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Eddies much larger than the puff move it bodily without distorting it; eddies
much smaller than the puff only wrinkle it. Eddies of the puff size
D
distort it,
strengthening its concentration gradients, increasing its effective surface area and
greatly enhancing molecular diffusion of the effluent; they dominate the ensemble-
mean puff growth. Their velocity scale is
u(D)
, so we write
dD
dt
(D)
1
/
3
.
∼
u(D)
∼
(7.20)
It follows that at times large enough that the initial puff size is not important,
D
behaves as (
Batchelor
,
1950
)
1
/
2
t
3
/
2
,
D
∼
(7.21)
which has been observed (
Gifford
,
1957
).
†
7.2.2 Structure-function parameters
The
structure functions
of a velocity component and a conserved scalar are
S
u
=
˜
r
,t)
2
,
c
=
˜
r
,t)
2
.
u
α
(
x
,t)
−˜
u
α
(
x
+
c(
x
,t)
−˜
c(
x
+
(7.22)
By its nature a structure function involves only the turbulent component of a homo-
geneous field, the mean being removed by the subtraction. The Russian school has
long used structure functions in theoretical developments in turbulence and in elec-
tromagnetic and acoustic wave propagation through turbulence. For separations
r
=|
|
in the inertial range of scales the dominant contributions to
S
u
and
S
c
come
from turbulent eddies of scale
r
=|
r
|
r
,sothat
C
1
2
/
3
r
2
/
3
C
v
2
r
2
/
3
,
S
u
=
=
≡
S
u
(, r)
(7.23)
C
2
χ
c
−
1
/
3
r
2
/
3
C
c
2
r
2
/
3
,
S
c
=
S
c
(χ
c
,,r)
=
≡
with
C
1
and
C
2
constants that can be determined formally from the difference
operator and the spectra (
Chapter 15
).
C
v
2
and
C
c
2
are called the
structure-function
parameters
for velocity and scalar, respectively; in the case of velocity there is
some dependence of the structure-function parameter on the relative orientations
of the velocity and the separation vector
r
(
Part III
).
Sawford
(
2001
) says that except for the cited analysis of Gifford, there are no convincing demonstrations of the
behavior predicted by
Eq. (7.21)
. He feels this is due to the difficulty of making reliable dispersionmeasurements
under the conditions required by the theory.
†
