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weighting of all contributions; in
filtering
the weighting function can vary with
position. The concept is to apply to the governing equations a spatial filter that
removes the smallest eddies and leaves the energy-containing ones unaffected,
producing equations for the larger-scale part of each variable. If
L
is the spa-
tial scale of the computational domain and
f
is the
cutoff scale
of the filter -
that is, the filter removes spatial variations of scale less than
f
-thenoforder
N
L/
f
computational grid points are required in each direction to resolve the
filtered fields, of order
N
3
grid points in all. For this reason the filtered variables are
often called
resolvable-scale
variables. Today
N
is typically in the range 30-300.
This modeling approach now goes by the apt name
large-eddy simulation
, or LES.
It was proposed by
Lilly
(
1967
) and first used successfully by Deardorff for turbu-
lent channel flow (
Deardorff
,
1970a
) and then for the atmospheric boundary layer
(
Deardorff
,
1970b
).
=
1.8 Physical modeling of turbulent flows
In a review paper on turbulence written at the dawning of the computer age,
Corrsin
(
1961
) estimated the number of grid points required in a numerical calculation of a
modest Reynolds number (
R
t
10
4
)
turbulent flow.
†
Upon presenting his result,
10
14
grid points (which is still well out of reach today) he wrote:
The foregoing estimate is enough to suggest the use of analog rather than digital
computation; in particular, how about an analog consisting of a tank of water?
4
×
Corrsin's suggestion of “an analog consisting of a tank of water” is now called
physical modeling
or
fluid modeling
. It allows the structure of both convective and
“mechanical” turbulence to be observed in scaled-down, laboratory flows. Some
of its most successful applications have been to turbulent dispersion of effluents.
Early studies in a 1-m scale convection tank revealed for the first time some of
the unusual dispersion properties of convective turbulence (
Deardorff and Willis
,
1975
). Another successful application is the turbulent dispersion of effluents from
sources in complex terrain (
Snyder
,
1985
).
1.9 The impact of Kolmogorov
Of the many scientists who have worked in turbulence, none stands taller than
Andrei Nikolaevich Kolmogorov. In a brief paper published in 1941 he laid out the
basis for our present-day understanding of turbulence as a dynamical system.
‡
†
To put this example in more physical terms, the Reynolds number
ud/ν
of turbulent flow in a stirred (at
u
=
10
−
1
ms
−
1
) cup (of diameter
d
=
10
−
1
m) of tea (
ν
=
10
−
6
m
2
s
−
1
)is10
4
. That flow, which has an
R
t
value
somewhat smaller than 10
4
because
<d
, probably can be computed through DNS today.
‡
The
Frisch
(
1995
) monograph on turbulence is subtitled “The Legacy of A. N. Kolmogorov.”
