Geoscience Reference
In-Depth Information
Figure 6.3
Control volumes near (a) inlet and (b) outlet.
where
a
P
=
a
E
+
a
S
+
a
N
+
p
P
−
p
P
)
F
e
−
A
P
/(
)
, and
S
p
=−
(
A
P
/(
)
−
(
F
w
+
g
t
g
t
F
n
−
F
s
)
. The flow calculation can then be carried out over all internal points. After
the internal pressure field has been obtained, the pressure at the
w
-face of each inlet
cell can be extrapolated from the pressure values at adjacent internal points and a new
inflow flux can then be obtained using Eq. (6.17). The above procedure is repeated
until a convergent solution is obtained.
The turbulent energy and its dissipation rate can be directly specified at the center of
the control volume near the inlet, according to Eq. (6.20). However, as an alternative,
their fluxes may also be specified at the inflow face. When the relevant equation is
integrated over the control volume near the inlet, the specified flux is moved into the
source term and the coefficient at the inflow face is set to zero.
At the outlet, the pressure (water level) is specified for the subcritical flow. It may
be specified at either the center or the outflow face of the control volume shown in
Fig. 6.3(b). In the former case, the pressure correction should be zero at center
P
. In the
latter case, an imaginary computational point (noted as
E
) without a control volume
is set up at the outflow face, at which the pressure correction is zero. The pressure
correction equation at point
P
is Eq. (6.28), but the coefficient
a
E
needs to be specially
treated, as it cannot be determined in analogy to Eq. (6.26). The former approach is
easier to implement.
The flow velocity, turbulent energy, and dissipation rate at the outlet can be extrap-
olated from the values at adjacent internal points. When the relevant differential
equation is integrated over the control volume shown in Fig. 6.3(b), the diffusion
flux at the outlet (face
e
) is zero due to the quantity's zero gradient; because the con-
vection terms are usually discretized using an upwind scheme, the coefficient
a
E
may
actually be zero.
6.1.3.2 Projection method
Semi-implicit projection method
Casulli (1990) proposed a semi-implicit finite difference method for solving 2-D
shallow water equations (6.1)-(6.3). The staggered grid shown in Fig. 4.24 is used.
