Geoscience Reference
In-Depth Information
The parabolic model uses Eq. (2.49). The 3-D mixing length model is
l
m
|
ν
=
S
|
(7.4)
t
|
S
1
/
2
, and
l
m
is the mixing length described in
where
|=[
(∂
u
i
/∂
x
j
+
∂
u
j
/∂
x
i
)∂
u
i
/∂
x
j
]
Section 2.3.2.
In the linear
k
-
turbulence models, the eddy viscosity is calculated by Eq. (2.54),
and the turbulent energy
k
and its dissipation rate
ε
are determined by Eqs. (2.55)
and (2.56). The turbulent stresses can also be determined using the nonlinear
k
-
ε
ε
turbulence model, algebraic Reynolds stress model, Reynolds stress model, etc. The
details and relevant references can be found in Section 2.3.
7.1.2 Boundary conditions
Water surface
In early developed full 3-D hydrodynamic models, the water surface is treated as a
rigid lid; thus, the computational domain is fixed and the problem is simplified. On
the rigid lid, the normal velocity must be set to zero, and the pressure is no longer
atmospheric. This rigid lid approach encounters difficulties in the case of long river
reach under unsteady flow conditions where the water surface varies in time and space.
Therefore, in several recently developed full 3-D models (e.g., Wu
et al
., 2000a; Jia
et al
., 2001), the variation of water surface is simulated as part of the solution. At
the water surface, the pressure is given the atmospheric value, and the free-surface
kinematic condition (2.71) is applied.
The free surface approach is physically more reasonable than the rigid lid approach.
However, the free surface approach requires more computational effort because the
user must solve a movable boundary problem.
When wind shear appears, the wind driving force is determined using Eq. (6.5).
Note that the wind driving force is added as a source term in the depth-averaged 2-D
model, whereas it is treated as a boundary condition in the 3-D model. In analogy to
Eq. (6.112), a flow velocity gradient that forms a shear stress equating to the wind
driving force is applied near the water surface.
In the absence of wind shear, the net normal fluxes of horizontal momentum and
turbulent kinetic energy at the water surface are set to zero, and the dissipation rate
ε
can be calculated using the relation given by Rodi (1993):
k
3
/
2
0.43
h
ε
=
(7.5)
River bed and banks
On the river bed, banks, and other solid boundaries, the wall-function approach
described in Section 6.1.2 is applied. In particular, the movable bed roughness is
quantified by the equivalent roughness height
k
s
. For a stationary flat bed,
k
s
is usually
set to the median diameter
d
50
of bed material, but in practice, higher values are also
