Geoscience Reference
In-Depth Information
The third moment in the first ter
m of
E
q. (14.35)
, being odd in
x
3
and
x
1
,alsovan-
ishes. Thus we conclude that the
(θ
,
1
)
3
budget is maintained by locally anisotropic
mechanisms.
Sreenivasan
et al
.
(
1977
)and
Tong and Warhaft
(
1994
) have found that the
skewness of the lateral derivative
θ
,
3
can be even larger than that of the streamwise
derivative. Here the general third-moment budget is, with scaling of terms indicated
(Problem 14.24)
∂(θ
,
3
)
3
∂t
=−
3
,
3
u
3
,
3
θ
,
3
θ
,
3
−
3
U
,
3
θ
,
3
θ
,
3
θ
,
1
us
2
λ
3
s
3
λ
3
s
u
(14.36)
−
(θ
,
3
θ
,
3
θ
,
3
u
3
)
,
3
−
3
θ
,
3
θ
,
3
θ
,j
u
j,
3
−
6
γ θ
,
3
j
θ
,
3
j
θ
,
3
.
s
3
λ
3
s
3
λ
3
u
u
λ
Again the lowest-order balance is between turbulent production and molecular
destruction, and again both vanish under local isotropy. Of the two mean-gradient
production terms, that involving the mean temperature gradient does not vanish
under local isotropy, but is smaller than the leading terms by a factor
/λ
∼
R
λ
.
It
is no
t clear
that
it
is large enough to maintain the observed values
of
(θ
,
3
)
3
.
In summary, the observational evidence indicates that shear flows with a mean
scalar gradient can have an equilibrium state of local anisotropy of the turbulent
scalar field. Scaling arguments suggest that the local anisotropy is produced through
the turbulent strain field acting on turbulent scalar gradients and removed at the same
mean rate through molecular diffusion. The roots of the anisotropy appear to lie
in the mean shear and scalar gradient in the parent flow, but the nature of their
statistical link to this local anisotropy is not evident.
On a closely related topic,
Durbin and Speziale
(
1991
) derived the evolution
equation for the dissipation-rate tensor and found that maintaining the dissipation
rate in the presence of a mean strain rate requires locally anisotropic turbulence.
This supports the notion that the dissipation-rate tensor can be anisotropic in shear
flows, which is consistent with evidence discussed in
Section 14.5.2
.
14.5.4 Gauging local anisotropy
Relatively few of the myriad implications of local isotropy have been tested. Claims
of local anisotropy are often based on an observed
1 derivative skewness that
would vanish under isotropy. The implicit assumption is that
S
∼
∼
1islarge.
