Environmental Engineering Reference
In-Depth Information
v
¼r'
(14.3)
There is an entire mathematical discipline dealing with solutions of the 2D
potential equation. Subject of
complex analysis
are
harmonic functions
that are
solutions of the 2D Laplace equation. In this chapter we deal with 1D parallel flow
that is represented by the potential
'ðx; yÞ¼'
0
þ '
x
x þ '
y
y
(14.4)
Additionally there are sources and sinks in the infinitely extended space, which
are represented by the potential:
Q
2
'ðx; yÞ¼
log
ð
j
r
r
0
jÞ
(14.5)
p
where
'
0
are constant numbers.
Q
denotes the source- or sink-rate, r
0
the location of the source or sink in 2D space and r
'
x
;'
y
and
¼ðx; yÞ
the vector towards the
current location. According to vector analysis, r
r
0
is the vector connecting
source/sink location with the current position.
j
r
r
0
j
is the length of the
q
x x
0
2
2
ð
Þ
þ y y
0
ð
Þ
connecting vector, equal to
.
In the following we examine potentials emerging from the superposition of
formulae (
14.4
) and (
14.5
). According to the
principle of superposition
, the sum
of the functions is a solution of the Laplace equation too. The principle is a trivial
consequence from the fact that the potential equation is linear. In MATLAB
such
functions can be visualized easily as demonstrated by the following command
sequence.
®
xmin = -1; xmax = 1;
% x-coordinates
ymin = 0; ymax = 2;
% y-coordinates
x0 = 0; y0 = .905;
% source/sink location
Qx0 = 0.1; Qy0 = 0;
% baseflow components
Q = 1;
% source/sink rate
% mesh generation
xvec = linspace(xmin,xmax,100);
yvec = linspace(ymin,ymax,100);
[x,y] = meshgrid (xvec,yvec);
% create mesh
% processing
r = sqrt((x-x0).^2+(y-y0).^2);
% distances to well
phi = -Qx0*x - Qy0*y + (Q/(2*pi))*log(r);
% potential
%post-processing
surf (x,y,phi);
% surface plot
With the first five instruction lines the parameter values are specified. In the next
step, 100 equidistant positions at the intervals on the
x
- and
y
-axis are computed
before the mesh is constructed and stored in the
x
and
y
arrays, using MATLAB
®
meshgrid
. Next, the array
r
is computed, which contains the distances to the
