Environmental Engineering Reference
In-Depth Information
The condition for orthogonality is then obtained through:
@'
@x
@C
@x
@'
@y
@C
@y
¼q
x
q
y
þ q
x
q
y
¼
0
(15.4)
There are explicit formulae for the streamfunction for special flow elements,
for example:
(
Q
x
0
y þ Q
y
0
x
for baseflow
Cðx; yÞ¼
Q
2
(15.5)
p
#ðx; yÞ
for a well
with
angle of the connecting vector between position (
x,y
) and well location.
The physical unit of the streamfunction is [m
3
/s], i.e. the dimension of volume
flux. This is easiest explained regarding baseflow in
x
-direction.
Q
x
0
is the volume
flux per unit width
ϑ
¼
¼
Q
x
0
y
, representing is the flux between the
x
-axis and its parallel in distance
y
.It
follows that between two horizontal lines at locations
y
1
and
y
2
and streamfunction
values
Dy ¼
1. According to the formula (
15.5
) holds:
C
(
x,y
)
C
1
and
C
2
the flux is given by
DC ¼ C
1
C
2
(15.6)
It is a general property of the streamfunction that for any two locations in the
model region the flux between these positions is given by the difference of their
streamfunction values. This characteristic property of the streamfunction follows
directly from (
15.1
) by integration along curves within the model region. The
restricting condition is that the model region has to be simply connected (Needham
1997
), which roughly means that it has no holes.
Imagine two arbitrary locations. If the streamfunction takes the same value at
these positions, the volumetric flux between these locations is zero, which results as
a special case from the above given property of the streamfunction. The property
is obvious if the connecting line is the streamline itself, but it holds for every
connecting line. Then, positive fluxes across the line are exactly outweighted by
negative fluxes. In a more general sense one may consider arbitrary closed curves in
the model region. Take an arbitrary point on this curve, representing both start
and end point of the curve. The streamfunction property states that the integral of
fluxes across this line is zero (because the streamfunction value at start and end
is the same). Physically speaking, negative and positive fluxes outweigh each other
exactly.
Because of the mentioned property, the streamfunction is a very appropriate
measure for fluxes. In order to demonstrate the application of the streamfunction,
we extend the M-file
'gw_ flow.m'
, which was developed in Chap. 14. That is done
in three steps:
