Geoscience Reference
In-Depth Information
Section 5.2.2, with slight modification to consider the variation of Manning
n
with
flow conditions.
The depth-averaged 2-D flow equations (10.43)-(10.45) can be solved using the
2-D SIMPLE(C) algorithm in Section 6.1.3.1 by defining the pressure as
=
ρ(
−
c
v
)
p
1
gz
s
(10.58)
and the fluxes at cell faces as
n
+
1
1
i
w
U
n
+
1
i
,
w
F
w
=[
ρ(
1
−
c
v
)
h
]
(
J
α
η)
(10.59)
w
n
+
1
2
i
ξ)
s
U
n
+
1
i
,
s
F
s
=[
ρ(
−
c
v
)
]
(
α
1
h
J
(10.60)
s
The details on the 2-D SIMPLE(C) algorithm for flow in vegetated channels can
be found in Wu and Wang (2004b). Similarly, Eqs. (10.37) and (10.38) can be
solved using the 3-D SIMPLE algorithm in Section 7.1.3.2 by replacing
ρ
with
ρ(
.
For low vegetation density, the 3-D equations (10.39) and (10.40) and 2-D
equations (10.46)-(10.48) can be solved directly using the numerical algorithms
developed for flow in non-vegetated channels by arranging the drag force terms as
source terms.
1
−
c
v
0
)
10.2.3 Examples
Numerous verifications and applications of the vegetation effect models can be found
in the literature. Two examples are cited here. One is the simulation of flow in
open channels with rigid, submerged vegetation performed by Shimizu and Tsujimoto
(1994) using a vertical 2-D model with the
k
-
turbulence closure. The experiments
were conducted in flumes under uniform flow conditions. The rigid cylinders of equal
height and diameter were placed at equal spacings in a square pattern on smooth flume
beds. Fig. 10.10 shows the measured and simulated mean flow velocities, Reynolds
shear stresses, and streamwise turbulence intensities along the flow depth for the run
with a flow depth of 7.47 cm, a depth-averaged flow velocity of 13.87 cm
ε
s
−
1
,an
energy slope of 0.00213, a vegetation height of 4.1 cm, a vegetation diameter of
0.1 cm, and a vegetation spacing of 1.0 cm. The flow was retarded by vegetation in
the lower layer, and the maximum shear stress and streamwise turbulence intensity
occurred at the top of the vegetation elements. The simulated results are in generally
good agreement with the measured data.
Note that Shimizu and Tsujimoto (1994) calibrated coefficients
c
fk
·
=
0.07 and
c
f
ε
=
0.16 through the above simulation. Lopez and Garcia (2001) also simulated
the flow over rigid, submerged vegetation under similar conditions and validated the
theoretically-based values
c
fk
1.33. To clarify this difference, Neary
(2003) re-simulated the case shown in Fig. 10.10 using the
k
-
=
1.0 and
c
f
ε
=
ω
turbulence model. He
found that both sets of
c
fk
and
c
f
ε
values give very close predictions for the mean flow
velocity and the Reynolds shear stress, while
c
fk
=
0.07 and
c
f
ε
=
0.16 give a better
prediction for the streamwise turbulence intensity than
c
fk
=
1.0 and
c
f
ε
=
1.33.
