Geoscience Reference
In-Depth Information
Substituting Eqs. (7.10)-(7.12) and three similar equations for
F
e
,
F
n
, and
F
t
into
the discretized continuity equation (7.13) results in the following equation for pressure
correction:
a
p
a
p
a
p
a
p
a
p
a
p
a
p
P
p
P
=
W
p
W
+
E
p
E
+
S
p
S
+
N
p
N
+
B
p
B
+
T
p
T
+
S
p
(7.14)
where
a
P
=
l
=
W
,
E
,
S
,
N
,
B
,
T
a
p
n
+
1
V
n
+
1
P
P
V
P
)/τ
−
(
F
e
−
F
w
+
l
, and
S
p
=−
(ρ
−
ρ
P
F
n
−
F
s
F
t
F
b
)
+
−
.
Water level calculation
The water level can be calculated using the free-surface kinematic condition (2.71)
or the 2-D depth-integrated continuity equation (6.1). It can also be determined
using the 2-D Poisson equation of water level proposed by Wu
et al
.
(2000a),
which is expressed as Eq. (6.47) with the term
t
added
into the source term
S
z
in the case of unsteady flows. However, all the depth-
averaged velocities and stresses appearing on the right-hand side of Eq. (6.47) are
determined by depth-integrating the local quantities computed from the present
3-D model.
To the author's knowledge, the Poisson equation (6.47) is more stable than
the free-surface kinematic condition (2.71) and the 2-D depth-integrated continuity
equation (6.1) in the calculation of water level. However, Eq. (6.47) is valid only for
gradually varied open-channel flows because it is derived under the hydrostatic pres-
sure assumption. Nevertheless, many tests have shown that it can approximately be
applied to rapidly varied flows where no obvious hydraulic jump occurs.
−
∂(∂
U
x
/∂
x
+
∂
U
y
/∂
y
)/∂
Grid adjustment
In general, to conform to the water surface profile, the computational grid should be
regenerated once the water level changes. If a boundary-fitted grid is used, the Poisson
equations (4.74) need to be solved at each time step. This is relatively time-consuming.
The local coordinate transformation (4.89) on moving grids can be extended to the
3-D case, which may be more efficient. For a channel with simple geometry, the
grid needs to be adjusted in only one or two directions or in part of the domain,
and thus, the grid can be regenerated using simple algebraic methods, such as the
σ
-coordinate (4.91).
Once the grid is adjusted, the parameters related to geometry should be updated.
7.1.3.3 Projection method
In analogy to the depth-averaged 2-D model in Section 6.1.3.2, Jia
et al
. (2001)
developed a 3-D model based on the projection method. The partially staggered grid
shown in Fig. 6.4 is extended to the 3-D case with the pressure stored on the base
grid (cell centers) and velocities
u
x
,
u
y
, and
u
z
on the staggered grid (cell corners). The
governing equations are discretized and solved using the same methods mentioned
in Section 6.1.3.2. The pressure-correction algorithm is slightly modified below to
achieve the coupling of velocity and pressure in the 3-D model.
