Geoscience Reference
In-Depth Information
If we carry this to third order we have
T
u
=
˜
r
,t)
3
u
α
(
x
,t)
−˜
u
α
(
x
+
≡
C
3
r,
(7.24)
with
C
3
a constant that again depends on the relative orientations of the velocity and
r
. For the special case where they are parallel we have “Kolmogorov's four-fifths
law” with
C
3
=
4
/
5
.
It is an exact result for isotropic turbulence (
Frisch
,
1995
;
Hill
,
1997
).
7.2.3 Subfilter-scale modeling
Eddy-diffusivity models are often used to represent the subfilter-scale fluxes
τ
ij
and
f
j
in LES (
Chapter 6
). Standard models are
K
c
∂c
r
τ
ij
Ks
ij
,
=
j
=−
∂x
j
,
(7.25)
with
K
u
s
,
u
s
being the velocity scale of the subfilter-scale eddies
and
the filter cutoff scale. If
lies in the inertial subrange,
u
s
∼
K
c
∼
=
u()
=
()
1
/
3
. Hence, these eddy diffusivities are of order
u()
4
/
3
1
/
3
.
The
effective Reynolds number of LES with eddy-diffusivity closure is hence
(/)
4
/
3
(Problem 7.8)
.
∼
7.3 The dissipative range
An isotropic field is one whose statistics are invariant to translation, reflection, and
rotation of the coordinate axes.
†
Since its production mechanisms are anisotropic
(
Chapter 5
) naturally occurring turbulence cannot be isotropic, but isotropic com-
putational turbulence can be generated (
Yeung
et al
.
,
2002
). In introducing his
hypothesis of
local isotropy
, or isotropy confined to the dissipative-range scales,
Kolmogorov
(
1941
)wrote:
…we think it rather likely that in an arbitrary turbulent flow with a sufficiently large
Reynolds number
(R
t
)
the hypothesis of local isotropy is realized with good approxima-
tion in sufficiently small domains…not lying near the boundary of the flow or its other
singularities …
Kolmogorov
(
1941
) proposed two further similarity hypotheses for turbulence
statistics in the dissipative range at large
R
t
. The first is that they are determined only
by
and
ν
. If so, the statistics of the fine-scale velocity field in large-
R
t
turbulence,
when made dimensionless with the length and velocity scales
η
and
υ
, are the same
†
We discuss the tensor implications of isotropy in
Chapter 14
.
