Geoscience Reference
In-Depth Information
1
=
∂
U
x
∂
U
x
,2
−
U
x
,1
(6.49)
y
y
1
=
−
∂
U
x
∂
3
U
x
,1
+
4
U
x
,2
−
U
x
,3
(6.50)
y
2
y
or other higher-order approximations.
6.1.4 Wetting and drying techniques
In the calculation of flows in open channels with sloped banks, sand bars, and islands,
the water edges change with time, and some part of the domainmight be dry. A number
of methods have been reported in the literature to handle this problem. They may be
classified into two groups. One group tracks the moving water edges and adjusts the
computational mesh to cover the wet domain. This group can use the boundary-fitted
grid at each time (iteration) step and achieve better accuracy around the water edges.
However, it results in complicated codes and perhaps requires more computational
effort. The other group uses the fixed grid that covers the largest wet domain and
treats dry nodes as part of the solution domain. The latter group includes the “small
imaginary depth,” “freezing,” “porous medium,” and “finite slot” methods.
The “small imaginary depth” method uses a threshold flow depth (a low value, such
as 0.02m in natural rivers and 0.001m in experimental flumes) to judge drying and
wetting at each time step. If the flow depth at a node is larger than the threshold value,
this node is considered to be wet, and if the flowdepth is lower than the threshold value,
this node is dry. The dry nodes are assigned zero velocity. The water edges between
the dry and wet areas can be treated as internal boundaries, at which the wall-function
approach may be applied. The dry nodes can be excluded from the computation in an
explicit algorithm, but must usually be included in an implicit algorithm. In the latter
case, the “freezing” method is often adopted.
The “freezing” method also adopts a threshold flow depth to judge wetting and
drying in the computational domain. At dry nodes, the Manning
n
or the coefficient
a
P
in Eq. (6.22) is given a very large value, such as 10
30
; therefore, the calculated
velocity is zero and the water level does not change (as it is frozen). The “freez-
ing” method can include dry nodes in an implicit algorithm. However, it should be
noted that the water level gradient may induce false flow motions at the dry nodes.
To avoid this problem, a horizontal water level profile at the dry nodes may be
assumed.
The “porous medium” method (Ghanem, 1995; Khan, 2000) assumes that the bed
at the dry nodes is a porous medium and the flow can extend into the dry bed. Based on
a specified minimum depth criterion, either the St. Venant or groundwater equations
are applied at a particular computational point. The “finite slot” method proposed by
Tao (1984) is similar to the “porous medium” method. In the “finite slot” method, a
dry node is cut into two slots (with infinitesimal width and infinite depth) parallel to
the
x
- and
y
-coordinates, respectively, in which the water is assumed to move. Thus,
the water depth is kept positive artificially, even if the bed is dry. Different momentum
equations are used at the dry nodes in the “porous medium” and “finite slot” methods,
