
u
∂c
w
∂c
∂
∂z
∂c
∂t
+
w
s
∂c
∂x
+
∂z
=
∂z
+
(5.120)
Galappatti (1983) assumed a flow field as shown in Figure 5.29 for
the derivation of his model for non-equilibrium sediment transport. The
equation can be solved when the velocity components (
u
,
w
), the fall
velocity
w
s
and the mixing coefficients
ε
x
and
ε
z
are known.
Search WWH ::
Custom Search