Geoscience Reference
In-Depth Information
channel flows. The Manning equation has the form:
1
n
S
0
.
o
R
0
.
67
v
=
(5.7)
h
where
R
h
is the hydraulic radius and
S
o
is the local slope angle. Recall that the hydraulic radius is defined
as the cross-sectional area of the flow,
A
, divided by the wetted perimeter,
P
. Thus, for flows that are
wide compared with their depth
R
h
=
A/P
≈
Wh/W
h
where
W
is the width of the flow and
h
is the
local flow depth. The discharge can then be calculated as
1
n
WS
0
.
o
h
1
.
67
Q
=
vhW
=
(5.8)
This has the general form of the power law storage-discharge relationship used in Box 5.7 (where an
equivalent expression is developed for the Darcy-Weisbach uniform flow equation)
q
=
bh
a
(5.9)
where the specific discharge
q
1
.
67. The
kinematic wave velocity or celerity
c
is equal to the rate of change of discharge with storage (here
dq
=
Q/W
and, for Manning's equation,
b
=
S
0
.
o
/n
and
a
=
dh
). It
is an expression of the rate at which the effects of a local disturbance propagate downslope or downstream.
For the power law,
c
=
(
abh
1
−
a
) and this wave velocity always increases with discharge if
a>
1
.
For
a
=
1,
q
is a linear function of
h
and both the flow velocity and the wave speed
c
are constant with
changing discharge.
Other relationships may show different types of behaviour. Wong and Laurenson (1983) show, for a
number of Australian river reaches, how the form of the relationship between wave speed and flow in a
river channel may change as the flow approaches bankfull discharge and goes overbank (Figure 5.10).
Figure 5.10
Wave speed-discharge relation on the Murrumbidgee River over a reach of 195 km between
Wagga Wagga and Narrandera (after Wong and Laurenson, 1983, with kind permission of the American
Geophysical Union): Q
b
1
is the reach flood warning discharge; Q
b
2
is the reach bankfull discharge.