Geoscience Reference
In-Depth Information
density. What assumptions are required about the field? What information
can be gained by this process?
16.2 Explain
Figure 16.1
physically, and in particular discuss why it shows
no activity in the wavenumber region between the energy-containing and
dissipative ranges.
16.3 Interpret and discuss
Figure 16.2
physically.
16.4 Discuss the essence of the arguments in
Subsection 16.1.2
that allow us
to use the results of DNS or LES to calculate spectra in the homogeneous
horizontal plane of the ABL and interpret them in much the same way that
we did for isotropic turbulence.
16.5 Discuss the physical interpretation of the response function for line averaging
of a scalar field,
Figure 16.3
.
16.6 Explain physically why
Figures 16.3
and
16.4
are so different.
16.7 Explain the concept of the array technique for measuring resolved and
subfilter-scale variables.
16.8 Discuss the concept and implications of probe-induced flow distortion, and
how and why it can be analyzed quite simply when the scale of the turbulence
is much larger than the scale of the probe.
Problems
16.1 Show that the first two terms on the rhs of
Eq. (16.33)
integrate over the
wavenumber plane to turbulent transport.
16.2 Assuming the pressure spectrum behaves as
κ
−
7
/
3
in the inertial subrange,
contrast the difficulties in measuring the variances of
∂u/∂x
and
∂p/∂x
.
16.3 Write an expression for the cross spectrum of a conserved scalar
c
and
vertical velocity
w
in the homogeneous horizontal plane. In practice we
cannot measure
w
and
c
at the same point in the plane; if the distance on
the plane between their points of measurement is
r
, write their covariance
as an integral of a transfer function times their cross spectrum. Show how
both the cospectrum and the quadrature spectrum contribute to the measured
covariance.
16.4 Express the two-point velocity difference variance
2
as
an integral of the spectral density tensor times a transfer function, under the
assumption of isotropy.Write the integral for the particular cases
r
[
u
1
(
x
+
r
)
−
u
1
(
x
)
]
=
(r,
0
,
0
)
and
r
=
(
0
,r,
0
)
. Evaluate the integrals. Show how this yields the dissipation
rate.
16.5 A linear array of sensors spaced
r
apart on the horizontal plane has been
used to produce a low-pass filtered variable by adding the sensor outputs
with weight
w
. An example is
