Geoscience Reference
In-Depth Information
Therefore, this iteration procedure forms a higher level of coupling for the sediment
model.
Fully coupled model
The depth-averaged 2-D equations of flow, sediment transport, bed change, and bed
material sorting can be solved in an iteratively or fully coupled form, as described in
Section 5.4 on 1-Dmodels. For non-uniform sediment transport, an iteratively coupled
model can be implemented more conveniently than a fully coupled model. The flow
model in Section 6.1 and the sediment model in this section can be designed as flow and
sediment loops, and then a simultaneous solution of flow and sediment transport can
be obtained using the iteration procedure introduced in Section 5.4.3. Such iteratively
coupled 2-D models have been reported by Spasojevic and Holly (1990) and Kassem
and Chaudhry (1998).
More generally, a fully coupled flow and sediment transport model should con-
sider the effects of sediment transport and bed change on the flow field, using
Eqs. (2.119)-(2.121) as the governing equations of flow. However, this type of fully
coupled 2-D model has rarely been applied in practice.
6.2.3.3 Implementation of sediment boundary conditions
The finite difference method approximates boundary conditions using difference
operators. Generally, for Dirichlet boundary conditions, the known values of the
suspended-load concentration and bed-load transport rate are specified at the bound-
ary points and nothing further is necessary. For Neumann or mixed-type boundary
conditions, the gradients must be evaluated using difference schemes, such as the
one-sided schemes similar to (6.49) and (6.50) or higher-order approximations. The
algebraic equations resulting from the discretization of boundary conditions are solved
together with the discretized governing equations at the internal points.
As described in Section 6.1.3.1, the finite volume method usually plugs the sediment
boundary conditions into the transport equations integrated over the control volumes
near boundaries in order to insure mass balance. The details are given below.
Near a rigid sidewall, when the suspended-load and bed-load transport equations are
integrated over the control volume shown in Fig. 6.2, as demonstrated in Eq. (4.130),
the convection and diffusion fluxes at the wall face should be zero, thus yielding a zero
coefficient
a
S
in the discretized equations.
At the inlet, the suspended-load and bed-load fluxes can be specified at the inflow cell
face, according to Eq. (6.63). When the relevant governing equation is integrated over
the control volume near the inlet shown in Fig. 6.3(a), the specified flux is arranged in
the source term and the coefficient
a
φ
W
is set to zero. Note that the specified suspended-
load flux is equal to the sum of the convection and diffusion fluxes, and the specified
bed-load flux is equal to the convection flux at the inflow face (
w
).
At the outlet, the suspended-load concentration and bed-load transport rate can be
extrapolated or copied from the values at adjacent internal points. When the relevant
governing equation is integrated over the control volume shown in Fig. 6.3(b), the
convection terms are usually discretized using an upwind scheme, and the suspended-
load diffusion flux is zero. Thus, the coefficient
a
E
may actually be zero.
