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the same idea of statistical equilibrium of the small-scale components was put forward
independently.
†
On looking back at this incident I recall that G.I. did not draw attention…to the closeness
of both Kolmogorov's theory and these ideas of Heisenberg and von Weisacker to some of
his own much earlier work. …The important idea that he [Taylor] appears to have missed
was that the statistical quasi-equilibrium of the small-scale motions depends on such a small
number of parameters, namely two, the rate of energy dissipation and the fluid viscosity,
that dimensional arguments alone yield explicit results.
Kolmogorov's hypotheses might now appear as straightforward or unremarkable,
but they were a profound advance in our conceptualization of turbulence dynamics.
7.1.3 The extent of the inertial subrange
We saw in
Section 2.5
that
u(r)
, the velocity scale of eddies of size
r
,is
(r)
1
/
3
.
u(r)
∼
(7.6)
The
eddy Reynolds number Re(r)
=
u(r)r/ν
is then
r
η
4
/
3
(r)
1
/
3
r
ν
Re(r)
∼
∼
,
(7.7)
where we have used the definition
(1.34)
of the Kolmogorov microscale
η
.
One expects that the inertial subrange ends at eddy scale
r
end
1
/κ
end
where
Re(r)
decreases to a minimum value
Re
min
. If so, then from
(7.7)
we have
r
end
η
∼
(Re
min
)
3
/
4
(Re
min
)
−
3
/
4
∼
;
κ
end
η
∼
(
velocity spectrum
).
(7.8)
If
Re
min
is in the range 10-100, from
Eq. (7.8)
that would correspond to eddy scales
r
end
6
η
-30
η
and
κ
end
η
0
.
03-0.2, which is consistent with observations.
7.1.4 The scalar spectrum
The spectral physics of a conserved scalar (
Chapter 6
) is broadly like that of velocity:
large-scale turbulence acting on amean scalar gradient generates scalar fluctuations;
distortion of these fluctuations by turbulence “cascades” scalar variance to smaller
scales; the smoothing effects of molecular diffusion bring the cascade to an end in
the smallest scales.
Obukhov
(
1949
)and
Corrsin
(
1951
) extended Kolmogorov's scaling arguments
to a conserved scalar, assuming that in the inertial range its spectral density
E
c
(κ)
†
Eyink and Sreenivasan
(
2006
) have written at length on Onsager's original and deep contributions to turbulence.
