Geoscience Reference
In-Depth Information
just beyond the energy-containing range; and their behavior in the dissipative
range beyond that, which is revealed by derivative moments
(Chapter 15)
.Thereis
now abundant evidence that flux cospectra
(Chapter 15)
in shear flows with mean
scalar gradients do fall faster than
κ
−
2
in the inertial subrange. This ensures that
the turbulent flux budgets have negligible molecular-destruction terms at large
R
t
(Chapter 5)
, as required in local isotropy.
But as we have discussed, measurements show that for moments of veloc-
ity and temperature derivatives in shear flows with mean scalar gradients, local
isotropy is only a first approximation. The isotropic forms
(14.19)
and
(14.23)
,
while attractive in their simplicity, could introduce non-negligible errors in shear
flows.
Questions on key concepts
14.1 Explain the notion of an
isotropic
turbulence field.
14.2 Explain why naturally occurring turbulence is not isotropic in its energy-
containing range.
14.3 In
Chapter 6
we discussed forcing turbulence with a stochastic body force
β
i
(t),
Eq. (6.16)
. If we wish the turbule
nc
e
to b
e iso
tropic
, how should
β
i
be constrained? Discuss, for example,
β
i
,
β
i
β
j
,and
β
i
u
j
. Explain your
reasoning.
14.4 Explain the meaning of local isotropy and the essence of the traditional
arguments for it.
14.5 Explain why the notion of local isotropy is useful in dealing with turbulence
moment equations.
14.6 What new type of term appears in second-moment budgets for dissipative-
range properties?
14.7 Explain the underlying difference in the scaling of second-moment budgets
for energy-containing-range properties and for dissipative-range properties.
14.8 Explain physically why the traditional arguments for local isotropy of scalar
fields seem not to hold in the presence of mean scalar gradients.
14.9 Explain physically why the skewness of the temperature derivative vanishes
under the assumption of local isotropy.
Problems
14.1 Prove
Eq. (14.19)
. Discuss how experimentalists use this result to estimate
from a time series of
u
1
measured at a point.
