Geoscience Reference
In-Depth Information
Table 5.1 Some typical values of the roughness coefficient
n
for natural channels
Channel type
n
Earth, straight, with short vegetation
0.02-0.03
Gravel bed, straight
0.03-0.04
Earth and gravel, winding with some weeds on banks
0.03-0.05
Sand and gravel bed with boulders or with brush and overhanging
trees on banks
0.035-0.06
Boulders and banks of exposed rock
0.05-0.08
Earth overgrown with weeds
0.07-0.09
the velocity is uniform along the vertical. This suggests that for a given channel
C
r
can
only be truly constant for highly turbulent flow, or for very high Reynolds numbers
Re
≡
/ν
.
Another very popular formula is the Gauckler-Manning (GM) equation, named after
the two engineers who contributed most to its development (see Powell, 1962; 1968;
Williams, 1970; 1971). It is usually written as
Vh
1
n
S
1
/
2
R
2
/
3
h
V
=
(5.41)
f
The constant
n
is referred to as the channel roughness coefficient, when the variables
are expressed in SI units. Numerous experiments have been carried out to determine it
for all kinds of channels and surfaces. Some values are shown in Table 5.1, but more
detailed results for a wider range of conditions can be found in Chow (1959) and in
Barnes (1967). Comparison with Equation (5.38) shows that the GM formula (5.41) can
be derived theoretically by assuming a power law such as (5.37) with
m
6. This
indicates that the GM formula can be expected to be valid over a range of lower Reynolds
numbers than Chezy's, which requires the extreme value
m
=
1
/
0 for a perfectly uniform
velocity profile. Equation (5.38) also shows that
n
is directly proportional to
z
1
/
0
;it
should be recalled from Section 2.5.2 that, as a first approximation,
z
0
may be assumed
to be of the order of one tenth of the size of the roughness elements of the wall. In any
event, the power law assumption, on which the GM equation is implicitly based, should
be adequate for most practical applications. This is illustrated in Figure 5.5, which gives
a comparison between the dependence of the average velocity
V
on the water depth
(
h
=
z
0
), as calculated with the logarithmic profile (5.36) and that calculated with the
power profile (5.38). The two curves display satisfactory agreement for the value of the
constant
C
p
=
/
4. Comparison of the GM equation (5.41) with (5.38) produces then
the following relationship between the channel roughness coefficient and the boundary
layer roughness height,
5
.
0690
z
1
/
6
0
n
=
0
.
(5.42)
in which
z
0
is expressed in metres.
