Geoscience Reference
In-Depth Information
adopted, e.g., 3
d
90
by van Rijn (1984c). For a sand-wave bed,
k
s
should be related
to the height of bed forms and can be determined using an empirical formula, such as
Eq. (3.58) proposed by van Rijn. However, the effects of large-scale bed forms, such
as point bars and even sand dunes, can be simulated using fine grids with a 3-D model.
This implies that
k
s
should consider only the roughness elements that are somewhat
smaller than the grid spacing. In addition, the near-wall values of turbulent energy
k
and dissipation rate
ε
are given by Eq. (6.15).
Inflow and outflow boundary conditions
Flow conditions at the inlet can be either the flow discharge or a detailed 3-D
distribution of flow velocity. For a given discharge at the inlet, the cross-stream distri-
bution of the depth-averaged velocity can be determined using Eq. (6.17), and then, the
vertical distribution of local flow velocity can be specified according to the logarithmic
or power law.
The inflow direction should also be specified, which essentially determines three
velocity components at each point of the inlet.
If the inlet is located in a nearly straight reach with simple geometry and far from
hydraulic structures, the turbulent energy and its dissipation rate can be determined
using the relations of Nezu and Nakagawa (1993):
z
h
−
1
/
2
E
1
U
3
4.78
U
2
∗
e
−
2
z
/
h
,
h
e
−
3
z
/
h
k
in
=
ε
=
(7.6)
in
where
z
is the vertical coordinate above the bed, and
E
1
is a coefficient related to the
Reynolds number. At moderate Reynolds numbers of 10
4
to 10
5
,
E
1
is approximately
equal to 9.8.
The specification of outflow boundary conditions in the 3-D model is similar to
that in 1-D and 2-D models. If the flow is subcritical, the water level is required
at the outlet. The gradients of flow velocity, turbulent energy, and dissipation rate in
the streamwise direction can be set to zero at an outlet located in a reach with simple
geometry and far from hydraulic structures.
7.1.3 Numerical solutions
The 2-D MAC, projection, and SIMPLE algorithms described in Section 4.4 can be
easily extended to solve the full 3-DNavier-Stokes equations (7.1) and (7.2). However,
for open-channel flows, special care has to be taken in handling the free surface.
A number of techniques have been used to solve this moving boundary problem. They
may be grouped under two main categories: surface tracking and volume tracking
(Shyy
et al
., 1996). A surface tracking method usually adopts a moving (adaptive) grid
in which at least one grid line is along the free surface so that the surface shape is exactly
simulated. Examples of the surface tracking method are given in Sections 7.1.3.2 and
7.1.3.3. A volume tracking method usually uses a fixed grid and defines the shape and
location of the free surface through the volume of fluid at each grid cell. Examples
include the MAC, volume-of-fluid (VOF), and level set methods. The details on the