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∼
R
0
.
34
, the exponent being one-third
t
λ
of that predicted by
Eq. (7.30)
.
7.4 Revised Kolmogorov scaling
Moffatt
(
2002
) has described the “sense of frustration” that afflicted G. K. Batchelor
and many others working in turbulence in the early 1960s:
These frustrations came to the surface at the now legendarymeeting held in
Marseille
(
1961
)
to mark the opening of the former Institut de Méchanique Statistique de la Turbulence. This
meeting, for which Batchelor was a key organizer, turned out to be a most remarkable
event. Kolmogorov was there, together with Obukhov, Yaglom, and Millionshchikov…;
von Karman and G. I. Taylor were both there - the great father figures of prewar research
on turbulence - and the place was humming with all the current stars of the subject - Stan
Corrsin, John Lumley, Philip Saffman, Les Kovasznay, Bob Kraichnan, Ian Proudman, and
George Batchelor himself, among many others.
One of the highlights …was when Bob Stewart presented …the first convincing mea-
surements to show several decades of a
κ
−
5
/
3
spectrum and to provide convincing support
for Kolmogorov's theory…. But then, Kolmogorov gave his lecture, which I recall was in
the sort of French that was as incomprehensible to the French themselves as to the other
participants. However, the gist was clear: He said that …Landau had pointed out to him a
defect in the theory …. Kolmogorov showed that the exponent
(
5
/
3
)
should be changed
slightly and that higher-order statistical quantities would be more strongly affected ….
I still see the 1961 Marseille meeting as a watershed for research in turbulence. The
very foundations of the subject were shaken by Kolmogorov's presentation; and the new
approaches …were of such mathematical complexity that it was really difficult to retain
that essential link between mathematical description and physical understanding, which is
so essential for real progress.
Given that Batchelor was already frustrated by the mathematical intractability of turbu-
lence, it was perhaps the explicit revelation that all was not well with Kolmogorov's theory
that finally led him to abandon turbulence in favor of other fields.
†
−
Kolmogorov
(
1962
) replaced the
ensemble mean
dissipation rate
in his 1941
hypotheses with a
locally averaged dissipation rate
r
defined as the average of the
instantaneous, local dissipation rate
over a sphere of radius
r
:
˜
4
πr
3
3
r
(
x
,t)
=
|≤
r
˜
(
x
+
h
,t)d
h
.
(7.31)
|
h
˜
is defined as
The local dissipation rate
∂u
i
∂x
j
+
∂u
i
∂x
j
+
,
ν
2
∂u
j
∂x
i
∂u
j
∂x
i
˜
=
(7.32)
†
Coincidentally, shortly after the Marseille meeting D. Lilly submitted a paper (
Lilly
,
1962
) that was prescient
in advocating and demonstrating the use of numerical simulation in turbulence research (
Wyngaard
,
2004
).
