Geoscience Reference
In-Depth Information
Fig. 5.5
Comparison between the
average stream velocity
V
,as
described by the logarithmic
equation (5.36) (heavy line) and
by the power equation (5.38)
(thin line). The constants in the
power equation are
m
=
(1
/
6)
and
C
p
=
5
.
4.
4000
3000
h/z
0
2000
1000
0
0
5
10
15
20
V/
(
gS
f
h
)
1/2
The equations of Chezy and Gauckler-Manning suggest that
a
in (5.39) normally lies
in a range between 0.5 and 0.7 under fully turbulent flow conditions. On the other hand,
Equation (5.32) shows that
a
2 for laminar flow. In some studies of high Reynolds
number sheet flow over surfaces covered with short vegetation, such as grass, it has
been concluded that
a
, as the power of
h
, may have an intermediate value close to unity.
Horton (1938) adopted
a
=
1 to derive the rising hydrograph; he conjectured that this
might represent a flow that is 75% turbulent and 25% laminar. Horner and Jens (1942)
derived this value of
a
=
1 from experimental data by different investigators. An analysis
by Henderson and Wooding (1964) of data published by Hicks (1944) confirmed that
a
may indeed be close to unity over a very rough or grass-covered surface. Wooding
(1965) interpreted this phenomenon by noting that, owing to fluctuations in depth and
roughness over an irregular surface, the flow regime can vary spatially and temporally
between laminar and turbulent; in addition, even when the flow near the water surface
is turbulent, the flow within the lower layers between the grass stems and leaves may be
more like laminar seepage through a porous medium.
=
5.3.3
Summary of friction slope parameterizations
Equation (5.39) can be used as a general expression for the friction slope
S
f
. For two-
dimensional flow or for wide channels, it assumes the form
C
r
h
a
S
f
V
=
(5.43)
which can also be formulated conveniently in terms of the rate of flow per unit width,
q
Vh
. The values of the parameters for laminar and turbulent flow are summarized in
Table 5.2
=
