Geoscience Reference
In-Depth Information
Lateral inflow, i
h
V
S
0
Upstream
boundary
Downstream
boundary
x=0
x=L
Fig. 6.1
Definition sketch of a plane with lateral inflow.
Using this approach Woolhiser and Liggett (1967) and Liggett and Woolhiser (1967)
have presented a thorough analysis of this problem for a steady uniform lateral inflow
i
.
For the general problem defined by the boundary conditions (6.3), it is convenient to
scale the variables with the length of the plane
L
and with the normal depth
h
0
L
and
the corresponding velocity
V
0
L
at the downstream end
x
(
iL
).
(Recall that the
normal depth
is the depth produced by uniform flow at a given dis-
charge rate
q
, as given by Equation (5.43) with
S
f
=
=
L
, where (
h
0
L
V
0
L
)
=
S
0
) This scaling leads to the
following dimensionless variables,
x
+
=
(
x
/
L
),
t
+
=
(
V
0
L
t
/
L
),
h
+
=
(
h
/
h
0
L
) and
V
+
=
(
V
/
V
0
L
). Equations (6.1) and (6.2) assume then the dimensionless form
∂
h
+
∂
+
∂
(
V
+
h
+
)
∂
−
1
=
0
t
+
x
+
and
(6.4)
V
+
(
h
a
+
1
1
/
b
∂
V
+
∂
V
+
∂
V
+
Fr
0
L
∂
1
h
+
V
+
h
+
+
+
+
Ki
0
−
+
=
0
t
+
∂
x
+
∂
x
+
)
In (6.4), the symbol Ki
0
represents the kinematic flow number at
x
=
L
, defined as
S
0
L
Fr
0
L
h
0
L
Ki
0
=
(6.5)
and Fr
0
L
is the corresponding Froude number (cf. Equation (5.63))
V
0
L
(
gh
0
L
)
1
/
2
Fr
0
L
=
(6.6)
Except for the presence of the Froude number, Ki
0
has nearly the same form as (
d
4
x
)in
(5.73) or (
e
1
x
+
) in (5.79), where it plays essentially the same role, in that it is a measure
of the rate of attenuation of the dynamic waves. In dimensionless equations all terms
normally tend to be of order one, except those that involve a dimensionless number like
Fr or Ki. Thus the second of Equations (6.4) shows that the motion becomes kinematic
when Ki
0
>>
1.
