Geoscience Reference
In-Depth Information
Comparing Eqs. (10.2) and (10.4) leads to a relation between
N
a
A
v
and
c
v
:
4
α
v
c
v
h
π
N
a
A
v
=
(10.6)
D
N
a
A
v
and
c
v
have also been used to represent the vegetation density by Shimizu and
Tsujimoto (1994) and Wu and Wang (2004b). To avoid confusion, only
N
a
is called
the vegetation density in this topic.
10.1.2 Flow resistance due to vegetation
Drag force on vegetation
The drag force exerted on a vegetation element is expressed as
1
2
C
d
ρ
F
d
=
U
v
A
v
|
U
v
|
(10.7)
where
C
d
is the drag coefficient of vegetation element,
U
v
is the vector of flow velocity
acting on the vegetation element, and
(or simply written as
U
v
) is the magnitude
of
U
v
. For emergent vegetation,
U
v
is the depth-averaged flow velocity
U
, while for
submerged vegetation,
U
v
should be the velocity averaged only over the vegetation
layer, as shown in Fig. 10.4.
Note that the drag force defined in Eq. (10.7) acts on the entire wetted vegetation
height, which may be used in the 1-D and depth-averaged 2-Dmodels. For the analysis
of local flow pattern in a 3-D model, the drag force per unit vegetation height is
defined as
|
U
v
|
1
2
C
d
ρ
f
d
=
a
v
|
|
u
u
(10.8)
where
u
is the vector of the local flow velocity acting on the vegetation element; and
a
v
is the projected area of the vegetation element per unit height, related to the stem
diameter by
a
v
v
D
.
The drag coefficient is the key parameter in Eqs. (10.7) and (10.8). White (1991)
summarized the experimental data of Wieselberger and Tritton, and obtained the drag
coefficient of a single cylinder in an ideal two-dimensional flow as a function of the
Reynolds number. This relation is shown in Fig. 10.6 and approximated by
=
α
10
R
2
/
3
C
d
0
=
1
+
(10.9)
e
where
C
d
0
is the drag coefficient for a single cylinder, and
R
e
=
/ν
U
v
D
. Eq. (10.9) is
10
5
. More sophisticated
curve-fit formulas for
C
d
0
can also be found inWhite (1991). Typically,
C
d
0
has values
from 1.0 to 1.2 for the Reynolds numbers between 10
2
- 2.5
in fair agreement with the measurement data up to
R
e
≈
2.5
×
10
5
.
×
