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where K
x
is the eddy diffusivity, a property analogous to the molecular diffusivity k
m
.
However, unlike k
m
, the eddy diffusivity is not a property of the fluid but of the flow
and any further advance requires some assumptions about the value of K
x
. The
simplest assumption is that K
x
is constant (K ) and has the same value for all three
flux components. If this is so, we have so-called Fickian diffusion for which the
advection-diffusion equation is:
@
S
@
2
S
t
¼
U
:r
S
þ
K
r
þ :
ð
4
:
40
Þ
Further simplification is possible if the flow is 2D or even 1D. A valuable reference
solution is that for diffusion of a conservative scalar property in one dimension (x)
only and without advection (U
0). Consider the instantaneous release of a quantity
of dye of mass M
s
into a canal at x
¼
0. The canal is narrow and shallow
so we can assume that the concentration of dye S (x, t) is uniform across each section
of the canal. In this case the advection-diffusion equation is just
¼
0 at time t
¼
2
S
@
S
@
K
@
t
¼
x
2
:
ð
4
:
41
Þ
@
The solution of
this equation is the well-known Gaussian function (Fischer
et al.,
1979
):
M
s
4pKt
e
x
2
=
4Kt
ð
;
Þ¼
p
:
ð
:
Þ
S
x
t
4
42
The behaviour of this solution is illustrated in
Fig. 4.11
. As the patch expands with
time, the peak concentration (at x
1
0) decreases as t
=
2
. The scale of the patch is set
¼
by the variance, s
2
(standard deviation
s
) of the dye from x
¼
0 which is given by:
Ð
1
x
2
Sx
ðÞ
;
t
dx
1
2
¼
dx
¼
2Kt
ð
4
:
43
Þ
Ð
1
Sx
ðÞ
;
t
1
so that the size of the patch increases as t
1
=
2
. Approximately 90% of the dye lies
within a distance 2s of the origin so the patch size is
4s.
This basic solution may be used to build solutions to more complex problems in
two or three dimensions (see Fischer, List, et al.,
1979
). For example, multiple inputs
of dye at different locations or at different times may be represented by simple
addition of the Gaussian patches from the individual inputs. After a long enough
time, the solution for such multiple inputs takes the form of a single Gaussian
function. (To see this and examples of multiple patches and the solution with
advection, see the supporting software at the topic website.)
∼
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