2
More on Conservation Laws
In this section, we expand on the tra
ﬃ
c flow model of section 1. First we see
how an entirely similar model can be applied to river flow.
In this context,
we see some of the model’s limitations, which motivates us to switch from a
single conservation law to systems. We do not pursue the mathematical study
of such systems any further though, since this would take us too far afield: the
use of partial di
ff
erential equations is only one of many possible approaches to
the modeling of natural phenomena, though a very fruitful one; we need to save
time and energy for the study of others.
2.1
Flood waves: a kinematic model
To see that the analysis developed in section 1 has applications far beyond tra
ﬃ
c
flow, we extend it to long waves in rivers. Here the principle of car conservation
is replaced by volume conservation along the river, which works well under the
assumption of negligible evaporation, infiltration and rain.
Since the volume
between two cross–sections of the river at positions
x
1
and
x
2
is given by
V
=
x
2
x
1
S
(
x, t
)
dx ,
where
S
is the cross–sectional area of the river up to the free surface of the
water, volume conservation takes the form
S
t
+
Q
x
= 0
,
(21)
entirely analogous to (2), with
S
playing the role of the car density
ρ
, and
Q
(
x, t
)
representing the volume flow per unit time through the river’s cross–section at
position
x
and time
t
:
Q
(
x, t
) =
S
(
x,t
)
u
(
x, y, z, t
)
dy dz.
Here
u
is the component of the fluid velocity normal to the crosssection. As
for tra
ﬃ
c flow, we have the kinematic constraint that
Q
=
S U ,
where
U
is the mean waver speed across the section.
In order to close the equation (21), we may want to invoke a relation between
Q
and
S
. Hydraulic engineers denote such a relation a
hydrological law
. This
is customarily measured in various cross–sections of many of the world’s main
rivers: a vertical stick measures the water height
h
(a surrogate for the area
S
, if the geometry of the river bed is well–known), while a variety of devices
are used for measuring the water speed
u
at various points; integrating these
velocity measurements across
S
yields an accurate estimate for
Q
.
15
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.5
1
1.5
2
2.5
S
Q = S
2
/2
A hydrological law Q(S)
Figure 7: Example (
Q
=
S
2
/
2) of a hydrological law for a river crosssection.
We shall assume, therefore, that a functional relation
Q
=
Q
(
S
) exists. An
important di
ff
erence with the tra
ﬃ
c model, is that
U
is typically an increasing
function of
S
, and so
Q
(
S
) is a convex function (see figure 7).
The reason
for this is that the mean speed
U
follows from a balance between two forces:
gravity, pushing the water downslope toward the sea, and lateral friction. Since
the latter is proportional to the wetted perimeter of the river, while the former
is a body force, proportional to the area, the mean speed grows as the water
level increases.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 TABAK

Click to edit the document details