Geoscience Reference
In-Depth Information
conservation laws are not satisfied when interpolating these quantities from the coarse
grid to the fine grid. Conservative corrections should be applied. If the interpolated
suspended-load concentration is denoted as
C
fi
and the interpolated bed-load transport
rate is
q
bfi
, the relevant corrections are
˜
i
max
U
fi
h
fi
l
fi
C
fi
=
C
fi
C
fi
(
U
c
h
c
l
c
C
c
)
(8.7)
i
=
1
i
max
q
bfi
= ˜
q
bfi
(
q
bc
l
c
)
1
˜
q
bfi
l
fi
(8.8)
i
=
If a cell face on the coarse grid does not exactly match to an integer number of cell
faces on the fine grid at the interface, the cells on the fine grid can be split into smaller
subcells. The above conservative correction can still be applied in the conversion from
the coarse cell to the subcells on the finer grid.
Interface treatment for governing equations
If the hybrid upwind/center scheme or the exponential scheme is used for discretizing
the convection terms and the center difference scheme for the diffusion terms in the
momentum and scale transport equations on the non-staggered grid, an overlapping
interface with one extended layer on each side of the interface is enough, as shown in
Fig. 8.4. However, if higher-order schemes, such as QUICK and HLPA, are used, more
extended layers on each side may be needed; otherwise, a lower-order scheme has to
be used to substitute the higher-order schemes near the interface. The convection flux
F
and diffusion parameter
D
at the interface are determined using the interpolated
variables, if needed.
As Shyy
et al
. (1997) described, when solving the pressure-correction equation on
each block, either the flow flux or the pressure correction interpolated from the adja-
cent blocks can be used as the boundary condition. If the interpolated flow flux is
used, as shown in Fig. 8.6(a), the solution of the pressure-correction equation is a
Neumann-type problem, and then the pressure correction is governed by Eq. (6.29) in
the depth-averaged 2-D model. If the interpolated pressure correction is used as the
boundary condition, as shown in Fig. 8.6(b), the problem is of the Dirichlet-type, and
the pressure-correction equation is the same as Eq. (6.28).
For the Dirichlet-type boundary, after Eq. (6.28) is solved, the flow flux
F
w
at the
interface in Fig. 8.6(b) is to be corrected using Eq. (4.196), while for the Neumann-
type boundary, the flow flux
F
w
at the interface does not need correction.
If the Neumann-type boundary is used on two sides of the interface, the pressure
fields in the neighboring blocks are independent and thus may be discontinuous. On
the other hand, if the Dirichlet-type boundary is used on both sides, the flow flux at the
interface is corrected twice, which may lead to inconsistency and discontinuity in flow
flux. Therefore, special care should be taken in these two cases to ensure continuous
(smooth) pressure and flux fields on two adjacent blocks. One remedy is to use the
Dirichlet-type boundary on one side of the interface and the Neumann-type boundary
on the other side.
