Geoscience Reference
In-Depth Information
Roughness of submerged rigid vegetation
For submerged, rigid vegetation, the “acting” flow velocity (which should be used for
computing drag) is the average velocity in the vegetation layer, as shown in Fig. 10.4(b).
This velocity can be determined using Stone and Shen's (2002) method:
v
U
h
v
h
1
/
2
U
v
=
η
(10.20)
where
v
is a coefficient of about 1.0.
Besides the bed shear stress, a shear stress exists at the top of vegetation elements.
The total bottom shear stress can be approximated by
η
c
fv
U
v
+
ρ
2
τ
b
=
ρ
c
fu
(
U
u
−
U
v
)
(10.21)
where
U
u
is the average resultant velocity in the water column above the vegetation
elements, which can be derived by applying the mass continuity equation
U
u
(
h
−
h
v
)
+
U
v
h
v
Uh
; and
c
fv
and
c
fu
are the friction factors on the channel bed and at the top of
vegetation elements, respectively. However, evaluation of the two coefficients
c
fv
and
c
fu
is relatively complicated. For simplicity, one may lump them into one parameter,
and determine
=
τ
b
using Eq. (10.16), with
R
s
given as
h
or by
h
v
l
n
2
h
v
R
s
=
l
n
+
h
−
h
v
(10.22)
+
Eq. (10.22) reduces to Eq. (10.17) for emergent vegetation (
h
v
=
h
), to
R
s
=
h
when there is no vegetation (
h
v
=
0), and to
R
s
≈
h
when the vegetation is sparse,
i.e.,
l
n
2
h
v
. For densely distributed, submerged vegetation, Eq. (10.16) may not be
as physically realistic as Eq. (10.21). However, in such a case the bed shear becomes
negligible in comparison with the vegetation drag force, and thus the use of Eq. (10.16)
does not induce significant errors in the flow calculation.
Therefore, substituting Eqs. (10.7), (10.15), (10.16), and (10.20) into Eq. (10.14)
yields
1
h
v
h
R
1
/
3
n
2
n
b
+
2
v
=
C
d
N
a
A
v
η
(10.23)
s
2
g
(
1
−
c
v
0
)
Similarly, Petryk and Bosmajian's equation (10.19) can be modified for submerged,
rigid vegetation as
A
vi
A
A
χ
4
/
3
v
h
v
h
1
2
g
C
d
n
2
n
b
+
2
=
η
(10.24)
where
h
v
represents the averaged height of vegetation elements blocking the flow.
