Geoscience Reference
In-Depth Information
The kinematic wave approach for free surface flow
In view of the form of the various expressions for the friction slope discussed in
Section 5.3, Equation (5.100) is tantamount to assuming that, at a given location
x
along the flow channel, the average velocity
V
is a function of the hydraulic radius only,
that is
V
=
V
(
R
h
)
(5.101)
For a given cross section, the hydraulic radius is uniquely related to the mean depth of
flow and to the cross-sectional area; thus
V
can also be expressed as a function of one
of those variables, and one can use
=
=
V
V
(
h
) r
V
V
(
A
c
)
(5.102)
as well.
As before, the most important features of the flow can be deduced by considering a
wide channel. Equation (5.102), which now represents the momentum equation, indicates
that the flow rate per unit width
q
=
(
Vh
) is a function of
h
only; thus, the continuity
equation (5.13) can be written as
∂
h
dq
dh
∂
h
t
+
x
=
i
(5.103)
∂
∂
Conversely, on account of (5.102) one has
h
h
(
q
) as well, and therefore substituting
this in Equation (5.13) one obtains in a similar manner
∂
=
q
dq
dh
∂
q
dq
dh
i
t
+
x
=
(5.104)
∂
∂
Equations (5.103) and (5.104) both have a structure, which is similar to that of the total
time derivative of
h
and of
q
, namely
∂
h
dx
dt
∂
h
dh
dt
t
+
x
=
(5.105)
∂
∂
and
∂
dx
dt
∂
q
dq
dt
q
t
+
x
=
(5.106)
∂
∂
Hence it follows that
dq
dh
=
dx
dt
(5.107)
defines a wave speed; for the sake of conciseness this can be denoted by
dx
/
dt
=
c
k
,so
that
dq
dh
c
k
=
(5.108)
The speed of this wave represents the rate of displacement of any point along
x
, where
the depth (or the cross-sectional area) and the rate of flow increase respectively as
dh
dt
=
dq
dt
=
i
and
c
k
i
(5.109)
