Geoscience Reference
In-Depth Information
7.1.5.3 The velocity spectrum
As wavenumber increases and the local eddy Reynolds number
Re(r)
decreases, the
three-dimensional velocity spectrum also falls away from its inertial-range form.
This departure is observed to occur in the range 0
.
01
0
.
1. There are
analytical models for the spectrum in this range, e.g., that of
Pao
(
1965
), who used
spectral-transfer arguments like those of
Corrsin
(
1964
)tofind
≤
κη
≤
α
1
/
3
κ
−
5
/
3
exp
2
α(κη)
4
/
3
.
3
E
=
−
(7.19)
7.1.6 Results from direct numerical simulations
The numerical calculation of turbulent flow directly from the Navier-Stokes equa-
tions, which today is called
direct numerical simulation
, or DNS, has advanced
steadily since the first trials of
Orszag and Patterson
(
1972
) with 32
3
10
4
3
×
numerical grid points. Today DNS uses as many as 4096
3
10
10
6
×
grid
points.
Figure 7.1
shows scalar spectra from DNS runs that produced stationary, homo-
geneous turbulence of large enough Reynolds number to evidence an inertial
subrange. The Schmidt numbers range from 1/8 to 64. The spectrum for
Sc
=
1
/
8 shows a falloff that is consistent with Corrsin's inertial-diffusive range,
Eq.
(7.16)
. That for
Sc
64 shows Batchelor's
κ
−
1
viscous-convective subrange,
=
Eq. (7.17)
.
7.2 Applications of inertial-range scaling
The Kolmogorov-Obukhov-Corrsin hypothesis that the governing parameters for
scalar turbulence in the scale range
η
reduce to
r
and the cascade rates
and
χ
c
has application to a wide range of problems.
r
7.2.1 Atmospheric diffusion
Batchelor
(
1950
) was one of the first to apply the
Kolmogorov
(
1941
) hypotheses to
atmospheric diffusion. He identified two types of problems: diffusion from a fixed,
continuous source, and diffusion of a discrete cloud (today called a puff) of effluent.
The first is the “Taylor problem” we discussed in
Chapter 4
;the
Kolmogorov
(
1941
)
hypotheses are not applicable to it
(Problem 7.23)
.
The second problem can be summarized as follows. An effluent puff released in
homogeneous turbulence is distorted as it is moved about. How does the ensemble-
mean puff diameter
D
vary with time in the period when
D
η
?
