Geoscience Reference
In-Depth Information
7.2
On what should the Kolmogorov microscales for 2-D turbulence depend?
Develop expressions for them. Determine the Reynolds number they define.
Use the microscales to evaluate the relative magnitude of the viscous
dissipation term in the TKE budget.
7.3
An
n
-th order velocity structure function is
[
u(
x
+
r
)
−
u(
x
)
]
n
where
u
is a
is in the inertial subrange of scales.
(a) How is it expected to behave under the original Kolmogorov hypothesis?
(b) How is it expected to behave under the revised Kolmogorov hypothesis?
(c) Using the Obukhov model for dissipation fluctuations, compare the two
predictions. How does the difference depend on
n
?
7.4 Assume that Taylor's hypothesis is valid for an eddy whose turnover time is
large compared to the time required to advect the eddy past the probe. Justify
this assumption. Then use it to develop a criterion for the validity of Taylor's
hypothesis for eddies of scale
r
. Interpret your result.
7.5 The inertial-range scaling result
(7.63)
has been interpreted for vorticity as
follows: vorticity is a conserved scalar in two-dimensional flow, so vorti-
city fluctuations should move to smaller scales without change in intensity.
Comment on this interpretation. What does the argument imply about scalar
fluctuations in three-dimensional turbulence? Is that interpretation correct?
7.6 Is the notion of preferred shapes for dissipative regions in turbulence in
conflict with local isotropy? Discuss.
7.7 Using an intermittency model as needed, discuss how local, instantaneous
values of velocity derviatives in large-
R
t
turbulence can differ from those
predicted by the
Kolmogorov
(
1941
) hypothesis.
7.8 Using the results of
Section 7.2.3
, show that the effective Reynolds number
of LES with eddy-diffusivity closure is of order
(/)
4
/
3
.
7.9 A commercial “
meter” determines
from the mean-squared difference of
wind speed measured at two points in space. Explain the concept.
7.10 Explain physically the feedback mechanism (
Section 7.1.2
) that adjusts the
dissipative eddy scales in turbulent flow.
7.11 Explain physically why and how the cutoff scale of the inertial subrange
of the scalar spectrum in a given turbulent flow changes as the molecular
diffusivity of the scalar changes.
7.12 Interpret the Corrsin spectrum,
Eq. (7.14)
, physically. Can you explain why
it is sometimes called the “leaky pipe” model?
7.13 Show that Corrsin's expression
(7.16)
is a solution of the system
(7.14)
and
(7.15)
.
7.14 Interpret physically the meaning of the negative exponent of
in
Eq. (7.23)
for the temperature structure-function parameter.
7.15 Show that
R
1
/
2
velocity component. Assume
r
=|
r
|
∼
R
λ
.
t
