Geoscience Reference
In-Depth Information
1
A
3
h
/
h
s
L
B
3
A
2
B
2
0.5
A
1
B
1
C
0
0
0.2
0.4
0.6
0.8
1
x
/
L
Fig. 6.4 Water depth profiles 0A
1
B
1
C, 0A
2
B
2
C, etc., during the buildup phase, obtained with the kinematic
wave approach (for fully turbulent flow with
a
=
2
/
3) ; the profiles are shown as functions of
downstream distance at different times after the start of the lateral inflow
i
. The water depth is
normalized with the equilibrium depth at
x
=
L
, which is given by Equation (6.19), or
h
s
L
=
(
iL
/
K
r
)
1
/
(
a
+
1)
.
On the other hand, the integral of (6.14) is
q
=
i
(
x
−
x
0
)
(6.17)
in which
x
0
is the starting point of the characteristic (i.e. the initial position of the
“observer” invoked above) at
t
=
t
0
=
0. Because
x
0
can assume any value over the
length of the plane 0
L
, there is an infinity of characteristics on which (6.17)
is valid, each depending on
x
0
. The boundary characteristic starting at
x
≤
x
≤
0, is
however of special interest. On that particular characteristic (6.17) assumes the form
=
x
0
=
q
=
ix
(6.18)
By virtue of Equation (6.8), (6.18) gives the position of a given depth,
x
=
x
(
h
), as
i
)
h
a
+
1
x
=
(
K
r
/
(6.19)
Thus on this particular characteristic starting at
t
=
0 and
x
=
0inthe
x
−
t
plane,
=
=
−
i.e. at
h
0 and
x
0 on the physical
h
x
plane, both (6.16) and (6.19) hold. This
−
trajectory on the
h
x
plane is shown in Figure 6.4 as going from 0 to A
1
,A
2
, etc., for
different values of
t
. For all the other characteristics, at
x
values larger than given by
(6.19), (6.17) is not very useful, because
x
0
is left unspecified, but (6.16) still indicates
the water depth
h
as a function of time, independently of
x
0
. Therefore, downstream
from the point
x
, given by (6.19),
h
is independent of
x
(see Figure 6.4).
Actually, Equations (6.18) and (6.19) also represent the continuity condition that
must be satisfied under equilibrium conditions, that is when the flow rate at any point
x
equals the total lateral inflow upstream from that point. This means that equilibrium
conditions are established upstream from any point
x
where the boundary characteristic
has passed and that the entire plane is at equilibrium as soon as that characteristic has
reached
x
=
L
. From then on, (6.18) and (6.19) are valid over the entire flow domain,
