Geoscience Reference
In-Depth Information
Figure 8.4
Interface between blocks for non-staggered FVM grid.
fine grids are extended one layer (or more layers if needed for methods of high accu-
racy) to the adjacent block for the convenience of information transfer and numerical
discretization at the interface.
To solve the flow and sediment transport equations in each block, the boundary
conditions of velocity, pressure (or pressure correction), suspended-load concentra-
tion, and bed-load transport rate are required at the interface. Other variables that
have to be transferred are flow depth, bed level, bed shear stress, sediment trans-
port capacity, bed change, and bed-material gradation. All these variables should
be interpolated from the corresponding points on the adjacent block. The linear
or quadratic interpolation method may be used. However, to obtain continuous
flow and mass fields, the interpolation should satisfy the conservation laws at the
interface.
Fig. 8.5 shows a portion of the interface corresponding to the width of a single
control volume on the coarse grid, and to the width of several control volumes, indexed
from
i
=
1to
i
max
, on the fine grid. The conservation law for flow flux reads
i
max
U
c
h
c
l
c
=
U
fi
h
fi
l
fi
(8.3)
i
=
1
where
l
c
is the length of the interface on the coarse grid,
h
c
and
U
c
are the flow depth
and velocity at
l
c
,
l
fi
is the length of the interface of cell
i
on the fine grid, and
h
fi
and
U
fi
are the flow depth and velocity at
l
fi
. Note that the velocities
U
c
and
U
fi
are normal
to the corresponding cell faces.
As information is exchanged from the fine grid to the coarse grid (Fig. 8.5), the
flow flux and, in turn, the flow velocity can be uniquely obtained with Eq. (8.3),
which satisfies the conservation law. However, it is not straightforward to obtain the
flow velocity from the coarse grid to the fine grid. The conservation law (8.3) does
