Environmental Engineering Reference
In-Depth Information
@
x
@t
¼ v
x
@
y
@t
¼ v
y
(15.7)
Streamlines, obtained as streamfunction contours, originate from a very different
idea. When
C
-contour levels are selected equidistantly, the same volume of fluid
flows between neighbouring streamlines. Between two locations on neighbouring
streamlines there is the same volumetric flux, independent of the locations, i.e.
C
1
C
2
. If the streamfunction is plotted using equidistant values of stream-
function levels (as in the example above), the same amount of fluid flows between
two neighbouring streamlines - that holds for all streamlines in the plot. Thus,
streamlines provide a very nice visualization of velocities: where streamlines are
dense, the velocities are high; where streamlines are far apart, velocities are low.
The streamline density illustrates the velocity of a flow field.
Figure
15.1
shows another advantage of the streamfunction: flowpaths are easily
followed into regions with high velocities, here into the immediate vicinity of the
well. As illustrated in the figure, the particle tracking algorithms have difficulties in
the vicinity of the well.
An advantage of the MATLAB
particle tracking is that markers can be used as
indicators of velocity size. A disadvantage of the streamfunction approach is that
cuts
may appear. In Fig.
15.1
there is a cut, depicted as a thick black line between
the well and the left side boundary. Such cuts appear where the stream function has
a singularity. Using the
atan2
function in the M-file the cuts appear if the angle is
# ¼ p
®
, which is 180
from the positive
x
-axis. Along the negative
x
-axis the
arcustangens function has a jump, which becomes visible in the contour plot as
a cut. If the
atan
function is called in the M-file, the flow pattern is still represented
correctly but with cuts from the well singularity in vertical direction.
100
5.4
80
5.3
60
5.2
40
5.1
20
5
0
4.9
-20
4.8
-40
-60
4.7
-80
4.6
-100
0
20
40
60
80
100
120
140
160
180
200
Fig. 15.1 Streamfunction contours and flowpaths for a single well in uniform baseflow
