Geoscience Reference
In-Depth Information
where
ψ
x
and
ψ
y
are weighting coefficients, which are positive and satisfy 0
≤
ψ
+
x
ψ
1.
As compared with Eq. (7.60), Eq. (7.61) can enhance the stability of the sedi-
ment model, but it may encounter numerical diffusion and mass imbalance. Wu
et al
.
(2000a) suggested giving the transverse coefficient
≤
y
ψ
y
zero and the longitudinal coeffi-
cient
x
a small value. However, the author's later work shows that if the adaptation
length
L
is given properly as recommended in Section 2.6.3, the model stability can
be enhanced and the use of Eq. (7.61) can be avoided.
As described in the depth-averaged 2-D model, the dicretized equation for the bed
material sorting in the mixing layer is Eq. (6.70).
ψ
7.3.3 Solution of discretized sediment transport
equations
As demonstrated in 1-D and depth-averaged 2-Dmodels, the bed-material gradation in
Eq. (7.50) can be treated explicitly, and then, a decoupled procedure can be established
to solve the discretized 3-D equations of sediment transport, bed change, and bed
material sorting. The sediment model is often decoupled with the flow model. If the
SIMPLE algorithm is used in the flow model, the fully decoupled calculations are
executed in the following sequence:
1) Start from the initial channel bed and flow field;
2) Solve the momentum equations with the estimated pressure
p
∗
and then the
pressure and velocity correction equations to obtain
p
n
+
1
and
u
n
+
1
i
;
3) Solve the
k
- and
t
;
4) Calculate the water surface profile and then adjust the grid if any change in water
surface occurs;
5) Treat the obtained pressure
p
n
+
1
as a new estimate, return to step (2), and repeat
the above flow calculation until a converged solution is obtained;
6) Calculate
c
n
+
1
b
ε
-equations and update the eddy viscosity
ν
and
q
n
+
1
b
k
using Eq. (7.50) with the known
p
bk
;
∗
k
∗
7) Calculate
c
n
+
1
bk
and
q
n
+
1
bk
using Eqs. (7.51) and (7.56);
8) Calculate
z
bk
and
z
b
using Eqs. (7.57) and (7.59);
9) Calculate
p
n
+
1
bk
using Eq. (6.70);
10) Update the bed topography using Eq. (7.60) and then adjust the grid if any change
in bed elevation occurs;
11) Return to (2) and repeat the above calculations for the next time step until a
specified time is reached.
If the bed-material gradation in Eq. (7.50) is treated implicitly, from Eqs. (7.50),
(7.57), (7.59), and (6.70), one can derive equations similar to (6.73) and (6.74) to
determine the fractional and total bed changes and establish a coupled procedure for
the 3-D calculation of sediment transport, bed change, and bed material sorting. This
coupled sediment model can be decoupled from the flow model to constitute a semi-
coupled model or coupled with the flow model to form a fully coupled model. The
calculation sequences in the 3-D semi-coupled and fully coupled flow and sediment
transport models are similar to those in the depth-averaged 2-D models described in
Section 6.2.3.2. The details are not repeated here.
