Geoscience Reference
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pass it over successively to pulsations of higher orders. The energy of the finest pulsations
is dispersed in the energy of heat due to viscosity.
In virtue of the chaotical mechanism of the translation of motion from the pulsations of
lower orders to the pulsations of higher orders, it is natural to assume that in domains of
space, whose dimensions are small in comparison with
(
1
)
, the fine pulsations of the higher
orders are subjected to [an] approximately space-isotropic statistical regime. Within small
time-intervals it is natural to consider this regime approximately steady even in the case,
when the flow as a whole is not steady.
Since for very large
R
the differences
u
i
(P
(
0
)
)
=
−
w
i
(P )
u
i
(P )
of the velocity components in neighboring points
P
and
P
(
0
)
of the four-dimensional space
are determined nearly exclusively by pulsations of higher orders, the scheme just exposed
leads us to the hypothesis of local isotropy….
In summary, Kolmogorov thought it plausible that the “chaotical mechanism of
the translation of motion from the pulsations of lower orders to the pulsations of
higher orders” that exists in high-
R
t
turbulence gives the “fine pulsations of the
higher orders …[an] approximately space-isotropic statistical regime,” the state of
local isotropy
.
Today we often state his argument in terms of the rms fluctuating strain rate
u(r)/r
associated with eddies of spatial scale
r
. As we discussed in
Chapter 2
,this
increases monotonically from of order
u/
in the energy-containing eddies to
υ/η
in the dissipative eddies. Since the mean strain rate is of order
u/
, the ratio of the
fluctuating and mean strain rates for an eddy of size
r
,
u(r)/ur,
increases with
decreasing eddy size
r
; it reaches
(υ/η)/(u/)
R
1
/
t
in the dissipative eddies.
At sufficiently large
R
t
, the argument goes, eddies beyond the energy-containing
range are not influenced by the large-scale, anisotropic mean strain rate because it
is much smaller than their own turbulent strain rate. Thus local isotropy is seen as
an asymptotic state reached at sufficiently large
R
t
.
A variant of this argument is that the
energy cascade
from the largest eddies
to the smallest ones (
Chapters 6
and
7
) occurs over such a large range of scales
in large-
R
t
turbulence that the anisotropy inherent in the energy-containing range
does not reach the smallest eddies. Equivalently, it is argued that the largest and
smallest eddies do not directly interact in large-
R
t
turbulence.
∼
14.5.2 Evidence
Batchelor
(
1960
, p. 110) wrote that Kolgmogorov's notion of local isotropy “has
now received considerable support.” He cited early measurements in laboratory
flows showing that certain predictions of
Eq. (14.18)
, for example
