Geoscience Reference
In-Depth Information
Example 5.7. Semi-infinite channel with known upstream inflow
In the diffusion approach the governing equation is now (5.88), but the boundary conditions
remain essentially the same as given in Equation (5.68) (or (5.59)). As was the case for
the complete formulation in Section 5.4.1, the solution of the linear diffusion analogy is
most conveniently given by its unit response. This is the outflow rate
q
p
(
x
,
t
)
=
u
(
x
,
t
)
at a distance
x
downstream from the point
x
=
0 where the flow rate perturbation is a
delta function or
q
p
(0
,
t
)
=
q
u
(
t
)
=
δ
(
t
). The unit response can then be used with (5.71)
to calculate the result for any arbitrary function
q
u
(
t
). The diffusion equation has been
thoroughly studied (see, for example, Carslaw and Jaeger, 1986) for various boundary
conditions. It can be readily shown that this unit response function can be written as
exp
−
x
(4
π
D
0
t
3
)
1
/
2
(
x
c
k0
t
)
2
4
D
0
t
−
u
=
(5.95)
This solution has a form which is closely related to the Gaussian or normal distribution in
which the position of the mean is
x
=
(2
D
0
t
). Thus
this unit response confirms that the main body of this wave, that is its centroid, moves
downstream with a celerity, given by the advectivity
c
k0
of Equation (5.91). It also shows
how the diffusivity causes dispersion or spreading of the wave as it moves along.
To generalize this result it is useful to express it in terms of the dimensionless variables
introduced in (5.78). Substitution of
c
k0
and
D
0
(with
b
=
1
/
2) from (5.77) and (5.89),
respectively, yields
=
(
c
k0
t
) and the spatial variance
σ
2
exp
−
[
x
+
−
(
a
+
1)
t
+
]
2
2
t
+
S
0
V
0
h
0
x
+
2
π
t
3
+
u
=
(5.96)
1
/
2
This solution of the diffusion approach is compared in Figure 5.9 with the main unit response
u
2
in the analogous solution of the complete shallow water equations, as given in (5.80). It
can be seen that there is little difference between the two formulations for larger values of
the dimensionless distance
x
+
. In fact, Figure 5.9 suggests that the diffusion approximation
should be adequate in practical calculations for
x
+
>
5. However, comparison of Equations
(5.79) and (5.80) with (5.96) also indicates that the agreement between the two sets of curves
in Figure 5.9 is perforce affected by the magnitude of the Froude number Fr
0
. As mentioned
earlier, the solution of the complete shallow water equations exhibits singular behavior as
Fr
0
≥
1; therefore it can be expected that the diffusion approach becomes less accurate as
the Froude number approaches unity, i.e. as the flow velocity
V
0
becomes critical. Finally,
Figure 5.9 illustrates how for small values of
x
+
the volume under the
u
2
curve is smaller
than unity; for instance, the
u
2
curve at
x
+
=
0 is lower than the corresponding diffusion
result. This merely illustrates the fact that for small values of
x
+
the dynamic wave part
u
1
is still not negligible; for instance, as mentioned above, at
x
+
=
1
.
0, in Equation (5.79) the
term exp(
−
e
1
x
+
) is still 0.411; but it rapidly decays with increasing
x
+
.
The main point of this analysis of the diffusion analogy of free surface flow is that it has
further illustrated how the inertia terms, i.e. the first two terms in Equation (5.22) which
represent acceleration, are the ones that generate the dynamic waves, shown in (5.64) and
(5.73). These waves are absent from the solution given by (5.95) and (5.96). The motion of
the main body of the wave is essentially controlled by the last three terms of Equation (5.22);
both the solution of the complete system and that of the diffusion analogy indicate that the
main body moves with the celerity of a kinematic wave; this will be further discussed in
Section 5.4.3.
1
.
