Geoscience Reference
In-Depth Information
Figure 10.7
Definitions of
U
v
and
U
vm
in a matrix of vegetation elements.
If the vegetation stems with a diameter of
D
are distributed uniformly in the
lateral direction with a spacing of
l
n
, then
U
v
=
U
vm
(
1
−
D
/
l
n
)
and Eq. (10.11)
can be written as
2
C
d
=
C
dm
/(
1
−
D
/
l
n
)
(10.12)
Furthermore, if the vegetation stems are arranged in a staggered pattern with equal
spacing in bo
th
the longitudinal and transverse directions shown in Fig. 10.7,
U
v
=
D
√
N
a
U
vm
(
1
−
)
. Using Eq. (10.5), one can write Eq. (10.11) as
1
2
−
4
c
v
0
/π
C
d
=
C
dm
/
(10.13)
Note that the drag force is expected to increase with the velocity squared in
Eq. (10.7). This is valid for rigid vegetation. Eq. (10.7) may still be used to com-
pute the drag force on flexible vegetation, but the projected area should be computed
using the deformed height as expressed in Eq. (10.3) (Tsujimoto and Kitamura, 1998),
or the drag coefficient has to be related to flow conditions. More methods for flexible
vegetation roughness are discussed later in this section.
Roughness of emergent rigid vegetation
Consider a steady, uniform flow in a channel with a plane bed covered with uniformly
distributed emergent, rigid vegetation. For the control volume over a unit bed area
extending from the stream bed to the water surface shown in Fig. 10.4, the total
resistance,
τ
, consists of the bed shear stress,
τ
b
, and the drag force of vegetation,
N
a
F
d
:
(
−
c
v
)τ
=
(
−
c
v
)τ
b
+
1
1
N
a
F
d
(10.14)
c
v
appears in Eq. (10.14) to account for only the water
column and bed area occupied by the flow. If the vegetation is relatively sparse, 1
Note that the factor 1
−
−
c
v
is close to 1 and can be eliminated from Eq. (10.14).
